Elastoviscoplastic Finite Element analysis in 100 lines of Matlab

Transcription

1 J. Numer. Math., Vol., No. 3, pp (22) c VSP 22 Elastoviscoplastic Finite Element analysis in lines of Matlab C. Carstensen and R. Klose Received 3 July, 22 Abstract This paper provides a short Matlab implementation with documentation of the P finite element method for the numerical solution of viscoplastic and elastoplastic evolution problems in 2D and 3D for von-mises yield functions and Prandtl-Reuß flow rules. The material behaviour includes perfect plasticity as well as isotropic and kinematic hardening with or without a viscoplastic penalisation in a dual model, i.e. with displacements and the stresses as the main variables. The numerical realisation, however, eliminates the internal variables and becomes displacement-oriented in the end. Any adaption from the given three time-depending examples to more complex applications can easily be performed because of the shortness of the program and the given documentation. In the numerical 2D and 3D examples an efficient error estimator is realized to monitor the stress error. Keywords: finite element method, viscoplasticity, elastoplasticity, Matlab. INTRODUCTION Elastoplastic time-evolution problems usually require universal, complex, commercial computer codes running on workstations or even super computers [,5]. The argument of keeping commercial secrets hidden inside a black-box, leaves the typical user without any idea what exactly is going on behind the program s userfriendly surface. The difference between a mathematical and a numerical model is fuzzy. Often, users do not care about nasty details such as quadrature rules or the exact material laws realised. But sometimes it does matter whether a regularised or penalised discrete model is solved, how the termination of an iterative process is steered, and what post-processing led to both, equilibrium and admissibility of the approximate stress field. In particular, if the solution process is part of guaranteed error control, it is quite important to know if discrete equilibrium is fulfilled exactly or not. Finally, the use of more than one black-box in a chain is likely to be less efficient. In summary, to aim efficiency (e.g. by well-adapting automatic mesh-refinements) and reliability we have to prevent, at least confess, all Institute for Applied Mathematics and Numerical Analysis, Vienna University of Technology, Wiedner Hauptstraße 8-, A-4 Vienna, Austria Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, D Kiel, Germany

2 58 C. Carstensen and R. Klose the numerical crimes. Those range from modelling errors (e.g. data errors, coarse geometrical resolution, wrong mathematical model, etc.) to variational crimes (e.g. material laws fulfilled at discrete points only or just in some approximate sense). This work aims a clear presentation of clean algorithms to compute a few model examples. Their implementation in Matlab (as one possible higher scientific computing tool) is the content of this paper. The provided software may serve as both, reference material for future numerical examples and as scientificly proved public domain source for modifications in other applications for research and educational purposes. This paper continues our efforts on the Laplace and Navier Lamé problem [2,3] towards more complicated material behaviour to advertise and to open the field of computational plasticity to a larger community of applied mathematicians and engineers. Within a small strain framework, the rheological models of this work include perfect plasticity, elastoplastic evolution with kinematic or isotropic hardening in the setting of von-mises yield conditions and a Prandl-Reuß flow rule [,5]. All variants are available for a viscoplastic regularisation with a Yosida-regularisation, known as viscoplasticity due to Perzyna. The time discretisation yields a minimisation problem under severe constraints for each time step. The discrete problem is solved with a Newton-Raphson scheme and Matlab s standard direct solver. The main principle of the presented method is, that, in each time step, the stress tensor is expressed explicitly and applied in Cauchy s first law of motion. We expect that the reader is familiar with Matlab as well as with linear elasticity and otherwise refer, e.g. to [3,3]. The paper is organised as follows. In Section 2 we introduce the viscoplastic and elastoplastic problem and their mathematical model. In Section 3 we perform a time discretisation and calculate the explicit expression of the stress tensor for the different models of hardening. In Section 4 the discretisation in space together with the data representation of the triangulation, the Dirichlet and Neumann boundary is considered. Furthermore the iterative solution of the discrete problem is explained. Section 5 concerns the calculation of the stiffness matrix. Section 6 discusses post processing routines, one for previewing the numerical solution and one for an a posteriori error estimation. Numerical examples in Section 7 illustrate the usage of the provided tools for four benchmark examples in two and three space dimensions. The programs are written for Matlab 6. but adaption for previous versions is possible. For a numerical calculation it is necessary to run the program ÑÒºÑ with (user-specified) files ÓÓÖÒØ ºØ, ÐÑÒØ ºØ, ÖÐغØ, ÒÙÑÒÒºØ, as well as the subroutines ºÑ, ºÑ, and Ù ºÑ. Their meaning and usage is explained throughout the paper and examples are provided in the later sections. The graphical representation is performed with the function ÓÛºÑ and the error is estimated in ÔÓ ØÖÓÖºÑ. All files with the code and the data for all examples can be downloaded from ÛÛÛºÑغØÙÛÒººØ»Ö ØÒ».

