The financial support via projects No. GA22-20181S and No. GF21-06569K provided by the
Czech Science Foundation is gratefully acknowledged

Frost, M.

Moskovka, A.

Sedlák, P.

Valdman, J.

DSpace at University of West Bohemia

Numerical implementation of incremental minimization principlefor materials with multiple rate-independentdissipative mechanismsM. Frosta, A. Moskovkab,c, P. Sedláka, J. Valdmanb,daInstitute of Thermomechanics, Czech Academy of Sciences, Dolejškova 5, 18200 Praha, Czech RepublicbInstitute of Theory of Information and Automation, Czech Academy of Sciences, Pod Vodárenskou věžı́ 4, 18200 Praha, Czech RepubliccDepartment of Mathematics, Faculty of Applied Sciences, University of West Bohemia, Technická 8, 30100 Plzeň, Czech RepublicdDepartment of Computer Science, Faculty of Science, University of South Bohemia, Branišovská 31, 37005 Č. Budějovice, Czech Republic1. IntroductionNew experimental techniques allow for a more comprehensive examination of the mechanicalresponse of materials, providing the opportunity for the development of more detailed materialmodels. The incremental energy minimization approach is a compact variational formulationof the evolutionary boundary value problem for constitutive models of materials with a rate-independent response, see e.g. [3]. Although it can be easily applied to many conventionalmodels, its main advantages arise when applied to models with multiple strongly coupled dis-sipation mechanisms, where the direct construction of the coupled yield conditions and flowrules may be challenging. However, the approach usually requires a more complex numericaltreatment of the resulting sequence of time-incremental boundary value problems resolved viathe finite element method.2. Incremental minimization principleIn what follows, we focus on a quasi-static evolutionary boundary value problem under isother-mal conditions. Let us introduce a partition of the time interval from time 0 to T in the form0 = t0 ≤ t1 ≤ · · · ≤ tN = T , N ∈ N. In this time-discrete setting, the mechanical response ofthe system at time tn+1, n ∈ {0, . . . , N − 1} can then be determined by solving the incrementalminimization probleminfχLτ (χ,χn) , (1)where χ represents the set of all thermodynamic descriptors of the system. The superscript τdenotes the time-discretized counterparts of time-continuous functionals of the correspondingweak formulation, and the subscript n denotes the values of the previous time step. Let usnote that the minimization can be subject to some additional (kinematic and physically-based)internal constraints, see [3] for details.The (Lagrangian) functional Lτ usually combines the incremental energy functional, Eτ ,the incremental dissipation functional, Dτ , and the incremental external work functional P τ .For many engineering-relevant solid materials, the objective function of the minimization from(1) takes the the following form:Lτ (u,α,αn) := Eτ (u,α,αn) +Dτ (α,αn)− P τ (u,α,αn). (2)52Here, u denotes the displacement and α encompasses thermodynamic descriptors of the mi-crostructure, which can be further constrained to an admissible set Aτ . The most commonexamples include (hyper)elastic materials, where α and Dτ disappear, or perfectly plastic ma-terial with α being the plastic strain andDτ being a one-homogeneous function of the differenceα−αn with an imposed constraint trα = 0.3. Numerical solution strategiesAt least two distinct methods can be utilised to solve the rate-independent problem stated in (2).In the first case, frequently used in finite element software, the minimisation problem is split intoa nested form, where a sequence of “structural” nonlinear minimization problems and “mate-rial” nonlinear minimization problems is resolved. The second approach benefits from the clearvariational structure of (1) and directly applies minimization.To illustrate these approaches, let us consider the following form of incremental functionalsfor a material body:Eτn+1(u,α;un,αn) =∫Ωfn+1(u,α)− fn(un,αn) dV, (3)Dτn+1(α;αn) =∫Ω(tn+1 − tn) δn+1(α;αn; tn+1 − tn) dV, (4)P τn+1(u;un) =∫ΩF voln+1 · (u− un) dV +∫ΓNF surfn+1 · (u− un) dS, (5)where fi and δi stand for the discretized density of the thermodynamical (free energy) potentialand dissipation (pseudo)potential, respectively. Moreover, the set Ω ⊂ R3 is the geometricrepresentation of the physical body in the space with (assumed Lipschitz) boundary ∂Ω, andΓN ⊂ ∂Ω represents its subset where the Neumann boundary condition is applied. The termsF ivol and Fisurf represent the corresponding time discretizations of the prescribed volumetric andsurface forces, respectively. Let us further assume that all necessary information on the systemat t0 = 0 (including u0,α0) is known and well defined. We can disregard the terms in (3)–(5),which are constant with respect to the minimization, and make profit from one-homogeneity ofthe dissipation (pseudo)potential to the eliminate the time increment in (4).The nested form of the boundary problem at a time instant tn+1 