The mortar method with approximate integration

8/11/2019 The mortar method with approximate integration

1/19


2/19

Domain decomposition

domain

split

(disjoint) subdomains

reduce solution of a problem in to solution of set of

smaller problems in the subdomains

preconditioning

parallelism

The mortar method with approximate integration: coupling nite elements on non-matching unstructured grids in 3D p.2/1


3/19

Conforming DomDec

continuity of the discrete solution across the interface

same discretization type in each SD

matching FE decomposition

Non Conforming DomDec

no strong continuity requireddifferent spaces in different subdomains allowed

non matching FE grids allowed



4/19

Mortar Method

weak continuity across interface: jump orthogonal tosuitable multiplier space

to x ideas:

on + Dirichlet B.C.

and

discretizations of

and

resp.

multiplier space trace on slave edge of the

discretization space in the subdomain(function on the slave edge determined from function on master edge via theconstraint)



5/19

Mortar method

+

SLAVE SIDE

MULTIPLIER

MASTER SIDE

constraint:

constrained space

set of discrete functionssatisfying weak continuity constraint (n.b.

)

problem:



6/19

Computation of the jump constraint

implementation requires computing the constraint

efcient computation particularly difcult in 3D

spaces of different type problems!

example: wavelet/FEM coupling:discretization = wavelet

multiplier = suitable dual wavelet space

problems in coupling with nite elements

Problem computing integrals of products of functions liv-ing on non-matching grids



7/19

Coupling non-matching grids

Problem computing inte-gral of the product of

two functions living on un-structured non matchinggrids

standard quadrature is no good!

intersecting triangles very technicalquadrature on master grid : bad for approximation

quadrature on slave grid : bad for consistency



8/19

Approximate Integration

Problem : compute

Idea

introduce auxiliary space

:

orthogonal projection onto :



9/19

( three elds formulation )

must be chosen in such a way that three things happen

the integrals

must be easier to compute than

the projector

must be easy to computethe integrals must be (easily) computablethe mass matrix must be easily invertible

error estimates must be available



10/19

The modied mortar method

new non conforming approximation space:

set ofdiscrete functions satisfying the approximate weakcontinuity constraint

Problem : nd

s.t. for all



11/19

Error Estimate (two subdomains)

Assumptions : direct inequality for

+ usual mortar assumptions

then : (

: broken

norm)

usual mortar error



12/19

Coupling non-matching FE

choice: = Q FE on tensor product grid of mesh-size

error estimate: smooth

usual mortar error

error balancing: choose s.t.

. . .

. . .The mortar method with approximate integration: coupling nite elements on non-matching unstructured grids in 3D p.12/1


13/19

Choosing the auxiliary mesh

P1 FE mortar discretization :

FE mesh-size

the auxiliary grid can be coarse



14/19

Computation of

: Q function on a tensor product uniform grid

: function on the unstructured grid

intersection of triangles in with tensor productrectangles computed directly

each trianglesees at most rectangles

most of the trian-gles fall inside a rectangle m

(m+1)

n (n+1)



15/19

Distribution of triangles

good triangles:Grid Q1 Q2 Q3

5 x 5 20 % 60 % 60 %

10 x 10 20 % 60 % 80 %20 x 20 40 % 70 % 80 %

40 x 40 45 % 80 % 90 %

triangles interacting with an edge or with a X-point# El. Q1 Q2 Q3

50 40 20 20

200 160 80 40800 560 240 160

3200 1760 640 320

# El. Q1 Q2 Q3

50 4 1 1

200 25 4 1800 49 9 4

3200 121 16 4



16/19

Computation of

denition of

:

computing : the same arguments apply as for computing

inverting the mass-matrix

nodal basis for

: tensor product structure

inverting

inverting two 1D banded mass matrices (tridiagonal, in thebest case)



17/19

Coding the Mortar Method

Requirements for a code: extendibility

Extendibility of a Mortar code: aim at a code s.t.

adding new discretizations is easy

minimal changes needed if the new discretization isalready implemented in the monodomain framework

Difculty: need to be familiar with all discretizationsalready implemented



18/19

Adding a new discretization

What does adding a new discretization

require ?

Case A: Mortar code with discretizations

,

, . . . ,

.

in the monodomain framework

integrals of functions in

times the correspondigmultipliersmortar projection


,

, . . . ,

times themultipliers


times the multipliers for

,

, . . . ,



19/19

Case B: Mortar with approx. integration code withdiscretizations

,

, . . . ,

and auxiliary space and .

in the monodomain framework


times the correspondig

multipliersmortar projection

integrals of functions in times the multipliers


times the functions in

The code is more easily extended !!


Date post:	02-Jun-2018
Category:	Documents
Upload:	slirpa
View:	223 times
Download:	0 times

The mortar method with approximate integration

Documents