3 Elastoviscoplastic FE analysis RHEOLOGICAL MODEL 2.. Material models The material body occupies a bounded Lipschitz domain Ω in Ê d and deforms any time t, t T. Given a displacement field u u tµ ¾ H Ωµ d (i.e., the functional matrix Du exists almost everywhere in Ω, is measurable, and satisfies Ê Ω Du2 dx ) the total strain reads ε uµ Du Du T µ2 ε uµ jk u jk u k jµ2 jk dµ In the engineering literature, ε uµ is known as the linear Green strain tensor. The default model for small-strain plasticity involves an additive split ε e σ µ p ξ µ of the total strain in an elastic part, e σ µ σ, and an irreversible part p ξ µ. The plastic strain p ξ µ depends on further internal variables ξ. The stress field σ σ tµ ¾ L 2 Ω; Ê d d sym µ of the Cauchy stress tensor σ xtµ is linked to the elastic, reversible strain. The linear relation is provided by, the compliance tensor, which is the inverse of the fourth order elasticity tensor. In the numerical examples, a two-parameter model (λ and µ are the Lamé constants) in linear isotropic, homogeneous elasticity, e λ tr eµi 2µ e, is employed. The evolution law for the plastic strain p is more involved and requires the concept of admissible stresses, a yield function, and an associated flow rule. The kinematic variables p and ξ form the generalised strain P pξ µ. The corresponding generalised stress reads Σ σ αµ, where α ¾ Ê m m sym describes internal stresses. We denote by K the set of admissible stresses, which is a closed, convex set, containing, and is defined by K Σ : Φ Σµ The yield function Φ describes admissible stresses by Φ Σµ and models either perfect plasticity or hardening. The dissipation functional ϕ Σµ, defined by ϕ σ αµ 2ν inf σ τα β µ 2 : τβ µ ¾ Ê d d sym Ê m m sym Φ τβ µ (2.) approaches the indicator function of the set K, namely Φ Σµ ϕ Σµ Φ Σµ as ν. The time derivative of P is given by the flow rule (2.2) Ṗ ¾ ϕ Σµ σ α µ ¾ Ê d d sym Ê m m sym : τβ µ ¾ Ê d d sym Ê m m sym ϕ σ αµ σ : τ σ µ α : β αµ ϕ τβ µ (2.3)

4 6 C. Carstensen and R. Klose Here, A : B m jk A jkb jk denotes the scalar products of (symmetric) matrices AB ¾ Ê m m. For ν the flow rule results in Ṗ ṗ ξ µ ν σ τα β µ (2.4) for τβ µ uniquely determined as the projection of σ αµ onto K, i.e., τβ µ ¾ K with τβ µ : Π σ αµ uniquely determined by dist σ αµkµ σ τα β µ σ αµ Π σ αµ inf σ τ ¼ α β ¼ µ : τ ¼ β ¼ µ ¾ Ê d d sym Ê m m sym Φ τ ¼ β ¼ µ (2.5) The following examples involve the von-mises yield function in various modifications. Example 2. (Viscoplasticity). In the case of perfect viscoplasticity, there is no hardening and the internal variables ξ are absent. The yield function is given by Φ σ µ dev σ µ σ y (2.6) With (2.6) in (2.4) and sµ : maxss ¾ Ê we obtain ṗ ν σ y dev σ µµ dev σ µ (2.7) Example 2.2 (Viscoplasticity with isotropic hardening). Isotropic hardening is characterised by the scalar hardening parameter α and the constant H, the modulus of hardening. The yield function is With (2.8) in (2.4) we obtain ṗ ξ ν H 2 σy 2 Φ σ αµ dev σ µ σ y Hαµ (2.8) αhµσ y dev σ µ dev σ µ (2.9) Hσ y dev σ µ In our model the plastic part of the free energy is given by ψ p 2 H ξ 2. The internal force is defined by α ψ p ξ, which means α H ξ, where H is a positive hardening parameter. Example 2.3 (Viscoplasticity with linear kinematic hardening). In kinematic hardening the yield function is given by Φ σ αµ dev σ µ dev αµ σ y (2.) With (2.) in (2.4) we obtain ṗ ξ 2ν σ y dev σ αµ (2.) dev σ αµ dev σ αµ