then reads asun+1,p+1 = argminu∈Uτn+1{Eτn+1(u,αn+1,p;αn)− P τn+1(u)}, (6)αn+1,p+1 = argminα∈Aτn+1{Eτn+1(un+1,p+1,α;αn) +Dτn+1(α;αn)− P τn+1(un+1,p+1)}. (7)Whereas the first subscript denotes the time incrementation introduced above, the second onedenotes the iteration p + 1 (p ∈ N ∪ 0) within the resolution process at time tn+1, where thethe subproblems (6), (7) are solved consecutively and repeatedly until suitable convergence cri-teria are met, i.e. so-called alternating minimization; αn+1,0 = αn. After a discretization inspace, the first, “structural” subproblem formally corresponds to the principle of minimum po-tential energy. Its solution procedure is analogous to resolving an elastic body problem, whichis a rather standard problem for finite element method. The second, “material” subproblemcorresponds to minimization only with respect to internal variables, i.e. finding the local ther-modynamical balance, and represents the constitutive response of a material point with fixeddisplacement. This is usually tackled via numerical procedures specific to the particular consti-tutive law or with some universal tool of mathematical programming, cf. [1]. A link betweenthe two solution procedures is provided by the material tangent stiffness operator, which must53be derived from the constitutive law as well.In the alternative approach, the problem is tackled via “complete” minimization, i.e., mini-mizing all control variables at once,(un+1,αn+1) = argminu∈Uτn+1,α∈Aτn+1{Eτn+1(un+1,α;αn) +Dτn+1(α;αn)− P τn+1(un+1)}. (8)This monolithic approach may be especially advantageous for material models with multiple,strongly coupled rate-independent dissipative processes. In such a situation, the evolution of in-dividual thermodynamic descriptor cannot be addressed independently, benefiting from conven-tional active set search strategies. Solely a single “global yield function” driving the evolutioncan be derived, which may render the conventional numerical treatment elaborate and depen-dent on the model’s particular mathematical structure. In contrast, resolving directly (8) allowsto avoid transferring the results from one subproblem to the other, involving the construction ofthe material Jacobian matrix (material tangent modulus) and its incorporation into the structuralJacobian matrix. On the other hand, it requires to cope with with a non-smooth minimizationproblem with non-linear constraints, which provides a complex computational challenge.4. ConclusionRecently, a vectorized MATLAB tool for minimization of nonlinear (mathematically well-behaved) functionals discretized by the finite element method was developed [4]. The currentwork builds on it and investigates both the nested and monolithic approaches for a constitutivemodel with two strongly coupled dissipation process occurring in shape memory alloys, cf. [2].The principal benefits of the monolithic approach stated above are counterbalanced by somedrawbacks. One is related to the limits of the variational framework: some models might re-quire some adaptation of the sketched treat or they even might not fit to (1) at all. Second, thedimension of the “complete” minimization problem is naturally higher than in the case whenthe problem is split in the nested formulation, which poses a challenge for computational re-sources. Third, applying universal minimization methods is often less efficient than employingcustomized algorithms and may lead to prohibitive computational time consumption for morecomplex evolutionary boundary problems. However, the advance of new optimization meth-ods and increased computational power complemented with parallelization might provide someremedies in the future.AcknowledgementsThe financial support via projects No. GA22-20181S and No. GF21-06569K provided by theCzech Science Foundation is gratefully acknowledged.References[1] Frost, M., Benešová, B., Sedlák, P., A microscopically motivated constitutive model for shapememory alloys: Formulation, analysis and computations, Mathematics and Mechanics of Solids21 (3) (2016) 358-382.[2] Frost, M., Valdman, J., Vectorized MATLAB implementation of the incremental minimizationprinciple for rate-independent dissipative solids using FEM: A constitutive model of shape mem-ory alloys, Mathematics 10 (2022) No. 4412.[3] Miehe, C. J., A multi-field incremental variational framework for gradient-extended standard dis-sipative solids, Journal of the Mechanics and Physics of Solids 59 (4) (2011) 898-923.[4] Moskovka, A., Valdman, J., Fast MATLAB evaluation of nonlinear energies using FEM in 2D and3D: Nodal elements, Applied Mathematics and Computation 424 (2022) No. 127048.54

Numerical implementation of incremental minimization principle for materials with multiple rate-independent dissipative mechanisms

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Numerical implementation of incremental minimization principle for materials with multiple rate-independent dissipative mechanisms

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