On the Concept of Algorithms by Design with Illustrations toComputational Structural Mechanics/Dynamics
X. Zhou∗ and K. K. Tamma†
Abstract
A novel procedure to design time operators under the notion of algorithms by design is formulated withemphasis on application to the broad area of computational mechanics but with focus on structural mechan-ics/dynamics. The algorithms by design concept capitalizes on: (i) the recently developed unified theoryunderlying computational algorithms by the authors [Zhou and Tamma, 2004], and (ii) newly establishedalgorithmic measures [Zhou and Tamma, 2005]. The objective is to foster the design of computational algo-rithms for time dependent problems with improved algorithmic attributes in the sense of accuracy, stabilityand other characteristics including algorithmic complexity. The design process behind the design of com-putational algorithms is explained in the sense of the algorithms by design concepts via selected numericalillustrations of practical scenarios.
1. Introduction
Algorithms by design is a seminal concept to designtime operators for effectively transient analyzing ap-plications. Every engineering application has its ownemphasis and analysis requirements. Design of op-timal algorithms for engineering applications is nottrivial. Alternately, how to determine whether an al-gorithm design is optimal for a particular engineer-ing application and how to foster the design of anoptimal algorithm for a particular engineering ap-plication, if such an optimal does not exist is a de-sirable goal and a challenging task. We term themeans to help achieve the aforementioned objectivesas the notion of algorithms by design. The notion ofalgorithms by design consists of three important per-spectives: (i) a unified theory recently developed [1]describing the underlying principle of the evolution,classification, and design of time operators; (ii) thedesign spaces and algorithmic measures [2] for eval-uation and comparison and for qualifying the opti-mality of an algorithm with respect to a particularengineering application; and (iii) an educated designprocedure utilizing (i) and (ii) above for designingtransient algorithms that meet the particular quali-fications.
In this exposition we focus on the process of de-signing time operators that meet the desired proper-ties rather than the design of particular algorithms.The traditional or classical approach for designingalgorithms is based on the following procedure:Procedure 1 (Traditional/Classical Approach)
1. Come up with an idea (the idea could be a phys-ical interpretation, a mathematical construct,or a deficiency of an already existing method).
2. Formulate the algorithm design.3. A-posteriori study the resulting algorithmic
properties.
4. If the algorithmic properties are good (in con-trast to existing state-of-the-art), propose thealgorithm design.
The traditional or classical algorithm design pro-cedure possesses the following deficiencies: (i) Thenew algorithm development has strong correlationto the previous work and therein the resulting de-velopment most likely inherits the limitations of theprevious work; (ii) Until the complete algorithm de-sign procedure is finished, there are no guaranteesthat the resulting development will possess improvedalgorithmic properties in contrast to the previouswork; and (iii) This algorithm design procedure mayserve to an extent to improve the algorithmic proper-ties of an existing algorithm, but it does not possessthe flexibility to design an algorithm that is mostsuitable to the problem at hand. This so-called clas-sical algorithm design procedure is the initial pro-cedure we also briefly embarked upon when we firststarted our earlier efforts for designing new compu-tational algorithms. Alternatively, from our past ex-periences in developing a unified theory underlyingcomputational algorithms for time dependent prob-lems and more recent efforts on design spaces andmeasures for evaluating time operator [1], we havenow shifted the burden to instead foster the notionof algorithms by design to circumvent the deficienciesand to knowledgeably lead to more meaningful andwell educated guidelines.
Procedure 2 (Algorithms By Design)1. A-priori design the algorithmic measures with
features what inherit the desired attributes(based on a wish list) corresponding to the ap-plication at hand.
2. Utilizing the underlying unified theory [1] iden-tify the limitation and the feasibility of the pos-
∗Research Associate, AHPCRC, [email protected]†Professor/Technical Director, to receive correspondence, Department of Mechanical Engineering/AHPCRC, University of
sible approach and establish a general algorith-mic design framework.
3. Impose the wish list to design the algorithmfrom a knowledge of the design spaces and al-gorithmic measures [2] and seek also to reducecomputational effort if possible through estab-lishing a mapping relation of the design spaces.
4. Propose the algorithm by design which meetsthe desired wish list.
The algorithms by design procedure possesses thefollowing advantages in contrast to the traditionalpractices: (i) It allows one to design the algorithmsuitable for the application at hand, (ii) No trial anderror design iterations are necessary, and (iii) Thedesired algorithmic properties are naturally embed-ded in the design procedure itself. However, the al-gorithms by design procedure can not be formulatedsolely based on past traditional practices and avail-able theories existing in the literature; it requires twokey ingredients: one is the established design spacesand algorithmic measures for evaluation and compar-ison of algorithms [2], and the other is an underlyingtheory of computational algorithms that govern thealgorithmic structure, relationship, and limitationsof the computational algorithms [1] which we havepreviously developed.