5 Elastoviscoplastic FE analysis 6 For linear kinematic hardening, the plastic part of the free energy has the form ψ p 2 k ξ 2. The internal force is defined by α ψ p ξ, which means α k ξ, with a positive parameter k. Remark 2. (Elastoplasticity for ν ). In the present situation the positive number ν may be seen as a viscose penalty which leads to equalities (such as (2.7), (2.9), (2.)). The elastoplastic limit for ν leads in (2.3) to a variational inequality [,5]. This model will be included subsequently as one may consider ν in the formulae below Equilibrium equations The stress field σ ¾ L 2 Ω; Ê d d sym µ and the volume force f ¾ L2 Ω; Ê d µ are related by the local quasi-static balance of forces, divσ f (2.2) On some closed subset Γ D of the boundary with positive surface measure, we assume Dirichlet conditions while we have Neumann boundary conditions on the (possible empty) part Γ N. The two components of the displacement u need not satisfy simultaneously Dirichlet or Neumann conditions, i.e., Γ D and Γ N need not be disjoint. With M ¾ L Γ D µ d d, w ¾ H Ωµ d, and a surface force g ¾ L 2 Γ N µ we have Mu w on Γ D σn g on Γ N (2.3) Let Π denote the projection onto the set of admissible stresses K. The plastic problem is then determined by the weak formulation: Seek u ¾ H Ωµ d that satisfies Mu w on Γ D that, for all v ¾ H D Ωµd : v ¾ H Ωµ d : Mv on Γ D, σ uµ : ε vµdx f vdx g vds Ω Ω Γ N ε uµ σ σ Πσ a.e. in Ω ν α Πα ξ αµ 3. TIME DISCRETISATION AND ANALYTIC EXPRESSION OF THE STRESS TENSOR 3.. Time discretisation scheme (2.4) A generalised midpoint rule serves as a time-discretisation. In each time step there is a spatial problem with given variables u tµσ tµα tµµ at time t denoted as u σ α µ and unknowns at time t t k denoted as u σ α ). Time derivatives are replaced by backward difference quotients. The time discrete problem reads: Seek u ϑ ¾ H Ωµ d that satisfies Mu ϑ w on Γ D that, for all

6 62 C. Carstensen and R. Klose v ¾ H D Ωµd : v ¾ H Ωµ d : Mv on Γ D, σ u ϑ µ : ε vµdx f ϑ vdx g ϑ vds Ω Ω Γ N ε u ϑ u µ σ ϑ σ µ (3.) σ ϑ Πσ ϑ ϑk ξ αt ϑ µ ξ αt µ ν α ϑ Πα ϑ where σ ϑ ϑ µσ ϑσ with 2 ϑ. We define With this notation in the flow law, we have uϑ u A : ε σ ϑk ϑk (3.2) A σ ϑ ϑk ν id Πµσ ϑ (3.3) To obtain an expression of σ ϑ in terms of trσ ϑ and devσ ϑ we note that σ ϑ λ tre ϑ I 2µ e ϑ. There are constants α, β, γ and δ such that σ ϑ γ tre ϑ I δ deve ϑ e ϑ α trσ ϑ I β dev σ ϑ (3.4) Inserting of (3.4a) in (3.4b) we have e ϑ dαγ tre ϑ I βδ deve ϑ α d 2 γµ β δ (3.5) According to (3.4a) the stress tensor σ ϑ reads δ σ ϑ γ tre ϑ I δ e ϑ d tre ϑ Iµ γ tre ϑ I δ e λ tre ϑ I 2µ e ϑ d (3.6) This shows δ 2µ and γ λ 2µd. Inserting this in (3.4a) we deduce σ ϑ 3.2. Analytic expression of the stress tensor d 2 λ 2dµ trσ ϑ I 2µ devσ ϑ (3.7) In this subsection we derive explicit expressions for the stress tensor σ ϑ for the different cases of hardening. With the connection σ ϑ σ ϑ A u ϑ µµ between σ ϑ and u ϑ the elastoplastic problem is determined by a nonlinear variational problem: Seek u ϑ ¾ H Ωµ d that satisfies Mu ϑ w on Γ D and, for all v ¾ H D Ωµd : v ¾ H Ωµ d : Mv on Γ D, Ω σ ε u ϑ u µ σ µ : ε vµdx Ω f ϑ vdx We solve (3.3) for σ ϑ in the flow laws of Examples Γ N g ϑ vds (3.8)

7 Elastoviscoplastic FE analysis 63 Theorem 3. (Viscoplasticity and plasticity without hardening). For perfect viscoplasticity and perfect plasticity there exist constants C, C 2, and C 3 such that the stress tensor reads σ ϑ C tr ϑkaµi C 2 C 3 devϑkaµdevϑka (3.9) For perfect viscoplasticity these constants are C : λ 2µd C 2 : ν βν ϑkµ C 3 : ϑkσ y βν ϑkµ (3.) and in the limit ν for perfect plasticity these constants read The plastic phase occurs for In the elastic phase the stress tensor reads C : λ 2µd C 2 : C 3 : σ y (3.) dev ϑka σ y 2µ (3.2) σ ϑ C tr ϑkaµi 2µ devϑka (3.3) Proof. The discretised version of the flow law reads A σ ϑ σ y ϑk devσ ϑ (3.4) ν dev σ ϑ We consider the plastic phase σ y devσ ϑ µ. With σ ϑ α trσ ϑ I β devσ ϑ we obtain ϑk dev A β ϑk σ y devσ ϑ devσ ϑ ϑk νdev A σ yµ ν dev σ ϑ ϑk βν (3.5) Inserting (3.5b) in (3.5a) shows devσ ϑ C 2 C 3 dev ϑkaµdev ϑka (3.6) From (3.4) we obtain trσ ϑ C d trϑka. The expression for devσ ϑ in the elastic phase is obtained from (3.4) with σ y dev σ ϑ. The plastic phase occurs for σ y devσ ϑ µ. With (3.5b) this is equivalent to devϑka σ y 2µµ. Theorem 3.2 (Viscoplasticity and plasticity with isotropic hardening). For viscoplasticity and plasticity with isotropic hardening there exist constants C, C 2, C 3, and C 4 such that the stress tensor reads σ ϑ C tr ϑkaµi C 3 C 2 dev ϑkaµ C 4 C 2 µdev ϑka (3.7)

8 64 C. Carstensen and R. Klose For viscoplasticity with isotropic hardening these constants are C : λ 2µd C 2 : βν H 2 σ 2 y µ ϑk βh H 2 σ 2 y µ C 3 : ϑkσ y α Hµ C 4 : H H 2 ϑkσ 2 y ν H2 σ 2 y µ (3.8) and in the limit ν for plasticity with isotropic hardening these constants read The plastic phase occurs for C : λ 2µd C 2 : ϑk βh H 2 σ 2 y µ C 3 : ϑkσ y α Hµ C 4 : H H 2 ϑkσ 2 y (3.9) In the elastic phase the stress tensor reads dev ϑka β α Hµσ y (3.2) σ ϑ C tr ϑkaµi 2µ devϑka (3.2) Proof. The discretised version of the flow law is A σ ϑ ϑk α ϑ Hµσ y ν dev σ ϑ H α ϑ α ϑk ν H 2 σy 2 H 2 σ 2 y devσ ϑ (3.22) α ϑ Hµσ y Hσ y dev σ ϑ (3.23) dev σ ϑ σ ϑ We consider the plastic phase α ϑ Hµσ y dev σ ϑ. With α trσ ϑ I β devσ ϑ in (3.22) we infer ϑk dev A β ϑk α ϑ Hµσ y ν H 2 σy 2 devσ ϑ (3.24) dev σ ϑ and so devϑka β ϑk ν H 2 σ 2 y α ϑ Hµσ y dev σ ϑ (3.25) devσ ϑ The second component of the flow rule, (3.23), can be solved for α ϑ. Inserting of α ϑ in (3.25), solving for dev σ ϑ and inserting the resulting expression in (3.24) yields an expression for devσ ϑ. We obtain devσ ϑ C 3 C 2 devϑka C4 C 2 devϑka (3.26) From the first component of the flow rule we conclude trσ ϑ C d trϑka. With devσ ϑ α ϑ Hµσ y the expression for devσ ϑ in the elastic results from (3.22). The plastic phase occurs for dev σ ϑ α ϑ Hµσ y. With (3.26) this is equivalent to β α Hµσ y dev ϑka.

9 Elastoviscoplastic FE analysis 65 Theorem 3.3 (Viscoplasticity and plasticity with kinematic hardening). For viscoplasticity and plasticity with kinematic hardening there exit constants C, C 2, and C 3 such that the stress tensor reads σ ϑ C tr ϑkaµi C 2 C 3 dev ϑka βα µµdev ϑka βα µ devα (3.27) For viscoplasticity with kinematic hardening these constants are ϑkk 2ν C : λ 2µd C 2 : ϑk ϑkk 2µµ νµ C ϑkσ y 3 : ϑk ϑkk 2µµ νµ (3.28) and in the limit ν for plasticity with kinematic hardening these constants read C : λ 2µd C 2 : The plastic phase occurs for In the elastic phase the stress tensor reads ϑkk ϑk ϑkk 2µµ C ϑkσ y 3 : ϑk ϑkk 2µµ (3.29) dev ϑka βα µ βσ y (3.3) σ ϑ C tr ϑkaµi 2µ devϑka (3.3) Proof. The discretised version of the flow law reads A σ ϑ ϑk 2ν σ ydev σ ϑ α ϑ µµ dev σ ϑ α α ϑ α ϑ µ k ϑk (3.32) We consider the plastic phase σ y dev σ ϑ α ϑ µ. With σ ϑ α trσ ϑ I β devσ ϑ, α ϑ α k ϑka α trσ ϑ I β devσ ϑ µ (3.33) This yields in the flow rule that dev ϑka βσ ϑk ϑ µ 2ν σ y dev σ ϑ α k ϑka k βσ ϑ µ dev σ ϑ α k ϑka k βσ ϑ µ The absolute value on both sides results in an equation for dev σ ϑ α k ϑka k βσ ϑ µ. We solve for this modulus and substitute it in the above identity. This shows dev ϑka βσ ϑk σ y ϑ µ 2ν 2ν ϑkµdev ϑka βσ ϑ µ σ y dev σ ϑ α k ϑka k βσ ϑ µ (3.34)

10 66 C. Carstensen and R. Klose Multiplying both sides of (3.34) with 2νµ ϑkµdev ϑka βσ ϑ µ σ y together with some basic transformations shows 2ν ϑk k dev ϑka βσ ϑ µ σ y dev ϑka βσ ϑ µ dev ϑka βσ ϑ µdev σ ϑ α µ (3.35) On both sides of (3.35) we build the absolute value and obtain dev ϑka βσ ϑ µ dev σ ϑ α µ σ y k 2ν ϑkµ (3.36) which can be inserted in (3.35), dev σ ϑ α µdev ϑka βσ dev σ ϑ α µ σ y ϑ µ dev σ ϑ α µ (3.37) k 2ν ϑkµ Adding the term dev σ ϑ α µβ devα on both sides of (3.37) yields with some basic transformations dev σ ϑ α µdev ϑka βα µ βdev σ ϑ α µ dev σ ϑ α µ σ y k 2ν ϑkµ dev σ ϑ α µ (3.38) Because of 2νϑk k and with dev σ ϑ α µ σ y from (3.36) we can calculate the absolute value of dev σ ϑ α µ as dev σ ϑ α µ ϑkk 2νµdev ϑka βα µ ϑkσ y (3.39) ϑk βϑkk 2βν We insert (3.39) in (3.38) to obtain devσ ϑ C 2 C 3 dev ϑka βα µµdev ϑka βα µ devα (3.4) From the first component of the flow rule we have trσ ϑ C d tr ϑkaµ. The expression for devσ ϑ in the elastic phase is obtained from (3.32) with σ y dev σ ϑ α ϑ µ. The plastic phase occurs for σ y dev σ ϑ α ϑ µ. With (3.36), (3.37) and (3.39) this is equivalent to dev ϑka βα µ βσ y. 4. NUMERICAL SOLUTION OF THE TIME-DISCRETE PROBLEM This section is devoted to the framework of the spatial discretisation of one timestep (3.) with data structures in Subsection 4. and the iteration in Subsection 4.2. The heart of the matter is the assembling of the tangential local stiffness matrix explained in Section 5.

11 Elastoviscoplastic FE analysis 67 Γ N 2.8 Γ D Γ N.8.6 Γ N Γ D Figure. Plot of initial triangulation for the example of Subsection Discretisation in space Suppose the domain Ω has Ë a polygonal boundary Γ, we can cover Ω by a regular triangulation Ì, i.e., Ω T ¾Ì T, where the elements of Ì are triangles for d 2 and tetrahedrons for d 3. Regular triangulation in the sense of Ciarlet [] means that the nodes Æ of the mesh lie on the vertices of the elements, the elements of the triangulation do not overlap, no node lies on an edge of an element, and each edge E Γ of an element T ¾ Ì belongs either to Γ N or to Γ D. Matlab supports reading data from files given in ASCII format by ºØ files. Figure shows the initial mesh of a two dimensional example and Figure 2 shows the corresponding data files. The file ÓÓÖÒØ ºØ contains the coordinates of each node. The two real numbers per row are the x- and y-coordinates of each node. The file ÐÑÒØ ºØ contains for each element the node numbers of the vertices, numbered anti-clockwise. The files ÒÙÑÒÒºØ and ÖÐØºØ contain the two node numbers which bound the corresponding edge on the boundary. In the discrete version of (2.4), H Ωµ and H D Ωµ are replaced by finite dimensional subspaces Ë and ËD V ¾ Ë : V on Γ D. The discrete problem

12 68 C. Carstensen and R. Klose ÓÓÖÒØ ºØ ÐÑÒØ ºØ ÖÐØºØ ÒÙÑÒÒºØ Figure 2. Data files ÓÓÖÒØ ºØ, ÐÑÒØ ºØ, ÖÐغØ, and ÒÙÑÒÒºØ for the initial triangulation in the example of Subsection 7.. The files Ù ºÑ, ºÑ, and ºÑ are listed in Section 7. reads: Seek U ϑ ¾ Ë with Ω σ ε U ϑ U µ σ µ : ε Vµdx Let Ì denote a triangulation of Ω and Æ Ω f ϑ Vdx g ϑ Vds V ¾ Ë (4.) Γ the set of all nodes in Ì, then let ϕ ϕ dn µ ϕ e ϕ e d ϕ N e ϕ N e d µ be the nodal basis of the finite dimensional space Ë, where N is the number of nodes of the mesh and ϕ z is the scalar hat function of node z in the triangulation Ì, i.e., ϕ z zµ and ϕ z yµ for all y ¾ Æ with y z. Then (4.) is equivalent to F p : Ω σ ε U ϑ U µ C σ µ : ε ϕ p µdx Ω f ϑ ϕ p dx Γ g ϑ ϕ p ds (4.2) for p dn. F p can be decomposed into a sum of a part Q p which depends on u ϑ and a part P p which is independent of u ϑ. Thus we have F p : Q p P p with Q p : P p : 4.2. Iterative solution Ω Ω σ ε U ϑ U µ C σ µ : ε ϕ p µdx (4.3) f ϑ ϕ p dx Γ g ϑ ϕ p ds (4.4) This subsection describes the iterative solution of (4.2) by a Newton Raphson scheme which is realised in the Matlab program ѺÑ, listed at the end of this

13 Elastoviscoplastic FE analysis 69 section. One step of the Newton iteration reads DF U k ϑ µuk ϑ DF U k ϑ µuk ϑ F Uk ϑ µ (4.5) The discrete displacement vector U k ϑ is expressed in the nodal basis as Uk ϑ dn U k ϑ ϕ, U k ϑ U k ϑ dn µ are the components of Uk ϑ. The matrix DF is defined as DF U k ϑ U k ϑ dn µµ pq : F p U k ϑ U k ϑ dn µ U k ϑ q (4.6) At the end of this section the Matlab program ѺÑ, a finite element program for the two dimensional case is listed. In the following line ½ ÙