A Reduced Basis Element Approach for the Reynolds Lubrication Equation Lösen der Reynolds Schmierungs Gleichung mit dem Reduzierten Basis Ansatz Master’s Thesis in Mathematics Institut für Geometrie und Praktische Mathematik RWTH Aachen submitted by Eduard Bader Master of Science in Mathematics Matrikel number: 280359 First supervisor: Prof. Dr. Martin Grepl Institut für Geometrie und Praktische Mathematik, RWTH Aachen Second supervisor: Prof. Dr. Siegfried Müller Institut für Geometrie und Praktische Mathematik, RWTH Aachen Date of submission: 27th September 2012
Transcript
A Reduced Basis Element Approach for the ReynoldsLubrication Equation
Lösen der Reynolds Schmierungs Gleichung mit dem Reduzierten Basis Ansatz
Master’s Thesis in Mathematics
Institut für Geometrie und Praktische Mathematik
RWTH Aachen
submitted by Eduard BaderMaster of Science in MathematicsMatrikel number: 280359
First supervisor: Prof. Dr. Martin GreplInstitut für Geometrie und Praktische Mathematik, RWTH Aachen
Second supervisor: Prof. Dr. Siegfried MüllerInstitut für Geometrie und Praktische Mathematik, RWTH Aachen
Date of submission: 27th September 2012
Eidesstattliche Erklärung
Hiermit versichere ich, dass ich die vorliegende Arbeit selbstständig verfasst habe und keine anderen alsdie angegebenen Quellen und Hilfsmittel verwendet habe.
7.3 SCRBE with Port Reduction applied to the plain bearing with a different setting . . . . . . . 51
8 Conclusions 55
References 56
List of acronyms and symbols
COM index of subdomainsG-P global portL-P local portMBS Multi Body systemOFFLINE method stage which is done beforehandONLINE method stage which is done with given parameter µPDE partial differential equationPR Port ReductionRB Reduced BasisRBE Reduced Basis Element (method)RLE Reynolds Lubrication equationSC Static Condensation (method)SCRBE Static Condensation Reduced Basis Element (method)a(u, v;µ) bilinear form, left-hand side of weak formulationA(µ), A(µ) Schur Complement matrix and its approximationα(µ), αLB
i (µ) parameter dependent coercivity constantsbhf,i(µi) bubble-functions depending on f , component i and the parameter µibhk,L-P,i(µi) bubble-functions depending on mode number k, L-P, component i and µiBM(i);0 Hilbert space on a subdomain with homogeneous Dirichlet boundary conditions∆U? (µ), ∆U
? (µ) SC-error bound and the improved SC-error boundf(v;µ) linear functional, right-hand side of weak formulationF(µ), F(µ) right-hand side of Schur Complement System and its approximationΦk,G-P(µ) test and trial function for computing the Schur Complement SystemΓL-P,COM,ΓG-P part of subdomain boundaryh height, distance between bearing partsλmin(µ) minimal eigenvalue of A(µ)µ parameter tuple in RdµV viscosity constant of the lubricantnΓm number of L-Ps on one the mth reference domainnΓ number of G-Ps in the whole systemN Γ
L-P,m number of degrees of freedom of the mth reference domain “L-P”N Γ
G-P number of degrees of freedom on “G-P”N Γm number of relevant degrees of freedom on the mth reference domain boundary
ΩSYS domainΩCOM,Ωi subdomains of ΩSYSp pressureψk,L-P,i interface function depending on mode number k, L-P and subdomain iΨk,G-P global interface function depending on mode number k and G-Pρ density of the lubricantσ1(µ), σ2(µ), σ3(µ) parts of SC-erroru1, v1, w1 velocities of the inner bearing part in x-,y-,z-directionu2, v2, w2 velocities of the outer bearing part in x-,y-,z-directionU(µ), U(µ) solution of Schur Complement System and its approximationXSYS, Xm Hilbert spacesXh, Xh
m discretized Hilbert spaces
6 1 INTRODUCTION
1 Introduction
This master thesis deals with the Static Condensation Reduced Basis Element (SCRBE) method as describedin [1]. The objective is to introduce a method for solving parametrized elliptic coercive partial differentialequations (PDE) on several domains. Before describing the SCRBE method, we like to motivate the Re-duced Basis Element (RBE) method approach and its advantages. Instead of addressing very large problemswith the Finite Element method (FEM) the RBE method does many Reduce Basis (RB) approximation onsmaller subdomains with smaller dimensional parameters. Thus, the RBE method can deal with a largeramount of parameters than a RB approach on the full domain. One drawback is that the RBE method, inits preparation stage (OFFLINE), uses the static ordering of the subdomains beforehand, “top-down”. TheRBE method with RB approach never solves the full parameter dependent FEM problem. For domains thatconsist of many repeated subdomains the RBE method has to do the RB approximation only once for asingle subdomain and apply these RB approximations many times. For detailed description of RB and RBEmethod we review the paper [7].
Further, the paper [1] describes the Static Condensation Reduced Basis Element method and its advan-tages. The SCRBE method is based on the standard Static Condensation (SC) method with the followingelements: one subdivides the whole domain into subdomains, does some FEM-calculation on the interdo-main level, does some FEM-calculation to compute so-called bubble-functions for every subdomain and inthe end evaluates all entries for the so-called Schur Complement System. Although this system is muchsmaller than the FE problem, the time-consuming work is to compute the entries of this system, includingbubble-function on subdomains. After compiling and solving this system one can easily access the valuesof the FEM-solution of the problem, without applying FEM on the whole problem. The SCRBE methodis doing quite the same only that the bubble-functions will be RB approximated, which will imply hugecomputational advantages. We will use the RB approximations to perform products and summands of theSchur Complement System beforehand and store them.
SCRBE method extends the RBE in some important ways: in the RBE method the subdomains have tobe aware of the neighboring domains. Already in the OFFLINE stage one decides how the subdomains sticktogether and so all Reduced Basis approximations depend on the ordering of the subdomains. In SCRBEmethod the combination of the subdomains is only necessary when it comes to the application and someparameters are chosen in the so-called ONLINE stage. After doing some OFFLINE work for referencesubdomains one can stick these together many times.
For getting an a posteriori error in RBE method one needs the truth FEM-solution, but in SCRBEmethod we can compute one a posteriori error for the Schur Complement solution independently of thetruth FEM-solution.
The SCRBE method can be split into two parts: OFFLINE and ONLINE stage. In OFFLINE stagethe method involves solving sub-problems on reference subdomains for building the RB approximationsfor the bubble-function, which is done with FEM. Then we store this basis and some inner products in areference library. In ONLINE stage we are now free to assemble any system, “bottom-up”. The system isonly restricted by some interface properties of reference subdomains.
In this thesis we develop a slight modification of the SCRBE method. We will modify the treatment ofthe boundary of the reference subdomains, in the OFFLINE stage such that this modification will lead usto an extended a posteriori error estimate. Further, it will allow us to deal with even more general domainsthan already shown in [1].
In the beginning of Section 2 we introduce the Reynolds Lubrication equation (RLE) which modelsthe pressure distribution in a plain bearing. We will introduce a parametrization for RLE and use thisapplication to show the advantages of the SCRBE method. In Section 3 we will review [1] and give a detailedexplanation of the SC method with the modification for more general domain ordering. In Section 4 we willpremise on the third section and introduce the SCRBE method. The more general domain ordering makes
7
it necessary to extend the a posteriori error estimates in Section 5. After introducing the theoretical partsof the SCRBE method we address in Section 6 the Reynolds Lubrication equation with the SCRBE methodand see what kind of computational advantages this method brings. Moreover, we will use the method tooptimize the properties of a plain bearing by introducing shape-parameters for little dents, dimples, insidethe bearing parts. In Section 7 we introduce a framework for Port Reduction (PR) for the SCRBE method,from [3]. The goal is to reduce the number of ports in order to construct rapidly an approximate SchurComplement System of reduced size, which takes the main time in the SCRBE method. We will apply thisPR method combined with SCRBE method to the examples from the sixth section and see an additionalspeed-up.
8 2 REYNOLDS LUBRICATION EQUATION
2 Reynolds Lubrication Equation
2.1 Introduction of the application, background setting
In many industrial applications the conception of machinery parts involves computational simulations. Aplain bearing is a common component, for example, in a combustion engine which has to be designed forbearing huge forces while avoiding friction. The main question is then, how large the pressure in a plainbearing is, under different circumstances, which is usually answered by the incompressible Navier-Stokesequation:
ρDuDt
= −∇p+ µV ∆u,
div u = 0,
with the material derivative DuDt ≡ ∂tu + u1∂xu + u2∂yu + u3∂zu. Further, we enumerate some assumptions
which are generally satisfied for plain bearings:
1. incompressibility of the lubricant,
2. constant viscosity µV ,
3. the changing of the height between the bearing parts is very small,
4. the velocities of the lubricant normal to the bearing face is much smaller than the one tangential to thebearing face.
The physical circumstances in a plain bearing with these assumptions allow a simplification of the costlyNavier-Stokes equation to the RLE, for a derivations see [4]. Before we focus on the properties of the RLE(2.1) we exemplary give an application where RLE is used. The RLE is used to model and simulate thepressure in a plain bearing, e.g., in a crankshaft with connecting rod, see Fig. 2.1 for a schematic plan of aplain bearing. We list four steps which explain the components of a simulation:
1. The crankshaft and other parts of the engine are modeled as a Multi Body system (MBS) governed bya differential equation of second order. This equation describes the movement of the crankshaft andother parts. One external force acting on the MBS is derived from the pressure distribution p insidethe plain bearing, here: between crankshaft and connecting rod.
2. To this end, the distance and the velocities of the inner and outer bearing parts are determined fromthe MBS. These external variables are inserted into the RLE and it is solved for the pressure p. But,unfortunately, the solution of the RLE is often only partly useful. By imposing homogeneous Dirichletboundary conditions on the RLE the pressure distribution is point-symmetric to the zero-pressure line,which implies negative pressures. These pressures are physically not possible, since tensile forces donot occur in lubricants. One intuitive approach is to neglect the negative pressures and continue withthe positive part, which would lead to kinks in the pressure distribution. Avoiding these unphysicalkinks means to require p ≥ 0 and ∇p to be continuous. A way to suffice these conditions is todetermine heuristically where a mix of air and lubricant occur and where the pressure is so strong thatonly lubricant is between the bearing parts. For a detailed introduction of this separation algorithmsee [6]. In this thesis we will not face this hard separation problem but work only with the positivepart of the point-symmetric distribution.
3. Considering the RLE and the MBS, we have a coupled system of equations. In a discrete time settingwe need to solve the RLE at each time step and use the obtained pressure as an input for the MBS,
2.2 Reynolds Lubrication Equation 9
which updates the location and velocities of the crankshaft and the connecting rod. We feed the RLEwith the obtained locations and velocities and obtain a pressure, and so on.
4. For the plain bearing to function properly it is desirable for the inner and the outer bearing parts toavoid contact. Therefore, we want to influence the pressure distribution within the bearing with smalldents which are distributed with a regular pattern over the plain bearing, see Fig. 6.8 on p. 38.
Our goal is to efficiently solve the RLE and in the long term to optimize the location and the size ofdents. Because of the dents, we will have a frequently repeating pattern which will be strongly used by theSCRBE method.
2.2 Reynolds Lubrication Equation
In Fig. 2.1 we like to get familiar with the geometries of a plain bearing. We see a ring which can be eitherthe inside part or the outside part of the plain bearing. The further calculation will be done with the unfoldeddomain with a orthogonal coordinate system.
Figure 2.1: Schematic plan of an unfolded outer or inner bearing part.
In Fig. 2.2 we show a detail of a plain bearing with the inner part, which we enumerate with the index1, and the outer part with the index 2. Each part has a distance hi, i = 1, 2 to the center of the bearing andeach part has its velocities in each direction: velocities in x-direction are u1, u2, in y-direction are v1, v2, inz-direction are w1, w2.
All the variables are depending on position and time and they are obtained from the MBS, as describedin Section 2.1. Note that the RLE neglects the change of pressure in the y-direction such that the three-dimensional Navier-Stokes problem is reduced to an two-dimensional problem. We consider an unfoldedplain bearing with the domain Ω ⊂ R2 at a fixed arbitrary time t0. With the assumptions in Section 2.1 wecan derive the RLE:
∇ · ( ρh3
12µV∇p) = U∂x(ρh) +W∂z(ρh) + ∂t(ρh) (2.1)
with
• p(x, z, t0) is the pressure distribution within the lubricant,
10 2 REYNOLDS LUBRICATION EQUATION
v1
w1
u1
v2
w2
u2
h2
h1
x
z
y
Figure 2.2: We see the outer- and the inner-race of the plain bearing with their distancesh1, h2 to the center. Furthermore u1, u2 are velocities of the bearing parts in the x-direction,v1, v2 are velocities of the bearing parts in the y-direction (height) and w1, w2 are velocitiesof the bearing parts in the z-direction.
• ρ(x, z, t0) is the density of the lubricant,
• h(x, z, t0) is the distance in y-direction between the inner and outer bearing, h := h2 − h1,
• µV is the viscosity constant,
• U(x, z, t0) := u2+u12 and W (x, z, t0) := w2+w1
2 are the average velocities of the lubricant in the x-and z- direction, respectively.
2.3 Derivation of the weak formulation
We start with some assumptions, which are realistic for a plain bearing:
• we assume that the density ρ is set to 1 everywhere in the domain, since we assume lubricant betweenthe two bearing parts,
• we assume that the velocities U,W are constant at every time step t0 over the domain Ω.
Before we derive the weak formulation we like to comment on the term ∂th. By applying the materialderivative onto h1 and h2 we obtain
Dh1
Dt≡ ∂th1 + u1∂xh1 + w1∂zh1 = v1,
Dh2
Dt≡ ∂th2 + u2∂xh2 + w2∂zh2 = v2.
Looking at the difference ∂th2 − ∂th1 we get
∂th2 − ∂th1 =(u1∂xh1 + w1∂zh1 − v1
)−(u2∂xh2 + w2∂zh2 − v2
). (2.2)
2.3 Derivation of the weak formulation 11
We defined h = h2 − h1 as the distance between the two bearing parts, so we get ∂th =(2.2). Using (2.2),the RLE (2.1) is equivalent to (2.3):
∇ · ( h3
12µ∇p) =
1
2(u1 + u2)∂xh+
1
2(w1 + w2)∂zh
+((u1∂xh1 + w1∂zh1 − v1
)−(u2∂xh2 + w2∂zh2 − v2
)). (2.3)
Now we like to derive the weak formulation of RLE with homogeneous Dirichlet boundary conditions onthe sides of the bearing and periodic boundary conditions where the bearing is cut open (and unfolded). Wethus take the Hilbert space H1
0 (Ω) and start with the left-hand side of (2.1), multiply with a test function ψand integrate ∫
Ω
h3
12µ∇p · ∇ψdx︸ ︷︷ ︸
=:a(p,ψ)
+
∫ΓnTj
h3
12µ(∇p) · ψdx︸ ︷︷ ︸
=0, since ψ∈H10 (Ω)
.
For the right-hand side of the PDE (2.1) we obtain(∫ΩhU∂xψ + hW∂zψdx−
∫Ω∂thψdx
)︸ ︷︷ ︸
=:b(ψ)
+
∫Γ
nxhUψ + nzhWψdΓ
︸ ︷︷ ︸=0, since ψ∈H1
0 (Ω)
,
where nx, nz are the normal vectors for the boundary of the domain Ω. In summary, we have
We may now identify the height h and the velocities, ui, wi, vi, i = 1, 2, as the parameters of functionsfor the PDE. Note that we have to solve (2.4) at each discrete time step of the multi body simulation forprobably different parameters. Since the parameters are still functions, they are of little use for the ReduceBasis approach. Therefore, we simplify the application environment and so the parameter µ:
1. We introduce a reference level for the outer bearing part, which implies u2 = v2 = w2 = 0.
2. Furthermore, we assume that the distance h between the two parts changes linearly in the x- andz-direction on small domains (subdomains), i.e., we have h = h0 + hxx+ hzz on each subdomain.
3. Analogously, we assume that the velocity v1 is constant on every subdomain (in our application wewill use v1 = 0).
12 2 REYNOLDS LUBRICATION EQUATION
Thus we get: µ = (h0, hx, hz, u1, w1, v1) ∈ D ⊂ R6 and the equation simplifies to∫Ω
h3
12µV∇p · ∇ψdx =−
∫Ωh
1
2(u1∂xψ + w1∂zψ)dx +
∫Ωv1ψdx. (2.5)
Although the parameters thus also change for each PDE-solve, they presumably lie in a certain range andwe can now apply the RB approach.
Remarks:
1. For solvability of (2.4) or (2.5) we use the Lax-Milgram theorem. It is easy to show that the plainbilinear and the linear forms are bounded from above. The coercivity follows with the Poincare-Friedrichs inequality and the fact that h > 0.
2. Experiments with bearings have shown that one can improve the pressure build up by milling smalldents into the bearing parts. The larger the pressure is the more force is exerted and this keeps thebearing parts separated. The optimization question is, how many and how large those dents should beto improve the force such that the integral over the pressure on the domain is larger, i.e., F =
∫Ω∗ p dx,
with Ω∗ is the part of Ω where the pressure is positive.
13
3 Static Condensation method
In this section we give a very detailed and precise description of the SC method. The amount of definitionsand different indices could be distracting, such that we like to revise the seminar paper [5], where the SCmethod is explained by introducing the stationary heat conduction problem. By having a specific applicationin mind it might be easier to understand the SC method. One main idea of SC is to move inner degrees offreedom to the boundary of the domain. Then one solves slightly different problems than the originalproblem: one has interior and boundary problems. By introducing a subdivision of the domain one canreduce the complexity of those interior and boundary problems. In the end one gets the solution of theoriginal problem in a new basis, equivalent to the FEM-solution. The procedure includes a clever assemblingof subdomains and it reminds one of toy bricks: one assembles and connects toy bricks in many ways, whatcorresponds to our subdomains.
3.1 System level formulation; extended
We first introduce the PDE on a domain ΩSYS and in the end of this subsection we consider the weakformulation. Let ΩSYS ⊂ Rd be an open domain with d = 1, 2, 3, with the boundary ∂ΩSYS. Besidesthe applications in Sections 6.1–6.3 with parameter-independent domains, we will consider one applicationwith a parameter-dependent domain ΩSYS(µgeo), where µgeo is a geometry parameter, see Section 6.4. Wecontinue the introduction with a parameter independent domain ΩSYS and define the Hilbert space
XSYS ≡ v ∈ H1(ΩSYS) : v|∂ΩSYS,D = 0,
where ∂ΩSYS,D ⊂ ∂ΩSYS is the part of the boundary on which we impose homogeneous Dirichlet boundaryconditions. Since XSYS is a Hilbert space it is endowed with an inner product 〈·, ·〉XSYS and the norm‖·‖XSYS . In Fig. 3.1 we give an example of ΩSYS and we impose for the Hilbert space homogeneousDirichlet boundary conditions on the red lines.
Figure 3.1: A domain ΩSYS ⊂ R2 with homogeneous Dirichlet boundary conditions on thered boundaries.
Now we choose a subdivision of ΩSYS into I ∈ N subdomains
ΩSYS =⋃
COM∈IΩCOM. (3.1)
We call ΩCOM component domains or subdomains with the enumeration COM ∈
1, . . . , I
=: I . Westress out that the subdivision is not arbitrary: according to [1] the domain in Fig. 3.1 is only allowed tobe subdivided into vertical stripes. Our extension of [1] will allow to horizontally subdivide these verticalstripes, see Fig. 3.2. The shape of these components can repeat many times such that we define a libraryof reference components (reference subdomains), with M ∈ N components: LIB =
Ωref
1 , . . . ,ΩrefM
. For
the subdomains shown in Fig. 3.2 we would use only one reference subdomain, since all four subdomainsare of the same shape. We define a mapping M from I (component) to M :=
1, . . . ,M
(reference
component) with ΩCOM ≡ ΩrefM(COM). Granted, the reference domains have to be translated to the “right”
14 3 STATIC CONDENSATION METHOD
physical position, on which we do not want to comment. For these reference components we will do someOFFLINE-work and use it repeatedly, instead of doing the work for every component again and again.
As we have a subdivision of ΩSYS it is natural that each ΩCOM (or equivalently each ΩrefM(COM)) has at
least one boundary part which is connected to another ΩCOM′ , (ΩrefM(COM′)). We denote these boundary parts
as “ports”, denoted as ΓL-P,COM = ΓL-P,M(COM), 1 ≤ L-P ≤ nΓM(COM) with nΓ
M(COM) the number of portson the reference subdomain. L-P stands for local port, as we consider the ports of each component locally.
It is required in [1] that the intersection of any two ports in any given reference component is empty. Ifwe look on the original domain ΩSYS with its sub-components and its connecting ports we can define a setof global ports, denoted as ΓG-P, 1 ≤ G-P ≤ nΓ where each global port can be either the intersection of twolocal ports or a local port on ∂ΩSYS. The requirement in [1] implies that the definition of local and globalports is not applicable to the subdivision shown in Fig. 3.2, since the intersections of the black lines are notempty. Nonetheless, for domains in R2 we will show a way how to deal with these intersections of ports, asshown in Fig. 3.2 and so use the SCRBE method. We write our extensions of [1] in italic for R2 and thenthe extension can be easily formulated for domains in R3.
Figure 3.2: A domain ΩSYS ⊂ R2 with its subdivision into equal rectangles.
In our extension we focus on a subdivision of ΩSYS ⊂ R2 into rectangles, so we can have some kinds ofglobal ports: a global port can be the interior of the edge belonging to two subdomains: ΓG-P ⊂ ΩSYS, orit can be the interior of a global port on the domain boundary: ΓG-P ⊂ ∂ΩSYS, or it can be just a “point”-global port on the corners of rectangles. In Fig. 3.2 these “point”-global ports are the intersections of theblack straight lines. We can summarize the port connections with index sets πG-P, 1 ≤ G-P ≤ nΓ for the R2
extension:
1. Case of a global port with its (interior) straight segment corresponding to two local portsΓL-P,COM,ΓL-P′,COM′ takes the form πG-P =
(L-P,COM), (L-P′,COM′)
.
2. Case of a global port with its (interior) straight segment corresponding to one local port ΓL-P,COM ⊂∂ΩSYS takes the form πG-P =
(L-P,COM)
.
3. Case of a “point”-global ports on a point corresponding to up to four local pointsΓL-P,COM,ΓL-P′,COM′ ,ΓL-P′′,COM′′ ,ΓL-P′′′,COM′′′ takes the formπG-P =
We go on with the general formulation of the SC method and define a relation between global and localports with an index mapping G such that, e.g., πG-P =
(L-P,COM), (L-P′,COM′)
is equivalent to G-P =
GCOM(L-P) = GCOM′(L-P′); this mapping is invertible with L-P = G−1COM(G-P) and L-P′ = G−1
COM′(G-P′).Further, we address the system domain ΩSYS from (3.1) and add some geometric compatibility condi-
tions:
(i) The component domains must not intersect.
(ii) ΩCOM ∩ ΩCOM′ must either be empty or an entire local port in each of the two (distinct) instantiatedcomponent domains COM and COM′.
3.2 Finite Element Truth Formulation 15
(ii new) According to the subdivision of the domain in R2 (e.g., see Fig. 3.2) we will define additional globalports on single points such that (ii) is still true.
We associate to each of these component domains ΩCOM ≡ Ωrefm , (M(COM) = m) bilinear and linear
forms: for w, v ∈ Xm := H1(Ωrefm ), we define am(w, v, µcoef
m ), fm(v, µcoefm ), 1 ≤ m ≤ M , for a co-
efficient parameter µcoefm which lies in the associated coefficient parameter domain Dcoef
m ⊂ RP coefm . The
forms am(w, v, µcoefm ), fm(v, µcoef
m ) will be later derived as parts of the weak formulation of the PDE. Fur-ther, we denote the parameter domain Dm ≡ Dcoef
m , 1 ≤ m ≤ M with only coefficient parameters forthe mth reference domain. Analogously, we set µm ≡ µcoef
m and use µcoef = µ = (µ1, . . . , µI) ∈ D =DM(1) × · · · ×DM(I) to write down the weak formulation of the PDE. It is then natural to form the systembilinear and linear forms, defined with respect to XSYS, in terms of the corresponding components Ωref
m . Inparticular, for all w, v ∈ XSYS, we introduce
a(w, v;µcoef) =
I∑COM=1
aM(COM)(w|ΩCOM , v|ΩCOM ;µcoefCOM),
f(v;µcoef) =
I∑COM=1
fM(COM)(v|ΩCOM ;µcoefCOM).
Objective:Given for any µ ∈ D, a(·, ·;µ) : XSYS × XSYS → R a continuous, coercive bilinear form and given acontinuous linear functional f(·;µ) : XSYS → R, find u(µ) ∈ X which satisfies the weak formulation
a(uSYS(µ), v;µ) = f(v;µ), ∀v ∈ XSYS. (3.2)
We assume a and f satisfy all requirements to obtain a unique solution (e.g., with the Lax-Milgram theorem).In particular there exist α(µ) and γ(µ) such that
0 < α(µ) ≡ infv∈XSYS
a(v, v;µ)
‖v‖X2SYS
, γ(µ) ≡ supv∈XSYS
supw∈XSYS
a(v, w;µ)
‖v‖XSYS‖w‖XSYS
<∞.
3.2 Finite Element Truth Formulation
We saw in the last subsection the spaces Xm for the reference subdomains Ωrefm . Now we like to introduce
FE approximation spaces Xhm for the discretization of Ωref
m . For each m we endow Xhm with inner product
〈·, ·〉Xm and the norm ‖·‖Xm . We define discrete port spaces P hm,L-P with: ∀w ∈ Xhm ∃ v ∈ P hm,L-P with
v = w|Γm,L-P . Further, we face the so called “FE bubble-spaces” which are defined on the discretized domain
Bhm;0 =
v ∈ Xh
m : v|Γm,L-P = 0, 1 ≤ L-P ≤ nΓm
, 1 ≤ m ≤M,
and we have associated coercivity and continuity constants:
0 < αhm;0(µm) ≡ infv∈Bh
m;0
am(v, v;µm)
‖v‖2Xm
, γhm;0(µm) ≡ supv∈Bh
m;0
supw∈Bh
m;0
am(v, w;µm)
‖v‖Xm‖w‖Xm
<∞.
These constants are positive for all µm ∈ Dm for every reference component, because Xhm is endowed with
the semi-norm ‖·‖Xm and Bhm;0 has a non-empty Dirichlet boundary. The existence of these constants is
very important to ensure solvability of later defined subproblem, see (3.14) and (3.15). We focus on the truth
16 3 STATIC CONDENSATION METHOD
solution of (3.2) which lives on the system truth FE space Xh := (⊕ICOM=1XhM(COM)) ∩XSYS. Trough the
discretization we get the system truth coercivity and continuity constants:
0 < αh(µ) ≡ infv∈Xh
a(v, v;µ)
‖v‖2XSYS
, γh(µ) ≡ supv∈Xh
supw∈Xh
a(v, w;µ)
‖v‖XSYS‖w‖XSYS
<∞.
where α(µ) ≤ αh(µ) and γh(µ) ≤ γ(µ), since Xh ⊂ XSYS. We state the truth weak formulation:For µ ∈ D find uh ∈ Xh with
a(uh(µ), v;µ) = f(v;µ), ∀v ∈ Xh.
Until now we only introduced the domain ΩSYS, a subdivision of it, the corresponding Hilbert spaces andsome FE truth formulation for subproblems. We like to comment on this “top-down” procedure:
• “Top-down” procedure: we first focused on the whole physical domain ΩSYS, split it up into subdo-mains, introduced reference subdomains and made sure that discrete subproblems, as defined in (3.14)and (3.15), are still well defined.We can do the same with a “bottom-up” strategy we start with well-posed subproblems on referencedomains, then construct some physical domain using these reference domains. So both ways areapplicable.
• By doing the “bottom-up” strategy we can also introduce the above mentioned geometric parameterµgeo which can include dilation, shear or it can do the physically necessary translations, which weassumed implicitly. Then we would introduce aµgeo = (µ
geo1 , . . . , µ
geoI ) ∈ Dgeo
M(1) × · · · ×DgeoM(1) =: Dgeo such that the parameter for the PDE would
be µ = µcoef × µgeo ∈ Dcoef × Dgeo =: D. For all µgeo ∈ Dgeo we still have to satisfy the geometriccompatibility requirements with an addition: all connection ports have to change geometrically in aconforming way. In the system truth FE-formulation this means that discretization-nodes are stretchedor translated correspondingly on both sides of any port.
3.3 Static Condensation: OFFLINE
We now introduce a step by step introduction to the Static Condensation method and we will follow thetop-down procedure, shown in Section 3.1 and will repeat briefly the relevant parts of the domain decompo-sition. Again we will write the extension of a subdivision of ΩSYS ⊂ R2 into rectangles in italic.
1. Definition of local ports (L-P):After we subdivided ΩSYS into subdomains, we focus on every component COM ∈ I . For every COM
there is at least one local port (L-P) which is connected to another COM′. So we define ΓL-P,COM =ΓL-P,M(COM) as the local port hyper face with 1 ≤ L-P ≤ nΓ
M(COM), where nΓM(COM) defines the num-
ber of local ports in one reference domain. The L-Ps from one COM are not allowed to have commonsupport, which is very important because this restricts very much the kind of subdivisions one is allowed todo, according to [1].
We will soften this restriction in R2: if two L-Ps
L-P,L-P′
from one COM have common support (inR2 its always a point), we define this intersection as a new local point and “shorten” the others by thispoint. In the system truth formulation this means that we have a FE-node as a common node of two L-Pswhich now gets the status of a local port. The two L-Ps loose this FE-node which is now connected with upto four components in our applications.
3.3 Static Condensation: OFFLINE 17
2. Definition of Global Ports (G-P):As we look on these L-Ps globally we can assign to every L-P,COM a ΓG-P, 1 ≤ G-P ≤ nΓ, with nΓ
the number of all global ports for the whole system. Again, with ΓG-P we denote one hyper face which isassociated with at most two neighbors ΓL-P,COM and ΓL-P′,COM′ . This is derived from the restriction for L-Psof one COM from step 1.
We introduced special L-P-nodes, which can be interpreted globally as G-P-nodes, where again eachG-P-node is connected with up to four components.
3. Mapping from L-P,COM to G-P:We define an invertible local to global port index mapping G from L-P,COM to G-P which already
determines the system configuration. It maps G-P = GCOM(L-P) with L-P = G−1COM(G-P). The mapping
implies how the subdomains stick together and how they form the system domain ΩSYS. (Furthermore, weassume that ΓG-P does not lie on a boundary with homogeneous Dirichlet boundary conditions, since therethe values are just zero.) This mapping could also be defined later in the ONLINE stage, if we would followthe bottom-up strategy.
4. Component Bubble-Space:We define for each reference domain Ωref
m (equivalently for each COM ∈ I) FE bubble-spaces with
Bhm;0 =
v ∈ Xh
m : v|Γm,L-P = 0, 1 ≤ L-P ≤ nΓm
, 1 ≤ m ≤M.
5. Coercivity and continuity on subspaces Bhm;0 :
We make sure that the constants
0 < αhm;0(µm) ≡ infv∈Bh
m;0
am(v, v;µm)
‖v‖2Xm
, γhm;0(µm) ≡ supv∈Bh
m;0
supw∈Bh
m;0
am(v, w;µm)
‖v‖Xm‖w‖Xm
<∞,
are positive which ensures solvability of subproblems on the subspaces, see (3.14) and (3.15).
6. Eigenfunctions in Rd−1:We now introduce spaces that will affect the interaction between subdomains Ωref
m (respectively ΩCOM).This means that on the ports the information has to be transformed from one component to another. For eachreference domain and each L-P we introduce a discrete “port space” P hL-P,m, 1 ≤ L-P ≤ nΓ
m, 1 ≤ m ≤M ,which is the restriction of Xh
m to the port. Further, we define an eigenfunction spaceχk,L-P,m ∈ P hL-P,m :
1 ≤ k ≤ N ΓL-P,m
, whereN Γ
L-P,m denotes the degrees of freedom on this port space. The degrees of freedomare restricted by the kind of nodes on this port space and the kind of FE-method. We can compute theseeigenvalues with the generalized eigenvalue problem∫
ΓL-P,m
∇χk,L-P,m · ∇v = λk
∫ΓL-P,m
χk,L-P,m v, ∀v ∈ P hL-P,m, (3.3)
‖χk,L-P,m‖L2(ΓL-P,m) = 1. (3.4)
By solving this generalized eigenvalue problem we obtain the desired χk,L-P,m with corresponding eigen-values λk ∈ R with (λmin =)λ1 < ... < λNΓ
L-P,m. In applications with FEM one gets the equation:
Ax = λMx ⇔ (A − λM)x = 0. We search the null spaces of this system. In our case A has a nullspace of dim = 1 and the spectrum is σ(A) ≥ 0, M is symmetric positive definite. By varying λ one shiftsthe spectrum σ(A − λM) such that one has to ensure that there is always a nontrivial element in the nullspace. Further, the eigenfunctions satisfy
Note that (3.5) does not imply that χTk′,L-P,m ·χk′′,L-P,m :=∑NΓ
L-P,mk=1 (χk′,L-P,m)k(χk′′,L-P,m)k = δk′k′′ but on
the right-hand side of (3.3) we get a mass matrix M with the property χTk′,L-P,m ·M · χk′′,L-P,m = δk′k′′ .In our application, if we have a L-P-node on a component, we will slightly change the generalized
eigenvalue problem. Through a L-P-point in the corners of a domain we shortened the original straight-line-L-Ps. For these L-Ps we like to introduce a new “port space” P hL-P,m;0, 1 ≤ L-P ≤ nΓ
m, 1 ≤ m ≤ M
with P hL-P,m;0 := P hL-P,m ∩v ∈ P hL-P,m; v(∂ΓL-P,m) = 0
and have∫
ΓL-P,m
∇χk,L-P,m · ∇v = λk
∫ΓL-P,m
χk,L-P,m v, ∀v ∈ P hL-P,m;0, (3.6)
‖χk,L-P,m‖L2(ΓL-P,m) = 1, with: 1 ≤ k < N ΓL-P,m. (3.7)
We face an inequality k < N ΓL-P,m because we will get some χk,L-P,m which are zero. So we will take only
the eigenfunctions χk,L-P,m which are non-zero for the basis.
7. Elliptic Lifting:Next we search for functions that live on Xh
m and we use the obtained eigenfunctions χk,L-P,m, k =1, ...,N Γ
L-P,m, which are generally associated with two L-Ps and one G-P. We define the “lifted” interfacefunction Ψk,G-P which in general has support on two components as Ψk,G-P = ψk,L-P,COM + ψk,L-P′,COM′ .The function ψk,L-P,COM ∈ Xh
m is called “eigenmode” and is the restriction of Ψk,G-P to one COM. Nowwe define “lifting” as solving a Laplace problem for ψk,L-P,COM with homogeneous Dirichlet boundaryconditions on all local ports except “L-P”:∫
Ωrefm
∇ψk,L-P,m · ∇v = 0, ∀v ∈ BhL-P,m;0, (3.8)
ψk,L-P,m = χk,L-P,m, on ΓL-P,m, (3.9)
ψk,L-P,m = 0, on ΓL-P′,m,L-P′ ∈
1, . . . ,L-P− 1,L-P + 1, . . . , nΓm
. (3.10)
In (3.8) we chose∫∇ · ∇ as the inner product, however, others are also allowed. The obtained interface
functions Ψk,G-P ensure continuous interaction between subdomains and they especially contain the staticcondensation idea of getting degrees of freedom into the subdomains’ boundary.
If we have L-P-points we do the elliptic lifting for them as well by introducing so-called “cornerramps”=: ψ1,L-P,m ∈ Xh
m with L-P ∈
corner nodes⊂
1, . . . , nΓm
depending of the corner/node and
the reference subdomain. For a reference rectangle we pick one of the four corners and prescribe values forthe whole boundary: on the corner itself the eigenmode takes the value 1 and the values decay linearly onthe two neighbor edges to 0. The two edges opposite to the corner take the values 0, see Fig. 3.3.
Figure 3.3: Dirichlet boundary conditions for “corner-modes” on a reference domain in red.
Then one solves the problem ∫Ωref
m
∇ψ1,L-P,m · ∇v = 0, ∀v ∈ BhL-P,m;0
3.4 Static Condensation: ONLINE 19
with the described boundary conditions and gets the eigenmode ψ1,L-P,m. L-P is now one of the four cornernodes. Globally this implies for a G-P-point with connection to four components:Ψ1,G-P = ψ1,L-P,COM + ψ1,L-P′,COM′ + ψ1,L-P′′,COM′′ + ψ1,L-P′′′,COM′′′ .
We continue with some motivation why these eigenmodes ψk,L-P,COM or the global interface functionsΨk,G-P are that important: These functions will be like basis functions which have two important roles.Firstly, the eigenmodes are a main part of the solution on a subdomain. The solution will be completed byµ-dependent bubble-functions, we will define on the next pages. Secondly, the eigenmodes establish thedependence between subdomains.
These seven steps are prepared and computed in the OFFLINE stage, since the parameter µ is notinvolved.
3.4 Static Condensation: ONLINE
Here comes the core theory, which is now applicable also to our extended subdivision of ΩSYS ⊂ R2.With the above seven steps we would like to construct another representation of the truth solution: to
do this we especially use the step 4 and 7, see p. 17 and p. 18, to compute bubble-functions bhi (µ) and weuse the introduced interface-functions Ψk,G-P. For simplicity we now enumerate the set of components withi ∈ I , not COM. We express the truth solution as
uh(µ) =I∑i=1
bhi (µ) +nΓ∑
G-P=1
NΓG-P∑
k=1
Uk,G-P(µ)Ψk,G-P (3.11)
with bhi (µ) ∈ BhM(i);0 for each i ∈ I . The Uk,G-P(µ) with 1 ≤ k ≤ N Γ
G-P and 1 ≤ G-P ≤ nΓ arethe coefficients of the interface functions. This representation of uh(µ) means that firstly we use bubble-functions to represent the solution in the interior of subdomains and secondly we use interface functionsto ensure continuous interaction between these subdomains. One can ask why should we focus on thiscumbersome representation of the solution in (3.11)? The first reason is historically, that one can avoidby this addressing huge FEM-problems and using a sub-structural technique by chopping up the domainΩSYS into smaller domains. The second reason is that so one can involve the Reduced Basis approach forcomputing bhi (µ) in the ONLINE stage. This requires to invest more time in the OFFLINE stage to preparesome data sets, but it gives the advantage of getting approximate solutions relatively fast with a computableerror tolerance to the solution U(µ) of the truth Schur Complement System, see Section 5. This U(µ) is areordering of Uk,G-P(µ) and so is strongly connected to the truth solution uh(µ), see (3.11).
3.4.1 Compute bubble-functions bhi (µ) ∈ BhM(i);0 on component i
We assume that all functions in BhM(i);0 are extended on ΩSYS by zero, as we assume this for Ψk,G-P outside
its support. We focus on two spaces BhM(i);0 and Bh
M(i′);0 which have disjoint support and hence we caninsert (3.11) into (3.2) and test with space members to obtain
All these equations are well-posed by coercivity and continuity and they are computed ONLINE because ofthe µ-dependence. The steps (3.14) and (3.15) are expensive but we will use the RB approach in Section 4to compute these terms faster. For 1 ≤ G-P ≤ nΓ and for 1 ≤ k ≤ N Γ
G-P we define a new function
Φk,G-P(µ) ≡ Ψk,G-P +∑
(L-P,i)∈πG-P
bhk,L-P,i(µ) =∑
(L-P,i)∈πG-P
ψk,L-P,i + bhk,L-P,i(µ). (3.16)
3.4.2 New representation of the solution space
From the conclusions above we construct the “skeleton space” spanned by the new functions in (3.16)
XhS(µ) ≡ spanΦk,G-P(µ) : 1 ≤ G-P ≤ nΓ and 1 ≤ k ≤ N Γ
G-P ∩Xh,
and we endow this space with the inner product and norm
(v, w)S ≡nΓ∑
G-P=1
(v, w)L2(ΓG-P) ∀v, w ∈ XhS(µ), and ‖ · ‖2S = (·, ·)S .
With (3.11) we get on each i ∈ I a representation of uh(µ)
uh|Ωi(µ) = bhi (µi) +
nΓM(i)∑
L-P=1
NΓL-P,M(i)∑k=1
Uk,Gi(L-P)(µ)ψk,L-P,i. (3.17)
3.4 Static Condensation: ONLINE 21
Furthermore, with (3.13) and (3.17) we obtain for uh(µ)|Ωi a slightly different representation:
uh|Ωi(µ) = bhf,i(µ) +
nΓM(i)∑
L-P=1
NΓL-P,M(i)∑k=1
Uk,Gi(L-P)(µ)(bhk,L-P,i(µ) + ψk,L-P,i
). (3.18)
With the definition (3.16) and (3.18) we can express the global solution as
uh(µ) =I∑i=1
bhf,i(µ) +nΓ∑
G-P=1
NΓG-P∑
k=1
Uk,G-P(µ)Φk,G-P(µ). (3.19)
3.4.3 Compute coefficients for one basis of the solution space
To evaluate the coefficients (Uk,G-P(µ))k,G-P we will use XhS(µ) as a test space and insert (3.19) in (3.2) to
obtain
nΓ∑G-P=1
NΓG-P∑
k=1
Uk,G-P(µ) a(Φk,G-P(µ), v;µ) =
f(v;µ)−I∑i=1
a(bhf,i(µ), v;µ), ∀v ∈ XhS(µ),
to get the so-called Schur Complement System of size nsc ≡∑nΓ
G-P=1N ΓG-P which we rewrite as
A(µ)U(µ) = F(µ). (3.20)
Where:1. A(µ) ∈ Rnsc×nsc defined by A(k′,G-P′),(k,G-P)(µ) ≡ a(Φk′,G-P′(µ),Φk,G-P(µ);µ)2. U(µ) ∈ Rnsc the coefficients of basis elements.3. F(µ) ∈ Rnsc defined by Fk′,G-P′(µ) ≡ f(Φk′,G-P′ ;µ)−
∑Ii=1 a(bhf,i(µ),Φk′,G-P′ ;µ)
3.4.4 Assembly process of A(µ) and F(µ); ONLINE
To obtain the matrix A(µ) and the vector F(µ) in (3.20) one hast to go down to the component level andevaluate local bilinear and linear forms. We define Ai ∈ RN
9: end for10: end for11: end for12: end for13:end for
The abbreviation A+ = B means A := A+B. The system (3.20) is solvable, for a proof see [1], p. 12.Until now we introduced the general framework for Static Condensation. A major restriction in the paperwas that every G-P contains at most two L-Ps. But with the extension of this section we can apply thisOFFLINE ONLINE procedure for ΩSYS ⊂ R2 with a subdivision into rectangles and not only stripes, seeFig. 3.2.
23
4 Static Condensation with the Reduced Basis Element method
The Static Condensation method offers a new representation of the solution. A main drawback is that forbuilding the Static Condensation system equation (3.20) every bubble-function has to be evaluated whichmay cause some computational effort. Here the authors of [1] offer the idea of approximating the bub-bles with a RB approach, which means instead of evaluating bhf,i(µ) and bhk,L-P,i(µ) we will evaluate RBapproximations bhf,i(µ) and bhk,L-P,i(µ).
4.1 How to build the RB Approximations
We start with (3.14) and for every component i ∈ I we want to exchange the space bhf,i(µ) ∈ BhM(i);0 with
bhf,i ∈ BhM(i);0. The property of Bh
M(i);0 is
aM(i)(bhf,i(µ), v;µi) = fM(i)(v;µi), ∀v ∈ Bh
M(i);0 (4.1)
and the RB space BhM(i);0 is constructed for each component i with the common Greedy algorithm, de-
scribed in [2]. Analogously, we focus on (3.15) and build the spaces Bhk,L-P,M(i);0 for the RB approximations
bhk,L-P,i(µ) with the property
aM(i)(bhk,L-P,i(µ), v;µi) = −aM(i)(ψ
hk,L-P,i, v;µi), ∀v ∈ Bh
k,L-P,M(i);0. (4.2)
bhk,L-P,i(µ) is computed for each component i, for each L-P and each mode k.
4.2 Constructing an approximating Schur Compliment System
We follow the steps (3.16)-(3.20) in Section 3.3 and construct the system. We start with defining, as in(3.16), the basis functions of the new skeleton space
Φk,G-P(µ) ≡ Ψk,G-P +∑
(L-P,i)∈πG-P
bhk,L-P,i(µ) =∑
(L-P,i)∈πG-P
ψk,L-P,i + bhk,L-P,i(µ). (4.3)
So we build the skeleton space
XhS(µ) ≡ spanΦk,G-P(µ) : 1 ≤ G-P ≤ nΓ, 1 ≤ k ≤ N Γ
G-P
with the same inner product and norm as the space XhS(µ). Instead of the truth solution we get an approxi-
mation of the solution uh(µ):
uh(µ) =
I∑i=1
bhf,i(µ) +
nΓ∑G-P=1
NΓG-P∑
k=1
Uk,G-P(µ)Φk,G-P(µ). (4.4)
Here again the coefficient vector Uk,G-P(µ) satisfies
nΓ∑G-P=1
NΓG-P∑
k=1
Uk,G-P(µ) a(Φk,G-P(µ), v;µ) = (4.5)
f(v;µ)−I∑i=1
a(bhf,i(µ), v;µ), ∀v ∈ XhS(µ).
Using (4.5), we get a linear system of equations
A(µ)U(µ) = F(µ) (4.6)
with the original size of nsc. Next we like to derive an error bound for ‖U(µ)− U(µ)‖2.
24 5 A POSTERIORI ERROR ANALYSIS FOR RB APPROXIMATION
5 A Posteriori Error Analysis for RB Approximation
In this section we like to extend the error analysis from [1] to our more general subdivision of ΩSYS ⊂ R2.We will first cite some proofs from [1] and then extend them to our configuration. By doing the extensionwe will slightly improve the SC-error bound.
5.1 Properties of the Reduced Basis Approximation
For all components i ∈ I , the residual rf,i(·;µi) : BhM(i);0 → R for (4.1) is given by
We introduce the dual norms of (5.1) and (5.2) and call themRf,i(µi) andRk,L-P,i(µi):
Rf,i(µ) ≡ supv∈Bh
M(i);0
rf,i(v;µi)
‖v‖XSYS
,
Rk,L-P,i(µ) ≡ supv∈Bh
M(i);0
rk,L-P,i(v, µi)
‖v‖XSYS
.
From [2] we know that we have a posteriori error bounds for the bubble approximations. For all i ∈ I ,a given µi ∈ DM(i), 1 ≤ L-P ≤ nΓ
M(i) and 1 ≤ k ≤ N ΓL-P,M(i) we get:
‖bhf,i(µi)− bf,i(µi)‖XM(i)≤Rf,i(µi)αLBi (µi)
, (5.3)
‖bhk,L-P,i(µi)− bk,L-P,i(µi)‖XM(i)≤Rk,L-P,i(µi)
αLBi (µi)
, (5.4)
where αLBi (µi) satisfies
0 < αLBi (µi) ≤ αi(µi), ∀µi ∈ DM(i).
There are several ways, how to compute αLBi (µi): “min-θ-approach”, “Successive Constraint Method”. In
the FEM context we just compute the lower bound with the Rayleigh quotient: We determine the small-est eigenvalue of the generalized eigenvalue problem Ai(µi)x = λAi(µi)x, where Ai(µi) is the Gram-matrix of the bilinear form ai(u, v, µi), u, v ∈ Xh
M(i);0 and Ai(µi) is the Gram-matrix of the bilinear formai(u, v, µi), u, v ∈ Xh
M(i);0. The parameter µi defines the inner-product of the space XhM(i);0.
5.2 System Level Bounds
We cite the following Lemma 1, Proposition 1 and Corollary 1 from [1] with significant parts of the proofs.The Lemma 1 is explicitly for domain decompositions with the restriction that no more than two componentsmay share one global port (in applications it often means that a FEM node lying on the subdomain boundaryis part of at most two subdomains).
5.2 System Level Bounds 25
Lemma 1:For any µ ∈ D we can show that ‖F(µ) − F(µ)‖2 ≤ σ1(µ) and ‖A(µ) − A(µ)‖F ≤ σ2(µ), (‖·‖F denotesthe Frobenius norm).
σ1(µ) ≡
(2∑i∈I
(∆f,i(µ))2
(nΓM(i)∑
L-P=1
NΓL-P,M(i)∑k=1
(∆k,L-P,i(µi))2
))1/2
, (5.5)
σ2(µ) ≡
(2∑i∈I
(nΓM(i)∑
L-P=1
NΓL-P,M(i)∑k=1
(∆k,L-P,i(µi))2
)2)1/2
, (5.6)
where
∆f,i(µ) ≡ Rf,i(µ)/√αLBi (µ),
∆k,L-P,i(µ) ≡ Rk,L-P,i(µ)/√αLBi (µ).
Proof: The proofs for (5.5) and (5.6) are similar and we thus restrict our attention to the more involved case(5.6). To derive a bound for the error of the global system matrix ‖A(µ)− A(µ)‖F we have to consider theerror of the local stiffness matrix ‖Ai(µi)− Ai(µi)‖F . First we focus on one single entry of this error for acomponent i ∈ I . In is easy to show that
Now we can evaluate the Frobenius norm of the error in the local stiffness matrix for any component i:
‖Ai(µi)− Ai(µi)‖2F ≤nM(i)∑L-P=1
NΓL-P,M(i)∑k=1
nM(i)∑L-P′=1
NΓL-P′,M(i)∑k′=1
(∆k,L-P,i(µi)∆k′,L-P′,i(µi))2 (5.8)
=
(nM(i)∑L-P=1
NΓL-P,M(i)∑k=1
∆k,L-P,i(µi)2
)(nM(i)∑L-P′=1
NΓL-P′,M(i)∑k′=1
∆k′,L-P′,i(µi)2
)
=
(nM(i)∑L-P=1
NΓL-P,M(i)∑k=1
∆k,L-P,i(µi)2
)2
(5.9)
We return to the global system error ‖A(µ)− A(µ)‖F and look at one entry (j, j′) of E(µ) := A(µ)− A(µ)with Ej,j′(µ) 6= 0. This entry (j, j′) of E(µ) can be identified as
26 5 A POSTERIORI ERROR ANALYSIS FOR RB APPROXIMATION
Figure 5.1: A subdivision of ΩSYS ⊂ R2 into two equal rectangles with homogeneousDirichlet boundary conditions on the red boundaries.
for certain k, k′,G-P,G-P′. In Fig. 5.1 we give the simplest example of subdividing a domain into twoequal rectangles. The reader should have this figure in mind while building the matrix A(µ) or A(µ). Wedistinguish between two cases in computing Ej,j′(µ), depending on (k,G-P), (k′,G-P′):
1. G-P = G-P′, the global ports are the same. So we get πG-P =
(L-P, i), (L-P′, i′)
or πG-P =(L-P, i)
. This means that one G-P has at most two neighboring components.
2. G-P 6= G-P′, the global ports are different and we have the interaction of two modes on different localports. This means πG-P = L-P, i and πG-P′ = L-P′, i on one component i(j, j′).
Thus we know that Ej,j′(µ) consists of at most two summands from two different local stiffness matrixerrors:
Ej,j′(µ) = Ai(j,j′)
J (j),J (j′)(µ)− Ai(j,j′)
J (j),J (j′)(µ) + Ai′(j,j′)K(j),K(j′)(µ)− Ai
′(j,j′)K(j),K(j′)(µ)
with the discrete mappings J (respectively K) which maps from the global system size nsc to the local size
of the component NM(i) :=∑nΓ
M(i)
L-P=1N ΓL-P,M(i) (respectively NM(i′)):
J :
1, ..., nsc→
1, ...,NM(i)
,
K :
1, ..., nsc→
1, ...,NM(i′)
.
Now we use the inequality (a+ b)2 ≤ 2(a2 + b2) and get
|Ej,j′(µ)|2 ≤ 2|Ai(j,j
′)J (j),J (j′)(µ)− Ai(j,j
′)J (j),J (j′)(µ)|2 + |Ai
′(j,j′)K(j),K(j′)(µ)− Ai
′(j,j′)K(j),K(j′)(µ)|2
.
Then we consider that every entry of (Ai(µ)− Ai(µ))(l,l′), l, l′ ∈ 1, ...,NM(i) for all i ∈ I , is only once
involved in the global system error ‖A(µ)− A(µ)‖F , so we can add up the error:
and λmin(µ) is the smallest eigenvalue of the system matrix A(µ). For the proof of (5.10) see [1]. In onepart of the proof we use the inequality ‖
(A(µ)− A(µ)
)U(µ)‖2 ≤ σ2(µ)‖U(µ)‖2.
Further, [1] gives a sharper bound for the term ‖(A(µ) − A(µ)
)U(µ)‖2 which helps to decrease the
error:
Corollary 1:If λmin(µ)− σ2(µ) > 0 we can derive a sharper bound for ‖U(µ)− U(µ)‖2:
‖U(µ)− U(µ)‖2 ≤ ∆U? (µ) (5.11)
with
∆U? (µ) ≡ σ1(µ) + σ3(µ) + ‖F(µ)− A(µ)U(µ)‖2
λmin(µ)− σ2(µ)
where
σ3(µ) ≡
2∑i∈I
(nΓM(i)∑
L-P=1
NΓL-P,M(i)∑k=1
(∆k,L-P,i(µi))2)(nΓ
M(i)∑L-P=1
NΓL-P,M(i)∑k=1
∆k,L-P,i(µi)|Uik,L-P(µ)|)21/2
. (5.12)
We define Ui ∈ RNM(i) as the part of U with the entries Uik,L-P = Uk,Gi(L-P) for 1 ≤ L-P ≤ nΓM(i) and
1 ≤ k ≤ N ΓL-P,M(i).
Proof: Here we will cite the proof because it will be important for further extensions: We use the inequality(5.7) for deriving a bound for a single entry of (Ai(µi)− Ai(µi))Ui(µi):∣∣∣∣
Now we consider again that every entry of A(µ) − A(µ) has at most two summands from local stiffnessmatrix errors, we involve (a+ b)2 ≤ 2(a2 + b2) and get (5.12).
28 5 A POSTERIORI ERROR ANALYSIS FOR RB APPROXIMATION
5.3 System Level Bounds: sharper bounds
Now we will extend slightly the proof of Lemma 1:
Lemma 1 sharper:Under the assumption of Lemma 1 we derive a smaller σ2(µ) ≤ σ2(µ):
σ2(µ) ≡
(∑i∈I
nΓM(i)∑
L-P=1
NΓL-P,M(i)∑k=1
(∆k,L-P,i(µi))22
+∑i∈I
nΓM(i)∑
L-P=1
NΓL-P,M(i)∑k=1
(∆k,L-P,i(µi))22) 1
2
.
Proof: We focus again on the entry Ej,j′(µ) 6= 0 which belongs toa(Φk,G-P(µ),Φk′,G-P′(µ);µ)−a(Φk,G-P(µ), Φk′,G-P′(µ);µ) as seen before. We repeat the G-P,G-P′-relationsbut now we use the fact that some relations imply only one summand in Ej,j′(µ). This implication allowsus to avoid using the inequality (a+ b)2 ≤ 2(a2 + b2) for each entry of Ej,j′(µ).
1. G-P = G-P′, the global ports are the same. So we get πG-P =
(L-P, i), (L-P′, i′)
or πG-P =(L-P, i)
which means that one G-P has at most two neighboring components.
Remark: Here we have only one G-P and all the interactions between the modes: 1 ≤ k, k′ ≤N Γ
L-P,M(i)(= NΓL-P′,M(i′)). All those interactions for one G-P corresponds to a block-matrix on the
diagonal of E(µ). If this G-P has two neighbors i, i than these entries correspond to the interaction ofall modes k, k′ on one ΓG-P. We collect all coefficients (j, j′) of Ej,j′(µ) which satisfy case 1 in theset D. For simplicity we will add to D those coefficients which correspond in E(µ) to a block-matrixon the diagonal but do have only one neighbor.
2. G-P 6= G-P′, the global ports are different and one has the interaction of two modes on different localports, L-P, L-P′. This means πG-P = L-P, i and πG-P′ = L-P′, i on one component i(j, j′).
Remark: In this case we have two different G-Ps. This implies that the valuea(Φk,G-P(µ),Φk′,G-P′(µ);µ)− a(Φk,G-P(µ), Φk′,G-P′(µ);µ) belongs to a coefficient (j, j′) which doesnot lie in a block-matrix on the diagonal of E(µ). We collect coefficients (j, j′) which satisfy case 2with Ej,j′(µ) 6= 0 in the set ND.
By applying the inequality (a + b)2 ≤ 2(a2 + b2) only on entries of E(µ) with two summands (usingthe set D) we get:
‖A(µ)− A(µ)‖2F = ‖E‖2F =
nsc∑j,j′=1
Ej,j′(µ)2 (5.13)
≤nsc∑
j,j′=1
|Ai(j,j
′)J (j),J (j′)(µ)− Ai(j,j
′)J (j),J (j′)(µ)|2 + |Ai
′(j,j′)K(j),K(j′)(µ)− Ai
′(j,j′)K(j),K(j′)(µ)|2, if (j, j′) ∈ D
|Ai(j,j′)
J (j),J (j′)(µ)− Ai(j,j′)
J (j),J (j′)(µ)|2, if (j, j′) ∈ ND
+
nsc∑j,j′=1
|Ai(j,j
′)J (j),J (j′)(µ)− Ai(j,j
′)J (j),J (j′)(µ)|2 + |Ai
′(j,j′)K(j),K(j′)(µ)− Ai
′(j,j′)K(j),K(j′)(µ)|2, if (j, j′) ∈ D
0, if (j, j′) ∈ ND
(5.14)
As before we will get the term with local stiffness matrix errors:∑
i∈I‖Ai(µi)− Ai(µi)‖2F but withoutthe factor 2. Instead of taking this term twice, we additionally consider those entries of Ej,j′(µ) which havetwo error summands. So we have just to add them up by using the description of the set D in case 1. The
5.4 System Level Bounds for subdivision of Ω ⊂ R2 into rectangles 29
elements in D correspond to the interaction between the modes belonging to (L-P,i) for each component iand its L-Ps. Those entries lie in a block-matrix on the diagonal of E(µ).
(5.14) =∑i∈I‖Ai(µi)− Ai(µi)‖2F
new term+
∑i∈I
nΓM(i)∑
L-P=1
NΓL-P,M(i)∑k=1
nΓM(i)∑
L-P′=1,L-P′=L-P
NΓL-P′,M(i)∑k′=1
(∆k,L-P,i(µi)∆k′,L-P′,i(µi))2
(5.7)=∑i∈I‖Ai(µi)− Ai(µi)‖2F +
∑i∈I
nΓM(i)∑
L-P=1
NΓL-P,M(i)∑k=1
(∆k,L-P,i(µi))22
(5.9)≤∑i∈I
nΓM(i)∑
L-P=1
NΓL-P,M(i)∑k=1
(∆k,L-P,i(µi))22
+∑i∈I
nΓM(i)∑
L-P=1
NΓL-P,M(i)∑k=1
(∆k,L-P,i(µi))22. (5.15)
We see in the last line (5.15) that the second summand is smaller than the first one because it containsonly a part of the first one.
Corollary 1 sharper:Under the assumption of Corollary 1 we derive a smaller σ3(µ) ≤ σ3(µ):
σ3 ≡
∑i∈I
(nΓM(i)∑
L-P=1
NΓL-P,M(i)∑k=1
(∆k,L-P,i(µi))2)(nΓ
M(i)∑L-P=1
NΓL-P,M(i)∑k=1
∆k,L-P,i(µi)|Uik,L-P(µ)|)2
+∑i∈I
nΓM(i)∑
L-P=1
(NΓL-P,M(i)∑k=1
(∆k,L-P,i(µi))2)(NΓ
L-P,M(i)∑k=1
∆k,L-P,i(µi)|Uik,L-P(µ)|)21/2
. (5.16)
Proof: The approach to show this is equivalent to the approach in Lemma 1 sharper. We know which entriesof the vector (A(µ)−A(µ))U(µ) have one or two local stiffness matrix error summands, which depend onlyon the entries of A(µ) − A(µ). If we have two summands in one entry of (A(µ) − A(µ))U(µ) we have touse the equation (a+ b)2 ≤ 2(a2 + b2) and this implies that some terms are occurring with factor 2. Theseterms correspond to the modes k, k′, which belong globally to one G-P, elements in the set D. So there areno terms which would reflect the interaction of different local ports of one component i and therefore thesecond summand of (5.16) has only one sum over L-P.
5.4 System Level Bounds for subdivision of Ω ⊂ R2 into rectangles
Lemma 1 sharper can be easily extended to the subdivision with rectangles and corners ramps. First, we liketo stress that equations (5.7) and (5.9) are still valid for the new subdivision: if L-P is the corner node withits corner ramp we can use (5.4) and show properties (5.7) and (5.9).
Similarly to the cases in Section 5.3, we need to subdivide all entries of E(µ) and the correspondingindices (j, j′) to sets. Remember, we have corner nodes which touch at most four rectangles so we haveto consider up to four summands in Ej,j′(µ). Again we like to have a bound for the global system error‖A(µ)− A(µ)‖F . For this we focus on one entry Ej,j′(µ) 6= 0 and we identify the entry (j, j′) with
Remark: In this case we call the set of indices (j, j′) FOUR, because the entry Ej,j′(µ) containsat most 4 summands. For simplicity we collect in FOUR even corner ramps with less than fourcomponent connections.
C2. G-P = G-P′ and the G-P is not a G-P-point, but a straight segment. So we get the index set πG-P =(L-P, i), (L-P′, i′)
.
C3. G-P 6= G-P′, the global ports are different and at least one of the global ports is a G-P-point. Sowe have the interaction of two modes on different local ports. Because we have a corner rampincluded, we have at most two components i, i′ with πG-P =
(L-P, i), (L-P′, i′)
and πG-P′ =
(L-P′′, i), (L-P′′′, i′)
.
Remark: In the case C2 and C3 we have at most two summands so we collect those indices in the setTWO. For simplicity we collect in TWO coefficients which satisfy C2 and C3 and have less then twocomponent connections.
C4. G-P 6= G-P′ and none of the G-Ps is a G-P-point. The global ports are different and we have theinteraction of two modes k, k′ on different local ports, so πG-P =
(L-P, i)
and πG-P′ =
(L-P′, i)
with one component i(j, j′).
Remark: Here we have exactly one summand in Ej,j′(µ) so we call the set of indices ONE.
Analogously, to “Lemma 1 sharper” we formulate the Lemma 2 for our extended subdivision of ΩSYS:
Lemma 2:For any µ ∈ D, we show ‖F(µ)− F(µ)‖2 ≤ σ1(µ) and ‖A(µ)− A(µ)‖F ≤ σ2(µ) with:
(σ1(µ))2 ≡ 2∑i∈I
(∆f,i(µ))2
( nΓM(i)∑
L-P=1,L-P6=COR
NΓL-P,M(i)∑k=1
(∆k,L-P,i(µi))2
)(5.17)
+ 4∑i∈I
(∆f,i(µ))2
( nΓM(i)∑
L-P=1,L-P=COR
NΓL-P,M(i)∑k=1
(∆k,L-P,i(µi))2
),
(σ2(µ))2 ≡∑i∈I
(nΓM(i)∑
L-P=1
NΓL-P,M(i)∑k=1
(∆k,L-P,i(µi))2
)2
(5.18)
+ 1∑i∈I
nΓM(i)∑
L-P=1,L-P6=COR
(NΓL-P,M(i)∑k=1
(∆k,L-P,i(µi))2
)2
(5.19)
+ 1∑i∈I
nΓM(i)∑
L-P=1
NΓL-P,M(i)∑k=1
nΓM(i)∑
L-P′=1,L-P′=COR,L-P6=L-P′
NΓL-P′,M(i)∑k′=1
(∆k,L-P,i(µi)∆k′,L-P′,i(µi))2 (5.20)
+ 3∑i∈I
nΓM(i)∑
L-P=1,L-P=COR
NΓL-P,M(i)∑k=1
(∆k,L-P,i(µi))22. (5.21)
5.4 System Level Bounds for subdivision of Ω ⊂ R2 into rectangles 31
We denote L-P = 1, . . . , nΓM(i),L-P = COR as the set of L-Ps in the corner of one component i, L-P =
1, . . . , nΓM(i),L-P 6= COR denotes the set of L-Ps which are the straight segments of one component i and
L-P′ = 1, . . . , nΓM(i),L-P′ = COR,L-P′ 6= L-P denotes the set of L-Ps which are corners of one component
with L-P′ 6= L-P. We define again
∆f,i(µ) ≡ Rf,i(µi)/√αLBi (µ),
∆k,L-P,i(µ) ≡ Rk,L-P,i(µi)/√αLBi (µ).
Note: We could have computed σ1(µ) and σ2(µ) easily with the general statement that every entry of theglobal system matrix has at most four summands. So we would involve the inequality (a + b + c + d)2 ≤4(a2 +b2 +c2 +d2) and get the lines (5.17) and (5.18) with the factor 4 and would not consider which termsare really occurring four times. As in Lemma 1 sharper, through this proof we get σ1(µ) ≤ σ1(µ), σ2(µ) ≤σ2(µ).Proof: First we like to show equation (5.17): Similarly to Lemma 1 we can show that
|Fik,L-P(µ)− Fik,L-P(µ)| ≤ ∆f,i∆k,L-P,i(µi)
is valid, see [1]. We like to compute ‖F(µ)− F(µ)‖2: We have to consider how many summands out of thelocal load vector error Fik,L-P(µi)− Fik,L-P(µi) sum up to one entry of F(µ)− F(µ). Here we consider onlytwo cases F1 and F2:
F1. Either k,L-P belongs to a straight segment L-P in R1 such that there are at most two summands forevery entry in F(µ)− F(µ) or
F2. k,L-P belongs L-P in a corner with at most four summands in F(µ)− F(µ).
Again we use two inequalities (a + b)2 ≤ 2(a2 + b2), (a + b + c + d)2 ≤ 4(a2 + b2 + c2 + d2) andadd up terms which belong to F1 or F2, like in (5.13)-(5.14). Next we follow the arguments in the proof ofLemma 1 sharper and get (5.17).
For the second part of the Lemma 2, (5.18)-(5.21), we again follow the proof of Lemma 1 sharper andnow we use the discussed three index sets FOUR, TWO, ONE in the enumeration C1–C4. So again we getthe terms
∑i∈I‖Ai(µi) − Ai(µi)‖2F which we can bound with the line (5.18). Further, we have to add up
the elements in the sets TWO and FOUR. We use the characterization of those sets and get in lines (5.19)and (5.20) the elements corresponding to the set TWO. Using the properties of FOUR we get the line (5.21).
32 5 A POSTERIORI ERROR ANALYSIS FOR RB APPROXIMATION
Corollary 2:If λmin(µ)− σ2(µ) > 0 we can derive a sharper bound for ‖U(µ)−U(µ)‖2 by introducing a σ3(µ) ≤ σ3(µ).
(σ3(µ))2 ≡∑i∈I
(nΓM(i)∑
L-P=1
NΓL-P,M(i)∑k=1
(∆k,L-P,i(µi))2)(nΓ
M(i)∑L-P=1
NΓL-P,M(i)∑k=1
∆k,L-P,i(µi)|Uik,L-P(µ)|)2
+ 1∑i∈I
( nΓM(i)∑
L-P=1,L-P6=COR
NΓL-P,M(i)∑k=1
(∆k,L-P,i(µi))2)( nΓ
M(i)∑L-P=1,L-P6=COR
NΓL-P,M(i)∑k=1
∆k,L-P,i(µi)|Uik,L-P(µ)|)2
+ 1∑i∈I
nΓM(i)∑
L-P=1
NΓL-P,M(i)∑k=1
(∆k,L-P,i(µi))2
nΓM(i)∑
L-P′=1,L-P′=COR,L-P6=L-P′
NΓL-P′,M(i)∑k′=1
(∆k′,L-P′,i(µi)|UiL-P′,k′(µ)|)2
+ 3∑i∈I
nΓM(i)∑
L-P=1,L-P=COR
NΓL-P,M(i)∑k=1
(∆k,L-P,i(µi))2(∆k,L-P,i(µi)|Uik,L-P(µ)|)2.
We denote L-P = 1, . . . , nΓM(i),L-P = COR as the set of L-Ps in the corner of one component i, L-P =
1, . . . , nΓM(i),L-P 6= COR denotes the set of L-Ps which are the straight segments of one component i and
L-P′ = 1, . . . , nΓM(i),L-P′ = COR,L-P′ 6= L-P denotes the set of L-Ps which are corners of one component
with L-P′ 6= L-P.Proof:The proof of Corollary 2 is very similar to the proof of Lemma 2.
As described in Corollary 1 we use Lemma 2 and Corollary 2 to formulate the system error bound:
‖U(µ)− U(µ)‖2 ≤ ∆U? (µ) (5.22)
with
∆U? (µ) ≡ σ1(µ) + σ3(µ) + ‖F(µ)− A(µ)U(µ)‖2
λmin(µ)− σ2(µ).
33
6 Application of SCRBE method on RLE
6.1 Plain bearing and its general setting
In this section we apply the SCRBE method to the Reynolds Lubrication equation: We look on a plainbearing with eccentricity. Unfolded, it has length 0.1885m, width 0.02m and we set the distance, gapheight to h(x) = 30µm+ 20µm cos( x
0.1885), shown in Fig. 6.1.
0.03m
0.03m
h in 10−5m
rad1 2 3 4 5 6
123456
Figure 6.1: Left: Model of two bearing parts with eccentricity. Right: Gap height for aspecific example with maximal eccentricity of 6 · 10−5 m.
We fix the outer bearing part and give the inner part the velocity u1 = 4πm/s ≈ 12.57m/s, theviscosity of the lubricant is set to µV = 0.01Pa · s. Before we follow the SCRBE procedure given abovein Section 3 and 4, we like to make some comments on the variational formulation (3.2). We choose theheight h to be linear on every subdomain, through this we achieve some advantages for building the left-and right-hand side of (3.2). In Section 2 we introduced h ≡ h0 + hxx+ hzz. In the case of a plain bearingwith eccentricity we simplify h even more to h ≡ h0 + hxx and assume to have no velocities in the y-,z-directions: w1 = v1 = 0. We can compute h3, a polynomial, with coefficient parameters h0, hx and get anaffine parameter dependence of (3.2):
a(p, ψ;µ) =
Qa=4∑q=1
θqa(µ)aq(p, ψ)
with e.g.:
θ1a(µ) = h3
x, a1(p, ψ) =
∫Ω
x3
12µV∇p · ∇ψdx,
θ2a(µ) = h2
x, a2(p, ψ) =
∫Ω
x2
12µV∇p · ∇ψdx,
θ3a(µ) = hx, a3(p, ψ) =
∫Ω
x
12µV∇p · ∇ψdx,
θ4a(µ) = h0, a4(p, ψ) =
∫Ω
1
12µV∇p · ∇ψdx
34 6 APPLICATION OF SCRBE METHOD ON RLE
and the right-hand side
f(ψ;µ) =
Qf=2∑q=1
θqf (µ)f q(ψ),
θ1f = 0.5 · h0u1, f1(ψ) =
∫Ω
1∂xψdx,
θ2f = 0.5 · hxu1, f2(ψ) =
∫Ωx∂xψdx.
This affine decomposition makes it possible to compute rapidly the bilinear form a(ph, ψh;µ) withph, ψh ∈ Xh. We will choose the simple FE-method with hat functions on triangles for computing allnecessary functions in the OFFLINE stage.
Now we follow the steps from Section 3 and 4: First, we subdivide the domain ΩSYS into 18 × 4subdomains (tiles), see Fig. 6.2. Every tile has the same shape such that the OFFLINE stage will be doneonly for one reference subdomain Ωref.
(0, 0)
(0.1885, 0.02)
Figure 6.2: 18× 4 tiles configuration.
Considering Fig. 6.2, we can define the sets of L-Ps and G-Ps. Further, we impose on every tile 30× 16nodes, equidistant in each direction. Now we can define the component bubble space and compute theeigenfunctions on the straight segments and the corresponding eigenmodes, see Fig. 6.3.
6.2 Applying SCRBE method on a bearing with eccentricity
6.2.1 OFFLINE
In the OFFLINE stage we have to build the RB approximation spaces: BhM(i);0 and Bh
k,L-P,M(i);0. We willuse a test set Ξ for µ to get some “snapshots” for our application. We choose 56 combinations of h0, hx withh ∈ [10µm, 50µm] with hx 6= 0. This means we get 56 test parameters for the boundary bubble-functions,see (3.15). For u1 we take 7 parameters with u1 ∈ [12m/s, 13m/s]. Through this we get |Ξ| = 392 testparameters for the interior bubble-functions, see (3.14).
For the Greedy algorithm, see [2], we have to set a relative error bound ∆Enrel , which influences the number
of basis functions for each space Bh. Then we are able to represent the µ-dependent bubble-functions asa finite matrix-vector-product. For bhf,i(µi) ∈ Bh
M(i);0 we get bhf,i(µi) = Zf,i · Y hf,i(µi), where Zf,i is the
matrix with basis functions and Y hf,i(µi) is the vector with µ-dependent coefficients. The bubble-solution
vector Y hf,i(µi) will be computed in the ONLINE stage. Likewise we represent bhk,L-P,i(µi) ∈ Bh
k,L-P,M(i);0.After computing the Reduced Basis for all L-Ps and all modes k, we can compute all relevant inner productsand store them. We give some examples of inner products that can be computed beforehand. We look on
6.2 Applying SCRBE method on a bearing with eccentricity 35
Figure 6.3: Top: We see the eigenfunction with numbers 1, 2, 28. In the title is the cor-responding eigenvalue, which is rising with the number of eigenfunctions. Bottom: we seethe elliptical lifted eigenmodes, corresponding to the eigenfunctions.
the left-hand side of (4.5):
nΓ∑G-P=1
NΓG-P∑
k=1
Uk,G-P(µ) a(Φk,G-P(µ), v;µ) =
f(v;µ)−I∑i=1
a(bhf,i(µi), v;µ), ∀ v ∈ XhS(µ).
We see that we have the test and trial space XhS(µ), which is µ-dependent. But with the RB approxima-
tion we can extract the µ-dependence of the bilinear form:
1. We use the affine parameter dependence of a(u, v;µ) and focus on one aq(u, v).
2. Now we take Φk,G-P(µ), Φk′,G-P′(µ) and look at the local representation for one component i and get:ψk,L-P,M(i) + bhk,L-P,i(µi) and ψk′,L-P′,M(i) + bhk′,L-P′,i(µi), with the vector-representation:ψk,L-P,M(i)
+ Zhk,L-P,i · Y hk,L-P,i(µi) and ψ
k′,L-P′,M(i)+ Zhk′,L-P′,i · Yk′,L-P′,i(µi).
3. We insert these representations into aq(u, v) and get
aq(ψk,L-P,M(i)
+ Zhk,L-P,i · Y hk,L-P,i(µi) , ψk′,L-P′,M(i)
+ Zhk′,L-P′,i · Yk′,L-P′,i(µi))
= aq(ψk,L-P,M(i)
, ψk′,L-P′,M(i)
) + aq(ψk,L-P,M(i)
, Zhk′,L-P′,i · Yk′,L-P′,i(µi))
+ aq(Zhk,L-P,i · Y hk,L-P,i(µi) , ψk′,L-P′,M(i)
) + aq(Zhk,L-P,i · Y hk,L-P,i(µi) , Z
hk′,L-P′,i · Yk′,L-P′,i(µi)),
36 6 APPLICATION OF SCRBE METHOD ON RLE
then we use the linearity of aq(u, v):
= aq(ψk,L-P,M(i)
, ψk′,L-P′,M(i)
) + aq(ψk,L-P,M(i)
, Zhk′,L-P′,i) · Yk′,L-P′,i(µi)
+ (Y hk,L-P,i(µi))
T · aq(Zhk,L-P,i , ψk′,L-P′,M(i)) + (Y h
k,L-P,i(µi))T · aq(Zhk,L-P,i , Z
hk′,L-P′,i) · Yk′,L-P′,i(µi).
4. Finally, we get four inner products which we can store. This is done for every combination of i, L-P,k, which needs lot of storage but will significantly help to decrease the computational times.
This evaluation of inner products is done for the right-hand side of (4.5) too.
6.2.2 ONLINE
We have done the OFFLINE stage with ∆Enrel =1E-2 and focus on the ONLINE stage:
Through the choice of 18 × 4 tiles we have a piecewise linear height for the domain in x-direction, seeFig. 6.4. We set u1 = 4πm/s, µV = 0.01Pa · s and compute the matrix A(µ) ∈ R2574×2574 and the vector
Figure 6.4: Gap height approximated with 18 linear functions; periodic.
F(µ) ∈ R2574. The assembling of the matrix A(µ) takes the bulk of time to get the system-solution U(µ),which we can use to construct the FEM-like SCRBE-solution uh(µ) by using the representation (3.19).
Figure 6.5: SCRBE-solution for 523×61 nodes, equidistant in both directions, 18×4 tiles.
We computed the truth solution with FEM in its simplest version: hat functions on triangles. In Fig. 6.6we display the pointwise difference of these two solutions. We give some comments to Fig. 6.6: in the title
6.2 Applying SCRBE method on a bearing with eccentricity 37
Figure 6.6: Difference between the FEM- and SCRBE-solution.
we see already the discrete pointwise `∞-error: 3.2372E-5. The h1-relative error is around 9.8612E-6, andit is defined as follows:
‖v‖2h1= dx · dz ·
∑node∈Ωh
∑j∈x,z
D+j v(node) + ‖v‖2`2 with
dx, dz are the node distances for the equidistant mesh,
D+j v(node) :=
v(node+ dj)− v(node)
dj, if node+ dj ∈ Ωh.
The error plot itself is not point-symmetric what is due to our RB approximations: the initial basisvector for Bh is computed with a µ ∈ Ξ with a positive slope. The cosine function has a positive slope inthe second half, which is represented better by the RB-spaces due to our initial choice of the RB. So weobserve a smaller error in the second half of the unfolded bearing.
In Table 6.1 we computed all terms including the ˆ -terms, which enter the SC-error bound (5.22) fromSection 5.4. Moreover we computed the original σi terms which enter the old error bound.
Table 6.1: σi- and σi-values and the minimal eigenvalue of A(µ).
In this application σ1(µ) ≈ σ1(µ) which is due to the fact that the errors in the corner modes are dominatingthe term in this application. Further, we compute the SC-error bound as described in Section 5: (5.11) and(5.22) in the old and the improved version:
‖U(µ)− U(µ)‖2 ≤ ∆U? (µ) ≡ 181.1, (∆U
? (µ) ≡ 286.8),
‖U(µ)− U(µ)‖2‖U(µ)‖2
≤ ∆U? (µ)
‖U(µ)‖2≡ 1.35E-5, (
∆U? (µ)
‖U(µ)‖2≡ 2.1E-5).
In Tab. 6.2 we see the computational times for the whole OFFLINE stage, the ONLINE stage without com-puting the SC-error bounds, and one FEM computation. All computations were done on an Intel Core2DuoT5250 processor. In this application we already see a speed-up factor of 20 while the relative errors are verysmall and the OFFLINE stage took only twice as much time as one FE-solving.
38 6 APPLICATION OF SCRBE METHOD ON RLE
SCRBE-OFFLINE SCRBE-ONLINE FEM (without mesh generation)715 14 355
Table 6.2: CPU times for SCRBE method and FEM in seconds.
6.2.3 Resulting force
In Fig. 6.7 we see the positive part of the SCRBE-solution. As mentioned in the introduction, we like to
Figure 6.7: We see the positive part of the SCRBE-solution from Fig. 6.5.
improve the integral over the positive pressure, with the resulting force is F =∫
Ω∗ p dx. We computed theforce for this application and got:
∫Ω∗ p dx = 1915.6N , which we like to increase by milling small regular
dents.
6.3 A plain bearing with eccentricity and dents
6.3.1 OFFLINE
We use the same bearing configuration and the same setting of tiles and nodes per tile as in Section 6.2.Now we introduce dents in the center of each tile. The discretization of the dent is 10 × 6 nodes. See forthe schematic visualization of the dents in the domain Fig. 6.8. Further, we show the mesh of the reference
(0, 0)
(0.1885, 0.02)
Figure 6.8: We have 18× 4 tiles and 30× 16 nodes per tile. In the center we impose a dent which contains10× 6 nodes.
subdomain with its dent in Fig. 6.9.The dent size is fixed by the size of the bearing and the mesh to ≈ 3.24mm× 1.6mm. We choose the dentdepth hdent in the interval [5µm , 15µm].
6.3 A plain bearing with eccentricity and dents 39
(0,0)
(0.0105,0.005)
Figure 6.9: FEM-mesh of the reference subdomain with 30 × 16 nodes and a dent with10× 6 nodes in red.
We like to describe the realization of the affine parameter dependence of a(p, ψ;µ) and f(ψ;µ) on Ωref.For this we introduce Ωdent as the domain where the dent lies and we get:
a(p, ψ;µ) =
Qa=8∑q=1
θqa(µ)aq(p, ψ)
with e.g.:
θ1a(µ) = h3
x, a1(p, ψ) =
∫Ωref\Ωdent
x3
12µV∇p · ∇ψdx,
θ4a(µ) = h0, a4(p, ψ) =
∫Ωref\Ωdent
1
12µV∇p · ∇ψdx,
θ5a(µ) = h3
x, a5(p, ψ) =
∫Ωdent
x3
12µV∇p · ∇ψdx,
θ8a(µ) = h0 + hdent, a8(p, ψ) =
∫Ωdent
1
12µV∇p · ∇ψdx
and the right-hand side
f(v;µ) =
Qf=4∑q=1
θqf (µ)f q(v),
θ1f = 0.5 · h0u1, f1(ψ) =
∫Ωref\Ωdent
1 · ∂xψdx,
θ2f = 0.5 · hxu1, f2(ψ) =
∫Ωref\Ωdent
x · ∂xψdx,
θ3f = 0.5 · (h0 + hdent)u1, f3(v) =
∫Ωdent
1 · ∂xψdx,
θ4f = 0.5 · hxu1, f4(ψ) =
∫Ωdent
x · ∂xψdx.
We again choose 56 combinations of h0, hx with h ∈ [10µm, 50µm]. We choose 4 values in [5µm, 15µm]for the dent depth hdent. This means we get 224 test parameters for the boundary bubble-functions. For u1
40 6 APPLICATION OF SCRBE METHOD ON RLE
we take again 7 parameters with u1 ∈ [12m/s, 13m/s]. Through this we get |Ξ| = 1624 test parameters forthe interior bubble-functions.
6.3.2 ONLINE
Again, we computed the OFFLINE stage with ∆Enrel =1E-2. ONLINE stage: we set u1 = 4πm/s, µV =
0.01Pa·s and the parameter for dent depth to 10µm. In Fig. 6.10 we display the computed SCRBE-solutionuh(µ), where we see the influence of dents on the pressure: little hills.
Figure 6.10: SCRBE-solution for dent depth 10µm and for 523 × 61 nodes, equidistantdistributed in both directions ( 18× 4 tiles ).
Further, we computed the truth solution uh(µ) with FEM. In Fig. 6.3 we display the pointwise difference ofthese solutions and see that the relative `∞-error is 1.635E-4 and the relative h1-error is 6.224E-5. We again
Figure 6.11: We see the difference between the FEM- and SCRBE-solution with dent depth10µm.
computed all terms which enter the SC-error bound (5.22), see Tab. 6.3.
6.3 A plain bearing with eccentricity and dents 41
Table 6.3: σi- and σi-values and the minimal eigenvalue of A(µ).
Then we computed the SC-error bound which is relatively small
‖U(µ)− U(µ)‖2 ≤ ∆U? (µ) ≡ 8.28E+3, (∆U
? (µ) ≡ 1.34E+4),
‖U(µ)− U(µ)‖2‖U(µ)‖2
≤ ∆U? (µ)
‖U(µ)‖2≡ 6.74E-4, (
∆U? (µ)
‖U(µ)‖2≡ 1.1E-3).
In Tab. 6.4 we see the computational times where
SCRBE-OFFLINE SCRBE-ONLINE FEM (without mesh generation)6100 24 400
Table 6.4: CPU-times for SCRBE and FEM in seconds.
the OFFLINE-time increased because of the larger test set. The ONLINE-time increased especially becauseof the affine parameter dependence of the variational equation with Qa = 8 and Qf = 4. But we still havea speed-up factor around 17, while having small relative errors.
6.3.3 Resulting force
For the case of hdent = 10µm we compute the integral over the positive pressure, the force:∫
Ω∗ p dx =1772.6N . This is significantly smaller than for the application without dents: F = 1915.6N . In Fig. 6.12we display the pressure along the middle of the first and second rows of dents in x-direction.
Figure 6.12: We see in red the pressure along the first line of dents; in blue the secondline of dents with hdent = 10µm. In comparison, we see in black the corresponding lineswithout dents.
The conclusion from the figure, at least for the lines we considered, is that the pressure peaks can be higher
42 6 APPLICATION OF SCRBE METHOD ON RLE
in the dent-case. But we also can easily see that the integral over the positive pressure is less than in thedent-less case.
We computed with SCRBE method the integral over the positive pressures for several dent depths andget the forces:
hdent : 0µm 5µm 7.5µm 10µm 12.5µm 15µm
force in N: 1915.6 1818.7 1792.2 1772.6 1757.2 1744.3
Table 6.5: Resulting forces for several dent depths.
Moreover, we tried to find examples with different bearing settings where the force would increase byimposing dents. We changed the bearing properties slightly such that the minimal distance between thebearing parts is 2µm and the maximal distance is 10µm. We computed the solution of the RLE with andwithout dents and still saw that the resulting force is decreasing with larger dent depths. Since the RLEis only a simplification of the Navier-Stokes equation it might be that dents have a positive effect on thepressure but we could not prove it with the simple approach we used: consider only the positive part of p.
6.4 Changing the size of the dents on the plain surface
6.4.1 Affine mappings
We still keep the geometry of the bearing, the number of tiles and the number of nodes per tile. But welike to change the length and the height of the dents. We stretch the mesh inside every subdomain withoutchanging any angles in the mesh, see Fig. 6.13.
(0,0)
(0.0105,0.005)
Figure 6.13: FEM-mesh on one reference subdomain with 30 × 16 nodes. We raise thedistance between the parallel green lines to enlarge the size of the dent.
With the changing of size we have to subdivide the reference subdomain into 9 subdomains, see Fig.6.14. We enumerate the subdomains with I to IX and use affine mappings to describe the changing of thesubdomain size. The subdomains I-IX are changing size while keeping all straight lines parallel to eachother which means that the stretching of those domains can be done separately in x- and z-direction. LetµS = (µS1 , µ
S2 ) be the parameter for stretching the subdomain V in x- and z-direction, what implies a
stretching of all other domains I-IV, VI-IX too. We focus on the domain I and derive all necessary terms:ΩI ≡ [0, xI] × [0, zI], we rename the interval into [x1, x2] × [z1, z2]. Now we like to change the size of
this domain with affine mappings to [x1, x2] × [z1, z2]. We exemplary show for the x-direction how to useaffine mappings: An affine mapping maps x 7→ ax+ b = x and we have the two equations:
x1 = ax1 + b
x2 = ax2 + b⇔(x1 1x2 1
)(ab
)=
(x1
x2
)⇒(ab
)=
(x2−x1x2−x1
x1x2−x2x1x2−x1
).
6.4 Changing the size of the dents on the plain surface 43
(0, 0)
(0.0105, 0.005)
I II III
IV V VI
VII VIII IX
Figure 6.14: Subdivision of the reference domain into subdomains with varying size.
We define a =: ax and b =: bx since this is the x-part of the affine mapping. Let
T : ΩI,ref → ΩI(µ) : x :=
(xz
)7→(bxbz
)+
(ax 00 az
)(xz
)=: C +G ·
(xz
)=
(xz
)=: x
be the mapping from ΩI,ref to ΩI(µ). Let u0(x;µ) be a function on ΩI(µ). We like to have a relation betweenthe integral over this function and the integral over its correspondent u(x) on ΩI,ref. We use the classicaltransformation theorem with J(µ) = det(DT (µ)). Given u0 integrable over ΩI(µ) we know∫
ΩI(µ)u0(x;µ) dΩI(µ) = J(µ)
∫ΩI,ref
u(x) dΩI,ref, (6.1)
with u0(x) = u0(T (x)) = u(x). Now we have to focus on the bilinear and linear form and since there aresome derivatives involved we have to consider what happens to the affine mapping. We define the inverse ofT :
T :
(xz
)7→ C +G ·
(xz
)=
(xz
)⇐⇒ T−1 :
(xz
)7→ D +H ·
(xz
), with
H := G−1 and D := −G−1 · C.
Then we derive some rules for derivatives, where i, j ∈ 1, 2 denotes the directions:
∂u0(x)
∂xiusing chain rule
=2∑j=1
∂xj∂xi
∂u(x)
∂xj, with
∂xj∂xi
=∂(Dj +Hj,·x
∂xi= Hj,· ·
∂x∂xi
= Hj,i,
=⇒ ∂u0(x)
∂xi=
2∑j=1
Hj,i∂u(x)
∂xj
In our examples G, and also H, are diagonal matrices, which will simplify the evaluations of the derivatives.The integrals in the parametrized weak formulation of RLE (2.5) are of the type:
I1 :=
∫ΩI(µ)
u0(x) ·∂ψk,0(x)
∂xi∂ψl,0(x)
∂xidΩI(µ), (6.2)
I2 :=
∫ΩI(µ)
u0(x) · ∂ψ0(x)
∂xidΩI(µ), (6.3)
I3 :=
∫ΩI(µ)
u0(x) · ψ0(x) dΩI(µ). (6.4)
44 6 APPLICATION OF SCRBE METHOD ON RLE
We like to derive the correlation between µ-dependent integral and reference integral for (6.3) first:
I2 =
∫ΩI,ref
u(C +G · x) ·Hi,i∂ψ(C +G · x)
∂xiJ(µ) dΩI,ref. (6.5)
Now we use the fact that u0 and ψ0 are polynomials up to order one:
u0(x) :=
c1 +m1x1 orc2 +m2x2
=⇒ u(x) :=
c1 +m1 · (C1 +G1,1x1) orc2 +m2 · (C2 +G2,2x2)
and
ψ0(x) :=
d1 + n1x1 ord2 + n2x2
=⇒ ψ(x) :=
d1 + n1 · (C1 +G1,1x1) ord2 + n2 · (C2 +G2,2x2).
Lets say u0(x) = c1 +m1x1 and ψ0(x) = d1 + n1x1 so we get for i = 1:
(6.5) =
∫ΩI,ref
(c1 +m1 · (C1 +G1,1x1)
)·H1,1
∂(d1 + n1 · (C1 +G1,1x1)
)∂x1
J(µ) dΩI,ref
=
∫ΩI,ref
(c1 +m1C1 +m1G1,1x1
)·H1,1 · n1G1,1J(µ) dΩI,ref. (6.6)
So we can extract of (6.6) all coefficients and get the integrals∫
ΩI,ref1dΩI,ref and
∫ΩI,ref
x1dΩI,ref. Then, wegive the correlations for (6.2). We set i = 1 and get for I1:
I1 =
∫ΩI(µ)
u0(x) ·∂ψk,0(x)
∂x1·∂ψ`,0(x)
∂x1dΩI(µ).
We assume that u0 is a polynomial with order up to 3: u0(x) =∑3
j=0 pj xj1 so
I1 =
∫ΩI,ref
3∑j=0
pj(a1x1 + b1
)j ·H21,1
∂ψk(a1x1 + b1)
∂x1· ∂ψ`(a1x1 + b1)
∂x1J(µ) dΩI,ref.
We know that ψk and ψ` are linear hat functions:
I1 =
∫ΩI,ref
3∑j=0
pj(a1x1+b1
)jH2
1,1 ·∂(dk,1+nk,1(C1+G1,1x1)
)∂x1
·∂(d`,1+n`,1(C1+G1,1x1)
)∂x1
J(µ) dΩI,ref
=
∫ΩI,ref
3∑j=0
pj(a1x1 + b1
)j ·H21,1 · nk,1G1,1 · n`,1G1,1J(µ) dΩI,ref.
The remaining term (6.4) is transformed like in the transformation theorem (6.1).
We return to the application: Through the sub-division of reference domains the affine parameter de-pendence of a(u, v;µ), f(v;µ) changes too. We get Qa = 72: instead of 2 domains as in the fixed dentexample we now get 9 domains. On the right-hand side we have the same construction with 9 domainsinstead of 2, so Qf = 18.
6.4 Changing the size of the dents on the plain surface 45
6.4.2 Applying SCRBE method: OFFLINE and ONLINE
We continue to describe the application. For the OFFLINE stage we formulate the test set Ξ: we choose6 combinations of h0, hx with h ∈ [10µm, 50µm] and we choose 3 values in [5µm , 15µm] for the dentdepth hdent. We will enlarge the dent size in both directions with the factors [0.9, 1, 1.1]. For u1 we takeonly 4 parameters with u1 ∈ [12m/s, 13m/s]. Through this we get already |Ξ| = 1296 test parameters forthe bubble-functions. We have done the OFFLINE stage with ∆En
rel =1E-2. The computational time was 46CPU-hours, with the storage space of 4.3 GB.
ONLINE: We set u1 = 4πm/s and µV = 0.01Pa · s. The parameter for dent depth is: 10µm. Westretch the dent size in both direction with the factor 1.09 and get the SCRBE-solution uh(µ):
In Fig. 6.16 we display the pointwise difference between FEM- and SCRBE-solution: The relative `∞-error
Figure 6.16: We see the difference between the FEM- and SCRBE-solution.
is 0.1459 and the relative h1-error is 0.0561. Here we see that the relative errors are much worse than infurther examples. Through imposing dents and varying their size it is more difficult for the RB to describethe behavior of the solution and the result is that the approximate solution is less accurate. We focus on the
46 6 APPLICATION OF SCRBE METHOD ON RLE
SC-error bound (5.22) and compute all terms which enter it, see Tab. 6.6:
Table 6.6: σi- and σi-values and the minimal eigenvalue of A(µ).
Further, we compute the SC-error bound:
‖U(µ)− U(µ)‖2 ≤ ∆U? (µ) ≡ 3.713E+5, (∆U
? (µ) ≡ 6.751E+5),
‖U(µ)− U(µ)‖2‖U(µ)‖2
≤ ∆U? (µ)
‖U(µ)‖2≡ 3.08E-2, (
∆U? (µ)
‖U(µ)‖2≡ 5.5595E-2).
In Tab. 6.7 we show the computational times:
SCRBE-OFFLINE SCRBE-ONLINE FEM (without mesh generation)163160 170 400
Table 6.7: CPU-times for SCRBE and FEM in seconds.
We see that the computational times between ONLINE and FEM are almost comparable, which shows thatthe SCRBE method has its drawbacks and is not a method which should be used for every application. Onereason for the large ONLINE time is that the data which is loaded into the RAM storage is very large, theother reason is the large number of summands in the affine decomposition Qa = 72.
6.4.3 Resulting force
Although the computational times are not small, we computed the forces which result by integrating thepositive pressures with the SCRBE method. In Fig. 6.17 we display the forces which result from varying thedent size by factors 0.9, 0.95, 1.0, 1.05, 1.1 in each direction. The figure suggests that the force is growingby lowering the area of the dent. Unfortunately the maximum force is F = 1803.5 where the dent size issmallest and the minimum force is F = 1738.6, where the dent has the largest size.
6.4 Changing the size of the dents on the plain surface 47
Figure 6.17: We see the force, computed by integrating over the positive pressure. The x-and z-axis gives the different stretching factors of the dent. The dent depth is set to 1E-5m.
48 7 SCRBE WITH “PORT REDUCTION”
7 SCRBE with “Port Reduction”
7.1 Quasi-rigorous a posteriori error bounds
The size of the Schur Complement System in our application is related to the number of modes, we haveon each global port. To get eigenmodes for ports we computed beforehand eigenfunctions with eigenvalues,see Fig. 6.3. Especially for elliptic problems it is often the case that eigenfunctions with large eigenvalueshave very small influence on the problem in the interior of the domain. This leads to the idea of cutting ofthe eigenmodes (eigenfunctions) with larger eigenvalues. The hope is that already a few eigenmodes willproperly display the behavior of the solution.
We use one result of [3], which allows us to compute the quasi-rigorous SC-error bound without com-puting the whole matrix A(µ) of the problem.We start with equation (5.22):
provided that the condition λmin(µ)− σ2(µ) > 0 holds.Since we will not compute the whole system A(µ)U(µ) = F(µ), we will face two problems:
1. In general F(µ)− A(µ)U(µ) is not zero, since we do not compute all parts of it and so the residuumof this term has to enter the error estimation.
2. We cannot compute λmin so we will use an inequality to get a lower bound for the minimal eigenvalue:λmin,LB.
We are starting with the first problem: nsc is the size of the original system A(µ)U(µ) = F(µ) whichinvolves all eigenmodes on each port. We like to reduce nsc by using only nA ∈ N active modes andneglecting nI ∈ N inactive modes, with nsc = nA + nI, and so we get a reordered SCRBE-system:(
AAA(µ) AAI(µ)
AIA(µ) AII(µ)
)(UA(µ)
UI(µ)
)=
(FA(µ)
FI(µ)
).
Now we define U′(µ) ≡ [UA(µ),
nI zeros︷ ︸︸ ︷0, · · · , 0] ∈ Rnsc to approximate U(µ), where UA(µ) satisfies
for which we have to compute additionally AIA(µ) and the vector FI(µ). We face the second problem, howto compute λmin,LB. For this we like to denote the µ-dependent eigenpairs of A(µ):
λi, vi, 1 ≤ i ≤ nsc
with A(µ)vi = λivi. The same we do for AAA(µ):
λ′i, v
′i, 1 ≤ i ≤ nA
with
AAA(µ)v′i = λ′iv′i.
We assume that all these eigenvectors are orthogonal and normalized to 1 and we take the smallest eigenvalueλ′min of AAA(µ) with its associated eigenvector v′min. We define vmin =
[v′min, 0, · · · , 0
]T and it is shown in[3] that we can use the eigenvectors vmin, v
′min to get a lower bound for λmin:
λmin ≥ λmin,LB ≡ λ′min − ‖AIA(µ)v′min‖2. (7.2)
7.2 SCRBE method with Port Reduction 49
For using the quasi-rigorous a posteriori error bound of (5.22) we have to compute AAA(µ),AIA(µ) and F(µ),which are used to compute the residuum (7.1) and the eigenvector of AAA(µ) with minimal eigenvalue.
Further, we like to show, according to the applications in Section 6, how Port Reduction influencescomputation time and error bounds. Later we will use the same bearing geometry and introduce anothersubdivision of the system domain, see Section 7.3.
7.2 SCRBE method with Port Reduction
Now we will go through the examples of Section 6 and decrease the number of ports and focus on theSC-error bound, relative `∞- and h1-error.
7.2.1 Plain bearing with variation of velocity
We apply the SCRBE method with Port Reduction to Section 6.2. On one tile we had 28 ports in x-directionand 14 ports in z-direction. Now we reduce the number and take 25 ports in x-direction, 25(28) and 12ports in z-direction 12(14). The dimension of U reduces from 2574 to 2268. In Tab. 7.1 we computed allrelevant terms for the quasi-rigorous bound, including the reordered solution of UA, which we denoted asU′(µ).
Table 7.1: Terms for error bound with Port Reduction.
Further, we can compute the quasi-rigorous error bound:
‖U(µ)− U′(µ)‖2 ≤ ∆U? (µ) ≤ 1.4362E+6.
So we have a relative quasi-rigorous bound of ‖U(µ)−U′(µ)‖2‖U′(µ)‖2
≤ ∆U? (µ)
‖U′(µ)‖2≤ 0.107 and although it is around
10% the relative errors in the `∞- and h1-norm are much smaller:
‖uh − u′‖`∞/‖uh‖`∞ = 3.2371E-5,
‖uh − u′‖h1/‖uh‖h1 = 1.7126E-5,
where u′ denotes the approximate SCRBE-PR-solution. These small errors show, that the port reduction isbetter than the quasi-rigorous bound would suggest. We use Tab. 6.2 and add the CPU-time for SCRBEwith Port Reduction for the setting 25(28), 12(14).
Table 7.2: CPU-times for SCRBE method (with PR) and FEM in seconds.
In Tab. 7.3 we display how the quasi-rigorous bound (q-r-b), the relative `∞-error, the relative h1-error andthe CPU-time develop while the number of ports is steadily reduced. Some comments on Tab. 7.3: First ofall we see that by decreasing the number of active ports the relative `∞- and h1-errors are still small whilethe CPU-time decreases. The achieve a speed-up factor around 90 for the last line, while errors are small.The reason for such a good result is that the problem we focus, without dents, is very elliptic and smooth
Table 7.3: SCRBE method with PR: We see the number of active ports, dimension of the reduced system,quasi-rigorous bound (q-r-b), relative quasi-rigorous bound, `∞-, h1-errors and the CPU-time in seconds.
such that the RB approximation with ∆Enrel =1E-2 is suitable to represent the solution. The drawback is that
the quasi-rigorous bounds are strongly rising and this is mostly due to the residuum R(µ) which is muchlarger than the terms σ1, σ3. We see in the last line of Fig. 7.3 that the quasi-rigorous bound is not usefulanymore since λmin,LB < σ2 and in this case we can not compute the quasi-rigorous bound at all with themethods from Section 6 and 7.1.
7.2.2 Plain bearing with variation of dent depth
Here we apply the SCRBE method with PR to Section 6.3. We repeat the steps from above: reduce thenumber of ports and take 25 in x-direction and 12 ports in z-direction. We compute terms for the quasi-rigorous bound in Tab. (7.4).
Table 7.4: Terms for error bound with Port Reduction.
Further, we can compute the quasi-rigorous bound:
‖U(µ)− U′(µ)‖2 ≤ ∆U? (µ) ≤ 1.1764E+6.
So we have a relative quasi-rigorous bound ‖U(µ)−U′(µ)‖2‖U′(µ)‖2
≤ ∆U? (µ)
‖U′(µ)‖2≤ 0.0957, which is almost the same
as in the last application. Then we compute the relative error bounds in the `∞- and h1-norm:
‖uh − u′‖`∞/‖uh‖`∞ = 1.6347E-4,
‖uh − u′‖h1/‖uh‖h1 = 6.3877E-5.
Again, the errors are very small, although the quasi-rigorous bound is much larger. We use Table 6.4 andadd the CPU-time for SCRBE method with PR for the setting 25(28), 12(14).
Table 7.5: CPU-times for SCRBE method (with PR) and FEM in seconds.
Now we like to reduce the number of ports steadily and see how the quasi-rigorous bound, the relative`∞-, h1-errors and the CPU-time develop, see Table 7.6.
Some comments on Tab. 7.6: analogously, to the last application the relative `∞- and h1-errors are stillsmall while the CPU-time decreases. But by using to few ports, we cannot compute any quasi-rigorousbound.
7.3 SCRBE with Port Reduction applied to the plain bearing with a different setting 51
Table 7.6: SCRBE method with PR: We see the number of active ports, dimension of the reduced system,quasi-rigorous bound, relative quasi-rigorous bound, `∞-, h1-errors and the CPU-time in seconds.
7.2.3 Plain bearing with variation of dent depth and dent size
Here we apply the SCRBE method with PR to Section 6.4. We saw already that for this example the SCRBEmethod did not compute acceptable results, but still we like to apply PR: We repeat the steps from above:reduce the number of active ports and take 25 ports in x-direction and 12 ports in z-direction. We computedthe relevant terms for the quasi-rigorous bound in Tab. 7.7 so we can evaluate the quasi-rigorous bound
Table 7.7: Terms for error bound with Port Reduction.
‖U(µ)− U′(µ)‖2 ≤ ∆U? (µ) ≤ 8.0674E+6.
So we have a relative quasi-rigorous bound ‖U(µ)−U′(µ)‖2‖U′(µ)‖2
≤ ∆U? (µ)
‖U′(µ)‖2≤ 0.6674 and we compute the relative
error bounds in the `∞- and h1-norm:
‖uh − u′‖`∞/‖uh‖`∞ = 0.1447,
‖uh − u′‖h1/‖uh‖h1 = 0.054.
In this example the `∞- and h1-errors are great and so we will not decrease the port number further. Wenote the CPU-time for SCRBE with Port Reduction (PR) for the setting 25(28), 12(14) in Tab. 7.8.
Table 7.8: CPU times for SCRBE and FEM in seconds.
The decreasing ONLINE time is rather pointless when focusing on the great relative errors.
7.3 SCRBE with Port Reduction applied to the plain bearing with a different setting
Here we like to switch to another setting of domain subdivision to improve computational times even more.The main idea is to decrease the number of ports already in the system configuration. We choose the sizeof subdomains such that the ratio between the degrees of freedom in the interior and on the boundary ofa subdomain is large. This implies a smaller Schur Complement System, since the skeleton space has lessentries. We still use the geometry of the bearing introduced above in Section 6.3 which means we use thesame global mesh and the ordering and the size of the dents. In the last subsections we used the subdivisionof the domain into 18 × 4 tiles, and parametrized the problem on one tile with linear heights h0 + hxx.
52 7 SCRBE WITH “PORT REDUCTION”
(0, 0)
(0.1885, 0.02)
Figure 7.1: We have 1 × 4 tiles and 523 × 16 nodes per tile, schematic. In every tile the dent position andsize is the same as in Section 6.3.
Now we like to subdivide the domain into 1× 4 tiles and parametrize the height with the cosine function inx-direction, see Fig. 7.1.
We introduce four different parameters, which enter the affine parameter dependence:
1. h0 is the offset of the cosine function which in our application is set to 30µm.
2. A ∈ [5µm, 25µm] is the amplitude of the cosine function, such that the gap height is described byh(x, z) = h0 +A · cos( x
0.1885).
3. hdent ∈ [5µm, 15µm] is the depth of the dents shown in Fig. 7.1.
4. u1 the velocity of the inner bearing.
We like to comment on the realization of the affine parameter dependence of a(p, ψ;µ) and f(ψ;µ) on Ωrefand for this we introduce Ωdent as the domain where the dent lies:
a(p, ψ;µ) =
Qa=8∑q=1
θqa(µ)aq(p, ψ)
with
θ1a(µ) = A3, a1(p, ψ) =
∫Ωref\Ωdent
cos( x0.1885)3
12µV∇p · ∇ψdx,
θ4a(µ) = h0, a4(p, ψ) =
∫Ωref\Ωdent
1
12µV∇p · ∇ψdx,
θ5a(µ) = A3, a5(p, ψ) =
∫Ωdent
cos( x0.1885)3
12µV∇p · ∇ψdx,
θ8a(µ) = h0 + hdent, a8(p, ψ) =
∫Ωdent
1
12µV∇p · ∇ψdx
7.3 SCRBE with Port Reduction applied to the plain bearing with a different setting 53
and the right-hand side
f(v;µ) =
Qf=4∑q=1
θqf (µ)f q(v),
θ1f = 0.5 · h0u1, f1(ψ) =
∫Ωref\Ωdent
1 · ∂xψdx,
θ2f = 0.5 ·Au1, f2(ψ) =
∫Ωref\Ωdent
cos(x
0.1885) · ∂xψdx,
θ3f = 0.5 · (h0 + hdent)u1, f4(v) =
∫Ωdent
1 · ∂xψdx,
θ4f = 0.5 ·Au1, f4(ψ) =
∫Ωdent
cos(x
0.1885) · ∂xψdx.
In this application we used an Intel Core2Quad Q8400 CPU. In the OFFLINE stage we choose 4 param-eters for A ∈ [5µm, 25µm] and 4 parameters for hdent ∈ [5µm, 15µm]. This means we get only 16 test pa-rameters for the boundary bubble functions. For u1 we take again 6 parameters with u1 ∈ [12m/s, 13m/s].Through this we get |Ξ| = 96 test parameters for the interior bubble functions.
Now we do Port Reduction already in the OFFLINE-stage: instead of taking 521 modes in x-directionwe take only 54 and only 7 of 14 in the z-direction. This will imply a shorter computational time for theOFFLINE-stage. We computed the OFFLINE stage with ∆En
rel =1E-1 which already took 238 CPU-hours.We focus on the the ONLINE stage. The original Schur Complement System size would have been
1608 which we now decreased by the OFFLINE-PR to 186. We choose the parameters: h0 = 30µm, A =20µm, hdent = 10µm, u1 = 4πm/s, µV = 0.01Pa · s. We compute the SCRBE-PR-solution u′(µ) with54(521) and 7(14) modes (which means we take all OFFLINE-modes), see Fig. 7.2.
Figure 7.2: SCRBE-PR-solution for 523× 61 nodes, 1× 4 tiles with 54× 7 modes per tile.
We computed the truth solution with FEM by using the cosine function for the height. In Fig. 7.3 we displaythe pointwise difference of the FEM- and the SCRBE-PR-solution: The relative `∞-error is 0.0237 and therelative h1-error is 0.0093, what is still small. In Tab. 7.9 we summarize the computational times:
54 7 SCRBE WITH “PORT REDUCTION”
Figure 7.3: We see the difference between the FEM- and SCRBE-solution.
OFFLINE ONLINE 54(521), 7(14) FEM (without mesh generation)858350 5.79 400
Table 7.9: CPU-times for SCRBE method and FEM in seconds.
We like to emphasize that we are not able to compute any quasi-rigorous bounds since we did notcompute the RB approximations for all modes, but only for 54(521) and 7(14). Now we can decrease theport number even more and get faster results but the larger errors:
Taking 27(521), 4(14) ports: the relative `∞-error is 0.152 and the relative h1-error is 0.0585 with theCPU-time of 4.72. By considering the first setting with 54(521), 7(14) ports we achieved a very goodcomputational speed-up. This is mostly due to the fact that we used SCRBE method with Port Reduction.Further, Qa = 8, Qf = 4 and the number of tiles are very small, which is also important for the speed-up.
55
8 Conclusions
In this thesis we saw a detailed introduction to SC method and SCRBE method (mainly taken from the paper[1]). Further, we introduced sharper bounds for the SC-error bound in the case that a global port can have upto two neighboring components. After understanding this setting we extended for ΩSYS ⊂ R2 the kinds ofsubdivision of the domain: we can subdivide a rectangular domain into arbitrary smaller rectangles, see Fig.6.2. For this setting we extended the SC-error bound and applied it to several examples in Section 6 wherewe saw some fast computational ONLINE times. In the best case we got an speed-up from 355 sec. to 14sec. with relative errors: `∞ = 0.003%, h1 = 0.001%, see p. 38. On the other hand we parametriced thedent size and depth which led toQa = 72 summands for the bilinear form in the affine decomposition. Thesemany summands unmade the advantages of the SCRBE method and the computational time was 170 sec.for SCRBE method (FEM took 400 sec.) with unacceptable relative errors in `∞-, h1-norms. Concerningthe optimazation of pressure we observed that the resulting force in a plain bearing is decreasing with deeperand/or larger dents, which is probably due to the fact that the RLE is only a simplification of the Navier-Stokes equation.
In the preceding Section 7 we introduced the Port-Reduction idea, see [3], and we used it to speed-upthe computational times again. The main idea is to decrease the number of eigenmodes on global portssuch that the incomplete Schur Complement System can be faster computed. We applied this SCRBE-PRmethod to the examples introduced in Section 6 and saw that for those examples where SCRBE methodworked properly we could even improve the speed-up factor. Moreover, we saw that SCRBE-PR can giveacceptable solutions on the FEM mesh although the quasi-rigorous bound was very large and obviously notsharp enough.
In Section 7.2.1 we saw for the dent-less problem a speed-up from 355 sec. to 4 sec. with relativeerrors: `∞ = 0.06%, h1 = 0.02%. But on the other hand we experienced that for those examples whereSCRBE method is already failing to produce acceptable solutions SCRBE-PR method is doing even worse.Further, we introduced in Section 7.3 another subdivision of the domain ΩSYS for the same plain bearingconfiguration. The intention was to choose subdomains which have much more degrees of freedom in theinterior than on the boundary. We saved much time in the OFFLINE stage by considering only a ninth of alleigenmodes but doing so we lost information which was necessary to compute the quasi-rigorous bounds.Although we used so few modes we achieved in the ONLINE stage a speed-up from 400 sec. to 6 sec.,while having relative errors of `∞ = 2%, h1 = 1%, see p. 54. In this example we imposed dents withvarying depth which made it more difficult for the RB to display the solution. So in contrary to Section7.2.1 the speed-up factor was slightly worse, but for a harder problem. Nevertheless, the speed-up factor inSection 7.3 was still very good and this shows that SCRBE method with Port Reduction can improve thecomputational ONLINE times significantly with accaptable errors in `∞, h1.
56 REFERENCES
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[3] J.L. Eftang, D.B.P. Huynh, D.J. Knezevic, E.M. Rønquist, and A.T. Patera, Adaptive Port Reductionin Static Condensation. Proceedings of 7th Vienna Conference on Mathematical Modelling - MATH-MOD 2012.
[4] O. Pinkus, B. Sternlicht, Theory of hydrodynamic lubrication, published in 1961 by McGraw-Hill inNew York.
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[6] K.G. Murty, Note on a Bard-type Scheme for Solving the Complementary Problem. Opsearch, 11,1974.
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