2. An Underlying Unified Theory of Com-
putational Algorithms for Transient Anal-
ysisFollowing our previous efforts, and the unified the-ory underlying computational algorithms developedfor time discretized operators, these can be catego-rized into Type 1, Type 2, and Type 3 classifica-tion of algorithms. These classifications pertain todistinct design spaces for computational algorithms.The algorithmic classifications are defined throughthe relation between the designed algorithm to theexact solution in the sense of linear transient situa-tions in association with the corresponding algorith-mic structures. The total number of degrees of free-dom of the semi-discretized equations of motion istermed as one system size. To find the solution fora set of equations in the sense of Ax = b, where A
is a matrix and x, b are vectors, with one set of un-known variables, x, this is termed as one system solveor one solution step. In the sense of the linear first-order ordinary differential equation systems in a sim-ply connected region, the exact solution of the sys-tem of equations can be represented as the variation-of-constants-formula which is in terms of the fun-damental solution and an integral operator associ-ated with the inhomogeneous term. In the variation-of-constants-formula, introducing approximation tothe integral operator leads to the Type 1 classifica-tion of algorithms. All algorithmic properties of thevariation-of-constants-formula are preserved in theType 1 classification of algorithms through the fun-damental solution. The only difference between theType 1 classification of algorithms and the variation-of-constants-formula is that the Type 1 classification
of algorithms possess certain order of convergencerate which is dependent on the approximation of theintegral operator. From the algorithmic structureperspective, the design space of the Type 1 classifica-tion of algorithms are naturally in the single-system-zero-solve (SSZS) structure involving the evaluationof matrix exponentials or eigen-problems. Since theinterested linear first-order differential equation sys-tems are elementary, further introducing approxima-tion to the fundamental solution of the Type 1 clas-sification of algorithms by a matrix-valued real ana-lytic function leads to the design of the Type 2 clas-sification of algorithms. Furthermore, if the approxi-mated matrix-valued real analytic function is a ratio-nal function, the corresponding Type 2 classificationof algorithms are termed as the Type 2k(p,q) clas-sification of algorithms, where k=maxp,q, p andq are the maximum powers of the matrix associatedwith the known variables and the unknown variables,respectively. The design space of the Type 2 classi-fication of algorithms also has the algorithmic struc-ture of a single system size with a maximum of onesystem solve. In general, the Type 2 classificationof algorithms contain second or higher matrix powerterms. However, a Type 2 algorithm can be castinto the spectrally equivalent Type 3 classification ofalgorithms with the algorithmic structure of multi-ple system sizes and/or multiple solution steps. Theconcept of spectral equivalence is defined when anytwo given algorithms have identical principal roots.A Type 2k(p,q) algorithm can also be casted intospectrally equivalent Type 3k
q classification of algo-rithms of multiple system sizes and/or multiple solu-tion step algorithmic structure with matrices of max-imum power of only one, where k is the number ofsets of unknowns and q is the maximum number ofeither the number of equation systems or the numberof solves. Any Type 3k
q classification of algorithmscan be mapped into a corresponding spectrally equiv-alent Type 2k(p,q) algorithm. While formulating theunified theory for computational algorithms, fromthe observation of all the computational algorithmknown to the authors, we propose a conjecture thata necessary condition for a computational algorithmto converge is that the mode super-position methodmust be applicable to the underlying algorithm forthe linear case.
The formulation of the unified theory for com-putational algorithms is based on one simple idea,namely, that all computational algorithms for timedependent analysis emanate from the exact solutionof the corresponding ordinary differential equation,and the resulting computational algorithms whichare approximations introduced in the design and de-velopment process can be categorized according totheir algorithmic structure. The algorithmic struc-ture of the computational algorithms are closely re-lated to the computational effort according to theoperational count, or equivalently, the algorithmiccomplexity. All the computational algorithms seek
2American Institute of Aeronautics and Astronautics
to converge to the exact solution. It is computa-tionally intensive to employ the exact integral op-erator (EIO) because an exponential matrix is as-sociated with the fundamental solution, and the re-sulting computations are cumbersome. The Type 1classification of algorithms reduce the computationaleffort of the integral operator, however, they retainthe computational effort of the evolution of the fun-damental solution. On the other hand, the Type 2classification of algorithms further attempt to reducethe computational burden of the evolution of the fun-damental solution that is associated with the Type1 classification of algorithms; however, the Type 2classification of algorithms involve the computationof matrix multiplications. Further restraining thetype of function to approximate the fundamental so-lution associated with the Type 1 classification ofalgorithms yields the Type 2k(p,q) classification ofalgorithms. Since the Type 2 or the Type 2k(p,q)classification of algorithms are closely related to theType 1 classification of algorithms, the algorithmicproperties of the Type 2 or the Type 2k(p,q) clas-sification of algorithms can be easily proven and/orcontrolled. Consequently, the Type 3 or the Type3k
q classification of algorithms are the results of cast-ing of the Type 2 or the Type 2k(p,q) classificationof algorithms in terms of the algorithmic structurewhile preserving the algorithmic properties as thatof the corresponding Type 2 or the correspondingType 2k(p,q) classification of algorithms to furtherreduce the computational effort. Therefore, for anygiven Type 2 or Type 2k(p,q) classification of algo-rithms can be cast into the algorithmic structure ofa Type 3 or Type 3k
q classifications, and there ex-ists a corresponding spectrally equivalent relation-ship between the Type 3 or Type 3k
q classification al-gorithms to that of the corresponding Type 2 or Type2k(p,q) algorithm. In other words, the Type 3 or theType 3k
q classification of algorithms are related to theType 1 classification of algorithms, and are furtherrelated to the variation-of-constants-formula throughthe corresponding spectrally equivalent Type 2 orType 2k(p,q) classification of algorithms. Therefore,the algorithmic properties of the Type 3 or the Type3k
q classification of algorithms are controlled by thecorresponding spectrally equivalent Type 2 or Type2k(p,q) classification of algorithms.
According to the relationship amongst the Type1, Type 2, and Type 3 classification of algorithms,it is logical to design an algorithm in the Type 2design space algorithmic structure, but implementthe algorithm in the Type 3 design space algorith-mic structure. To design an algorithm in the Type 2design space algorithmic structure, the stability andthe limitation of the algorithms of the Type 2 de-sign space algorithmic structure need to be criticallyunderstood. Focusing attention on the Type 2k(p,q)algorithmic structure of algorithms, the approxima-tion functions to the fundamental solution are ra-tional functions. The Pade approximation functionis a particular case of the rational function which
approximates the fundamental solution, and the en-tries of the Pade approximation serve as particularcases of the Type 2k(p,q) approximation. In addi-tion to the accuracy property of the Pade approxi-mation function, the Ehle theorem governs the sta-bility property of the Pade approximation function.However, for the general case of the Type 2k(p,q)approximation function, we previously proposed ageneralized stability and accuracy limitation barriertheorem to govern the maximum order of conver-gence and the stability that is associated with thegeneral rational function approximation of the Type2k(p,q) algorithmic structure. The generalized sta-bility and accuracy limitation theorem states that:For the polynomial rational form of the Type 2k(p,q)classification of algorithms, a necessary condition fora Type 2k(p,q) algorithm to be unconditionally stableis p ≤ q = k; and there exists only one 2k-order ac-curate local A-stable algorithm in the polynomial ra-tional form Type 2k(p,q) classification of algorithms,provided k = q = p. For the non-polynomial ratio-nal form of the Type 2k(p,q) algorithms, there existsa maximum of 2k-order accurate local A-stable algo-rithms. As such, the Dahlquist barrier theorem [3]is a particular case of the generalized stability andaccuracy limitation theorem for the case of k ≤ 1.Therefore, the generalized stability and accuracy lim-itation theorem directly governs and serves as thebarrier of the stability and accuracy for the Type2k(p,q) classification of algorithms. Indirectly, itserves as the barrier of the stability and accuracy forthe Type 3k
q classification of algorithms through thespectral equivalent relationship between the Type3k
q classification of algorithms to the correspondingType 2k(p,q) classification of algorithms. The gener-alized stability and accuracy limitation theorem pro-vides a general guideline for identifying the possi-ble structure of the overall design of an algorithmwith given desirable order of convergence and stabil-ity properties.
3. Algorithmic Measures
With the knowledge of the algorithmic classification,relationship, and the accuracy and stability limita-tion of the algorithmic structure, one more piece ofinformation is additionally needed for fostering thenotion of algorithms by design. This piece of informa-tion is the algorithmic measures for computationalalgorithms as described in Reference [2]. The ob-jective for establishing the algorithmic measures liesin two aspects: (i) for meaningful comparison andevaluation of computational algorithms; and (ii) foridentification and description of the desirable com-putational algorithm. We have categorized the al-gorithmic properties associated with computationalalgorithms into two type of algorithmic measures:the primary measures and the secondary measures.The primary measures are the generic algorithmicmeasures associated with any computational algo-rithm. And, the secondary measures are the algo-rithmic measures associated with computational al-
3American Institute of Aeronautics and Astronautics
gorithms for a particular type of application. Theprimary measures of a computational algorithm con-sist of convergence and algorithmic complexity. Theconvergence of a computational algorithm is impliedby the consistency (the order of convergence) and thestability according to the Lax equivalence theorem.The algorithmic complexity of a computational al-gorithm is a concept that is borrowed from the com-puter science literature and is in the form of O(nm),where the ”big O” notation means the order, n isthe size of the input for the algorithm, and nm isthe operation count of the algorithm for each timestep. The algorithmic complexity is formulated asthe combined total effect of the type of algorithmicformulation and its algorithmic structure [2]. The al-gorithmic formulation here refers to the notion of anexplicit formulation, implicit formulation, or the so-called nonlinearly explicit formulation of a computa-tional algorithm. Traditionally, an explicit algorithmis that algorithm which does not need a solver andonly requires matrix-vector operations and no non-linear iterations are needed for general nonlinear dy-namic situations. An implicit algorithm is that algo-rithm which requires the solution of a system of equa-tions and nonlinear iterations are needed in transientnonlinear situations. In our recent advances underly-ing the theoretical development of computational al-gorithms [4], a new design formulation was presentedwhich does not strictly belong to either the implicitformulation or the explicit type of formulation in thetraditional sense; and it contains both the featuresof the implicit and the explicit formulation. Thisdesign formulation and the corresponding transientalgorithm is such that it requires the solution of asystem of equations at each time step but no nonlin-ear iterations are needed in transient nonlinear situ-ations. We have termed this as a nonlinearly explicitformulation. In linear dynamic situations, the non-linearly explicit formulation can be regarded as animplicit formulation. In general, the primary mea-sures of computational algorithms are the measuresfor the performance of how accurate and how fasta computational algorithm performs. The secondarymeasures of computational algorithms are the algo-rithmic properties that closely relate to the type ofapplication. For example, the secondary measuresof computational algorithms for structural dynamicsare the numerical dissipation, the numerical disper-sion, and the overshooting behavior. Therefore, theprimary measures of computational algorithms arethe measures for the algorithm performance qual-ity, and the secondary measures of computationalalgorithms are the measures for the more refined as-pects of the solution quality for the particular typeof application. The secondary measures of compu-tational algorithms are meaningful for comparison ifand only only if the comparable algorithms possessidentical primary measures. And, for computationalalgorithms to be comparable, the computational al-gorithm should at least possess the same order of theconvergence rate.
4. Algorithms By Design Procedure
With the underlying unified theory of computationalalgorithms together with the design spaces and thealgorithmic measures developed in [1,2], as describedin the previous sections, we are now in a positionto describe the concept(s) of the algorithms by de-sign procedure. In engineering applications of thetime dependent numerical simulations, to a certainextent, the current developments of computationalalgorithms no doubt enable the analyst to find thetransient response. However, a formal and well ed-ucated approach to design algorithms with desirableattributes is still a deficiency that would be desir-able to overcome, and indeed, there does not exist acomputational algorithm that is optimal for generalapplications. For the particular engineering applica-tion at hand, it would be best served if a computa-tional algorithm is also designed for optimality forthe specific type of application. The algorithms bydesign procedure is intended to fill the void to pro-vide the feasibility to design optimal computationalalgorithms for practical situations.
To design a computational algorithm, one shouldfirst identify and define what properties the targetedcomputational algorithm should possess, namely, awish-list of desired attributes. The wish-list shouldbe identified clearly in terms of the primary and thesecondary algorithmic measures. The primary algo-rithmic measures are the generic algorithmic prop-erties regardless of the underlying physics which in-clude convergence and algorithmic complexity. Thesecondary algorithmic measures are the specific prop-erties that are closely related to the underlyingphysics. Although there exists a great deal of flexi-bility for coming up with a wish-list for the targetedcomputational algorithm, the feasibility of utilizingor selecting combinations of the primary algorith-mic measures is not arbitrary. These combinationsof convergence and algorithmic complexity are gov-erned by the general stability and consistency lim-itation theorem [1]. Within the limits of the gen-eral stability and consistency limitation theorem, onecould have full control for selecting combinations ofconvergence and the algorithmic complexity for theprimary algorithmic measures. The flexibility of thesecondary measures depends on the selected combi-nations of the primary measures. For some combi-nations of the primary measures, one could have fullcontrol of the secondary measures; and for some com-binations of the primary measures, one could have nocontrol of the secondary measures.
Once the the wish-list is identified, then one onlyneeds to utilize the unified theory underlying compu-tational algorithms to decide the design space (the al-gorithmic type) for the targeted algorithm, and sub-sequently formulate a generalized algorithmic struc-ture of the resulting algorithm type with generalizedparameters. This is a crucial step in the algorithmsby design procedure. According to the wish-list,what algorithm type will likely contain the algorithm
4American Institute of Aeronautics and Astronautics
that meets the wish-list can be determined throughthe generalized stability and consistency limitationtheorem. After the design space or the algorithmtype is specified, then the general formulation of thealgorithm for the corresponding algorithmic type canbe subsequently formulated. In the general formula-tion, generalized parameters should be placed on allthe possible places.
We now have a wish-list for the algorithmic prop-erties and the general formulation of the algorithmwith generalized parameters. The next step is toemploy the wish-list to identify the specific designparameters in the general formulation of algorithm.The wish-list in terms of the primary and the sec-ondary algorithmic measures are first quantified bynumerous conditions that are related to the designparameters of the general formulation of the under-lying algorithm. The conditions are then imposed onthe general formulation of the algorithm to identifyall the design parameters. After all the design pa-rameters are identified, we then have the algorithmby design that indeed meets the wish-list.
For the case when the algorithm type of the tar-geted algorithm is in the design space of the Type 2classification, the targeted algorithm should be fur-ther cast into the design space of the Type 3 classifi-cation. According to the unified theory of computa-tional algorithms, the designed algorithm should ap-pear in either the Type 1 classification or the Type3 classification. Algorithms of the Type 2 classifi-cation of algorithms are in general computationallymore expensive compared to the corresponding spec-trally equivalent Type 3 classification of algorithms.However, there is no unique approach for casting theType 2 classification of algorithms into the corre-sponding Type 3 classification of algorithms.
5. Illustration of the Concept of Algo-
rithms by Design: The Design of the
Stress Updated Formulation for Hypo-
elasticityThe stress update formulation is an important in-gredient for computational finite deformation analy-sis [5]. The stress update algorithm needs to main-tain the objectivity, stability, and possess certain or-der of accuracy. In the updated Lagrangian finiteelement formulation for the finite deformation prob-lems, a stress update algorithm is needed to evaluatethe stress and strain relation. However, in the senseof the co-rotational stress rate hypoelasticity, onlythe logarithmic stress rate hypoelasticity is exactlyintegrable. A stress update formulation has been de-veloped for the corotational stress rate hypoelastic-ity [5]. We next explain our design processes for thedevelopments in the sense of the algorithms by designconcepts.5.1 The Design of Stress Update Formula-
tionThe generalized corotational hypoelasticity constitu-tive equations are expressed in the form
σ∗ = σ −Ωσ + σΩ = C : D (1)
where σ is the Cauchy stress, C is the elasticity ten-sor, and C = λI ⊗ I + 2µI for grade zero hypoelas-ticity, I is the second-order identity tensor, I is thefourth-order identity tensor, λ and µ are the Lamematerial coefficients.
The general expression for the spin tensor Ω isgiven by [6]as Ω = W + Υ(B, D), where W is thespin tensor of the velocity gradient tensor, Υ(B, D)is a skew-symmetric tensor-valued isotropic func-tion, and there exist infinite selections for the spintensor. The choice of the spin tensor Ω in equa-tion (1) results in different objective stress rates,and hence leads to different hypoelasticity models.The choice of the objective stress rate could bethe Zaremba-Jaumann-Noll stress rate, the Green-McInnis-Naghdi stress rate, the Logarithmic stressrate, the twirl tensor of Eulerian triad stress rate,and the twirl tensor of Lagrangian triad stress rate.
Design Wish ListFor the stress update formulation, the objectivityand the order of accuracy are of primary concern.The stress update formulation is performed at theGauss point of each element. The system size of thestress update formulation for each Gauss point con-sists of six equations with six unknowns. To achievethe maximum robustness for finite deformation anal-ysis, we want the error introduced by stress evalua-tion to be a minimum. Therefore, we wish the stressupdate formulation to not only preserve the prop-erties of the constitutive equations exactly, but alsothe error should be a minimum.
Identify the ApproachSince the system size of the problem size for eachGauss point is small, the Type 1 classification of al-gorithm would be the most logical and will preservethe objectivity of the constitutive equation under anycircumstance.
The Algorithm DesignUnder the assumption that within each time stepthe rotation is proportional, consider the ordinarydifferential tensor equation (1), and employ the pro-jection operator obtained by the Sylvester’s formulato project the equation (1) onto the invariant space.Thus, we can derive the closed form fundamentalsolution for equation (1). Further employing themid-pint rule for the particular solution, an objec-tive second-order accurate stress update formulationyields as the following.
For the logarithmic stress rate hypoelasticity we de-rive the close form particular solution. Finally, weobtain a stress update formulation for the logarith-mic stress rate hypoelasticity as the following.
Algorithm 2σ(t) = T[Ωlog; σ(tα); tα; t] + C : ∆E
LE
Where Ωlog is the logarithmic stress rate spin tensor,E
LE is the incremental logarithmic Eulerian strainmeasure.
5American Institute of Aeronautics and Astronautics
In the above expressions, T[Ω; σ; tα; t] = σ +1
sin (t− tα)[Ωσ−σΩ]− 12 sin2 (t− tα)ΩσΩ+
12 [1− cos (t− tα)][Ω2σ −σΩ2] + 1
23 [2 sin (t−tα) − sin 2(t − tα)][ΩσΩ2 − Ω2σΩ] + 1
24 [3 +cos 2(t − tα) − 4 cos (t − tα)]Ω2σΩ2.
5.2 Numerical Illustration: Simple Share
Problem in Solid MechanicsConsider the simple shear problem shown in Figure1(a). The body is undergoing the following motion,x(t) = X + γY , y(t) = Y , z(t) = Z and with theassumption that motion belongs to the single param-eter family. Figure 1 shows the numerical results ofthe Algorithm 1 and the Algorithm 2 for the log-arithmic stress rate grade zero hypoelasticity. Thenumerical results of Algorithm 1 are consistentlyof second-order accuracy. The numerical results ofAlgorithm 2 are exact. In both cases the objectiv-ity is always preserved.
6. Illustration of the Concept of Algo-
rithms by Design: The Design of an Ef-
ficient and Robust Computational Algo-
rithm for Nonlinear Structural Dynamics
For highly nonlinear structural dynamic analysis, theexplicit central difference method is the primary timeintegration algorithm that is widely used. How-ever, the explicit nature of the central differencemethod has a limiting time step size due to the sta-bility of the algorithm. On the other hand the im-plicit time integration algorithms require nonlinearNewton-Raphson iterations during each time stepwhich is time consuming. To design an efficient timeoperator which can finish the simulation accuratelyand faster than the existing implicit and explicit timeintegration algorithm is our objective.
6.1 The Design of an Efficient Computa-
tional Algorithm for Nonlinear Structural
Dynamic Analysis in the Sense of the Al-
gorithms by Design ConceptConsider the semi-discretized system of equationsof linear structural dynamic problems by space dis-cretization of the single field form.
Design Wish ListAfter a review of the current existing time integra-tion algorithms available in the literature, we con-clude that the computational bottle neck for the ex-plicit time integration algorithms is the time step sizelimitation and the computational bottle neck for theimplicit time integration algorithm is the nonlineariterations. Therefore, within the scope of the uni-fied theory for computational algorithms, a feasibledesign wish list in terms of the primary measuresfor designing efficient time integration algorithms fornonlinear structural dynamic analysis can be listedas the following: 1.) Unconditionally stability; 2.)Second-order accurate; 3.) No nonlinear iterationneeded; and 4.) No more than two system size or nomore than two system solves.
Identify the ApproachAccording to the Dahlquist barrier theorem or thestability and accuracy limitation theorem, an algo-rithm with the aforementioned design wish list willnot exist in the framework of LMS methods. How-ever, according to the unified theory of computa-tional algorithms, such an algorithm is possible inthe Type 2k(p,q) classification. Therefore, we inves-tigate the design of the algorithm in the Type 2k(p,q)classification and then cast the resulting algorithminto a Type 3k
q classification to further reduce thecomputational effort.
The Design of the AlgorithmConsider the semi-discretized nonlinear dynamic sys-tem of equations as Md + Cd + R(σ) = f , wherex(t), d, d are the nodal displacement vector, nodalvelocity vector, and nodal acceleration vector, re-spectively; M , C are the symmetric and positivedefinite mass matrix and damping matrix, respec-tively; and R, f (t) are the resultant internal forcesand the external force, respectively. The original ELalgorithm design described in [4] can be rewritten inthe following forward displacement form for compu-tational convenience (however the algorithm designstructure is not computationally attractive):
Algorithm 3At each time step, known dtn , dtn+∆t, d
tn
and dtn
and to find dtn+2∆t, dtn+∆t
and dtn+∆t
:
Phase 1: Compute the acceleration dtn+∆t
by:
(M + ∆tC/2)dtn+∆t
= ftn+∆t
−Rtn+∆t(σtn+∆t) − C(d
tn
+ ∆tdtn
/2)(2)
Phase 2: Solve for dtn+2∆t and dtn+∆t
:
Qtn+∆t(dtn+∆t
− ∆tdtn+∆t
/2) =
M (dtn
+ ∆tdtn
/2) (3)
Qtn+∆tdtn+2∆t =
M (dtn+∆t + ∆tdtn+∆t
+ ∆t2dtn+∆t
/2) (4)
where Qtn+∆t = M + (2ξ∆t)4
4!K tn+∆tM−1Ktn+∆t,
and ξ > 0.583838795422.
Although the complexity of Algorithm 3 is ofO(n3), it can be further readily casted into the fol-lowing spectrally equivalent forward displacementsecond-order accurate nonlinearly explicit L-stable(FDEL) structure to reduce the computational ef-fort.
Algorithm 4At each time step, known dtn , dtn+∆t, d
tn
and dtn
and to find dtn+2∆t, dtn+∆t
and dtn+∆t
:
Phase 1: Compute the acceleration dtn+∆t
by:
(M + ∆tC/2)dtn+∆t
= ftn+∆t
−Rtn+∆t(σtn+∆t) − C(d
tn
+ ∆tdtn
/2)(5)
6American Institute of Aeronautics and Astronautics
Phase 2: Solve for dtn+2∆t and dtn+∆t
:
(M + i2(ξ∆t)4Ktn+∆t/3)(dtn+∆t
−∆tdtn+∆t
/2) = M (dtn
+ ∆tdtn
/2) (6)
(M + i2(ξ∆t)4K tn+∆t/3)dtn+2∆t =
M (dtn+∆t + ∆tdtn+∆t
+ ∆t2dtn+∆t
/2) (7)
where ξ > 0.583838795422.
The comparison of the primary and secondary mea-sures of the FDCD method, the Newmark method,and the FDEL method are shown in Table 1.
Numerical Illustration: Finite Deforma-
tion Dynamics of a Framed Structure with
External LoadThe numerical example of a framed structure withexternal load is selected. The configuration of theexample is taken from the literature, and is shownin Figure 2. The cross-section of the frame is a10cm × 1cm rectangular, and length of each beamis 50cm. The frame is fixed in one end (the upperend) and the external force is applied on the free end.The frame is meshed with a total 387 nodes, 168 8-noded brick elements. The Young’s modulus of thematerial is 210 GN/m2; density is 7.8 g/cm3; Pois-son ratio is 0.3; yield stress is 200 MN/m2; plasticmodulus is 20 GN/m3; and the external force is 1.5GN.
Two constitutive models are employed for thenumerical example: the Jaumann stress rate gradezero hypoelasticity/hypoelasto-plasticity. The up-dated Lagrangian formulation is employed for thehypoelasticity/hypoelasto-plasticity models. Theforward incremental displacement central difference(FIDCD) method, the Newmark average accelera-tion method, and the forward displacement nonlin-early explicit second-order accurate L-stable (FDEL)method are employed for comparison. The Newton-Raphson and modified Newton-Raphson nonlineariteration method are employed for the Newmark av-erage acceleration method. The displacement com-parisons are shown in Figure 3, and the CPU com-parison are shown in Table 2-3. For large time step,the nonlinear iteration of the Newmark method failto converge for the number of iteration up to 1,000with the tolerance of 10−12. The results shows thatthe FDEL method is more efficient and robust.
7. Illustration of Algorithms by Design:
The Design of LMS Methods
Linear multi-step (LMS) methods are the mostwidely used class of time integration algorithms forintegrating the semi-discretized equation of motionfor structural dynamic problems. As mentioned inthe introduction, the LMS methods have been in-tensively studied in the past fifty years. However,with the development of the unified theory for com-putational algorithms and the established algorith-mic measures, it is indeed possible to design optimalLMS methods as follows.
7.1 Design of the Optimal LMS Method in
the Sense of Algorithms by Design Con-
ceptsConsider the semi-discretized system of equationsof linear structural dynamic problems by space dis-cretization of the single field form as
Mu(t) + Cu(t) + Ku(t) = f(t)
u(0) = u0 , u(0) = u0
(8)
where M is the mass matrix, C is the damping ma-trix, and K is the stiffness matrix.
Design Wish ListBounded by the special case of the stability and ac-curacy limitation theorem [1], namely, the Dahlquistbarrier theorem, and focus our attention to the de-sign of all the possible practical algorithms within theLMS methods, the design wish list is described as thefollowing: 1.) Unconditional stability; 2.) Second-order in time accuracy; 3.) No more than first-orderdisplacement or velocity overshooting behavior; 4.)No more than one set of single-field system of im-plicit equations to be solved at each time step.
Identify the ApproachSince, the scope has been limited to the LMS meth-ods, and LMS methods are of the Type 3k
1 classifi-cation, we only need to first formulate a generalizedformulation for the LMS methods which includes allthe possible cases of the LMS methods. We thenimpose the design wish list to design the desired al-gorithms.
The Design of the AlgorithmWithin the limit of second-order accuracy, the gen-eralize sidle step single solve (GSSSS) algorithmicstructure of the LMS method are expressed as thefollowing.Algorithm 5Given un, un, and un, find un+1, un+1, and un+1
from`
Λ6W1M + Λ5W2C∆t + Λ3W3K∆t2´
∆a
= −Mun − C (un + Λ4W1un∆t)
−K`
un + Λ1W1un∆t + Λ2W2un∆t2´
+ (1 − W1)fn + W1fn+1
(9)
with the associated updates
un+1 = un + λ1un∆t + λ2un∆t2 + λ3∆a∆t2(10)
un+1 = un + λ4un∆t + λ5∆a∆t (11)
un+1 = un + ∆a (12)
As described in [7], there are twelve sets of conditionscan be imposed on Algorithm 5. Depending on theovershooting behavior, we can formulate two sets ofconditions which meet the design wish list to im-pose on Algorithm 5. One set of conditions allowsthe zero-order displacement and zero- or first-ordervelocity overshooting behavior. We term this set ofcondition as U0 conditions, and this set of conditionsare described as the following: 1.) Second-order ac-curate: 5 conditions; 2.) Zero-order displacement
7American Institute of Aeronautics and Astronautics
and no more than first-order velocity overshootingbehavior: 2 conditions; 3.) Allow bifurcation of theprincipal roots: 1 condition: 4.) The placement ofthe spurious root at high-frequency limit: 1 condi-tion; 5.) Unconditional stability: 1 condition; 6.)Allow the prediction functions equal to update func-tions: 1 condition; 7.) Second-order approximationfor the load term: 1 condition.
Imposing the set of twelve U0 conditions into Al-gorithm 5 and following the detailed design pro-cedure described in [7], finally yields the followingalgorithm.
Algorithm 6 (U0 Algorithms)Given un, un, and un, find un+1, un+1, and un+1,the parameters described in Algorithm 5 are se-lected as W1Λ1 = 1
1+ρ3∞, λ1 = 1, W2Λ2 = 1
2(1+ρ3∞),
λ2 = 12, W3Λ3 = 1
(1+ρ1∞)(1+ρ2∞)(1+ρ3∞), λ3 =
1(1+ρ1∞)(1+ρ2∞)
, W1Λ4 = 11+ρ3∞
, λ4 = 1, W2Λ5 =3+ρ1∞+ρ2∞−ρ1∞ρ2∞
2(1+ρ1∞)(1+ρ2∞)(1+ρ3∞), λ5 = 3+ρ1∞+ρ2∞−ρ1∞ρ2∞
2(1+ρ1∞)(1+ρ2∞),
W1Λ6 = 2+ρ1∞+ρ2∞+ρ3∞−ρ1∞ρ2∞ρ3∞
(1+ρ1∞)(1+ρ2∞)(1+ρ3∞), where ρ1∞,
ρ2∞, and ρ3∞ are the first principle root, the sec-ond principle root, and the spurious root at thehigh-frequency limit, respectively, and they satisfy:0 ≤ ρ3∞ ≤ ρ1∞ ≤ ρ2∞ ≤ 1.Another set of conditions allows the zero- or first-order displacement and zero-order velocity over-shooting behavior. We term this set of conditionas V0 conditions, and this set of conditions are de-scribed as the following: 1.) Second-order accurate:5 conditions; 2.) No more than first-order displace-ment and zero-order velocity overshooting behavior:2 conditions; 3.) Allow bifurcation of the principalroots: 1 condition; 4.) The placement of the spuriousroot at high-frequency limit: 1 condition; 5.) Uncon-ditional stability: 1 condition; 6.) Allow the predic-tion functions equal to update functions: 1 condition;7.) Second-order approximation for the load term: 1condition.
Imposing the set of twelve V0 conditions into Al-gorithm 5 and following the detailed design pro-cedure described in [7], finally yields the followingalgorithm.
Algorithm 7 (V0 Algorithms)Given un, un, and un, find un+1, un+1, andun+1, the parameters described in Algorithm 5
are selected as: W1Λ1 = 3+ρ1∞+ρ2∞−ρ1∞ρ2∞
2(1+ρ1∞)(1+ρ2∞),
λ1 = 1, W2Λ2 = 1(1+ρ1∞)(1+ρ2∞)
, λ2 = 12,
W3Λ3 = 1(1+ρ1∞)(1+ρ2∞)(1+ρ3∞)
, λ3 = 12(1+ρ3∞)
,
W1Λ4 = 3+ρ1∞+ρ2∞−ρ1∞ρ2∞
2(1+ρ1∞)(1+ρ2∞), λ4 = 1, W2Λ5 =
2(1+ρ1∞)(1+ρ2∞)(1+ρ3∞)
, λ5 = 11+ρ3∞
, W1Λ6 =2+ρ1∞+ρ2∞+ρ3∞−ρ1∞ρ2∞ρ3∞
(1+ρ1∞)(1+ρ2∞)(1+ρ3∞), where ρ1∞, ρ2∞, and
ρ3∞ are the first principle root, the second princi-ple root, and the spurious root at the high-frequencylimit, respectively, and they satisfy: 0 ≤ ρ3∞ ≤ρ1∞ ≤ ρ2∞ ≤ 1.
Remark 11. For the same values of ρ1∞, ρ2∞, and ρ3∞,
Algorithm 6 and Algorithm 7 are spectrally
identical (the principal invariants of the ampli-fication matrix are identical).
2. When selecting ρ2∞ = 1, Algorithm 6 and Al-
gorithm 7 exhibits no overshooting behavior.
3. The sufficient and necessary condition forAlgorithm 6 and Algorithm 7 to beunconditionally stable, the parameter setW2Λ5 is within the range W2Λ5 ∈ [a
b, c
d],
where a = 2 + ρ1∞ + ρ2∞ + ρ3∞ −ρ1∞ρ2∞ρ3∞, b = 2(1 + ρ1∞)(1 + ρ2∞)(1 +ρ3∞), c = 8 + (7 − 2ρ1∞ρ2∞ρ3∞)(ρ1∞ +ρ2∞ + ρ3∞) + (2 + 3ρ1∞ρ2∞ρ3∞)(ρ1∞ρ2∞ +ρ1∞ρ3∞ + ρ2∞ρ3∞) + ρ2
1∞(1 − ρ2∞ − ρ3∞) +ρ22∞(1 − ρ1∞ − ρ3∞) + ρ2
3∞(1 − ρ1∞ − ρ2∞) +ρ1∞ρ2∞ρ3∞(5ρ1∞ρ2∞ρ3∞ − 8), and d = 2(1 +ρ1∞)2(1 + ρ2∞)2(1 + ρ3∞)2.
7.2 Numerical Illustration: Total Energy
of Modal EquationTo demonstrate the overshooting behavior, con-sider the undamped single-degree-of-freedom homo-geneous modal problem u + ω2u = 0, with initialconditions u|t=0 = 1, and u|t=0 = 1. Recent stud-ies [8] found that a more stringent stability criteriatermed as the energy stability [8] over the classicalspectral stability criteria. The energy stability is de-fined as the total energy of a isolated dynamic systemshould be bounded independent of the mesh size andthe time step size. The total energy shown in Figure4 demonstrated that only when selecting ρ2∞ = 1within Algorithm 6 and Algorithm 7, the result-ing algorithm satisfies the energy stability criteria.
ACKNOWLEDGMENTSThe authors are very pleased to acknowledge sup-port in part by Battelle/U. S. Army Research Of-fice (ARO) Research Triangle Park, North Carolina,under grant number DAAH04-96-C-0086, and bythe Army High Performance Computing ResearchCenter (AHPCRC) under the auspices of the De-partment of the Army, Army Research Laboratory(ARL) under contract number DAAD19-01-2-0014.Dr. Raju Namburu is the technical monitor. Thecontent does not necessarily reflect the position orthe policy of the government, and no official endorse-ment should be inferred. Other related support inform of computer grants from the Minnesota Super-computer Institute (MSI), Minneapolis, Minnesota isalso gratefully acknowledged.
References
[1] X. Zhou and K. K. Tamma. A New Unified The-ory Underlying Time Dependent Linear First-Order Systems: A Prelude to Algorithms by De-sign. International Journal for Numerical Meth-ods in Engineering, 60:1699–1740, 2004.
[2] X. Zhou, K. K. Tamma, and D. Sha. DesignSpaces and Measures for Evaluating Time Dis-cretized Operators and the Consequences Lead-ing to Improved Algorithms By Design for Struc-
8American Institute of Aeronautics and Astronautics
tural Dynamics. International Journal for Nu-merical Methods in Engineering (in press), 2005.
[3] G. Dahlquist. A Special Stability Problem forLinear Multistep Methods. BIT, 3:27, 1963.
[4] X. Zhou, D. Sha, and K. K. Tamma. ANovel Non-linearly Explicit Second-Order Accu-rate L-Stable Methodology for Finite Deforma-tion: Hypoelastic/Hypoelasto-Plastic StructuralDynamics Problems. International Journal forNumerical Methods in Engineering, 59:795–823,2004.
[5] X. Zhou and K. K. Tamma. On the Applicabilityand Stress Update Formulation for CorotationalStress Rate Hypoelasticity Constitutive Models.
Finite Elements in Analysis and Design, 39:783–816, 2003.
[6] H. Xiao, O. T. Bruhns, and A. Meyer. On Objec-tive Corotational Rates and Their Defining SpinTensors. Int. J. Solids Structures, 30:4001–4014,1998.
[7] X. Zhou and K. K. Tamma. Design, Analysis,and Synthesis of Generalized Single Step Sin-gle Solve and Optimal Algorithms for StructuralDynamics. International Journal for NumericalMethods in Engineering, 59:597–668, 2004.
[8] I. Romero. Stability Analysis of Linear MultistepMethods for Classical Elastodynamics. Com-put. Methods Appl. Mech. Engrg., 193:2169–2189,2004.
Table 1: Comparison of primary measures of comparable algorithmsconvergence Complexity
![Page 1: [American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Austin, Texas ()] 46th AIAA/ASME/ASCE/AHS/ASC](https://reader042.documents.pub/reader042/viewer/2022020615/5750952a1a28abbf6bbf729e/html5/page/1.jpg)
![Page 2: [American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Austin, Texas ()] 46th AIAA/ASME/ASCE/AHS/ASC](https://reader042.documents.pub/reader042/viewer/2022020615/5750952a1a28abbf6bbf729e/html5/page/2.jpg)
![Page 3: [American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Austin, Texas ()] 46th AIAA/ASME/ASCE/AHS/ASC](https://reader042.documents.pub/reader042/viewer/2022020615/5750952a1a28abbf6bbf729e/html5/page/3.jpg)
![Page 4: [American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Austin, Texas ()] 46th AIAA/ASME/ASCE/AHS/ASC](https://reader042.documents.pub/reader042/viewer/2022020615/5750952a1a28abbf6bbf729e/html5/page/4.jpg)
![Page 5: [American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Austin, Texas ()] 46th AIAA/ASME/ASCE/AHS/ASC](https://reader042.documents.pub/reader042/viewer/2022020615/5750952a1a28abbf6bbf729e/html5/page/5.jpg)
![Page 6: [American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Austin, Texas ()] 46th AIAA/ASME/ASCE/AHS/ASC](https://reader042.documents.pub/reader042/viewer/2022020615/5750952a1a28abbf6bbf729e/html5/page/6.jpg)
![Page 7: [American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Austin, Texas ()] 46th AIAA/ASME/ASCE/AHS/ASC](https://reader042.documents.pub/reader042/viewer/2022020615/5750952a1a28abbf6bbf729e/html5/page/7.jpg)
![Page 8: [American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Austin, Texas ()] 46th AIAA/ASME/ASCE/AHS/ASC](https://reader042.documents.pub/reader042/viewer/2022020615/5750952a1a28abbf6bbf729e/html5/page/8.jpg)
![Page 9: [American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Austin, Texas ()] 46th AIAA/ASME/ASCE/AHS/ASC](https://reader042.documents.pub/reader042/viewer/2022020615/5750952a1a28abbf6bbf729e/html5/page/9.jpg)
![Page 10: [American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Austin, Texas ()] 46th AIAA/ASME/ASCE/AHS/ASC](https://reader042.documents.pub/reader042/viewer/2022020615/5750952a1a28abbf6bbf729e/html5/page/10.jpg)
![Page 11: [American Institute of Aeronautics and Astronautics 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference - Austin, Texas ()] 46th AIAA/ASME/ASCE/AHS/ASC](https://reader042.documents.pub/reader042/viewer/2022020615/5750952a1a28abbf6bbf729e/html5/page/11.jpg)
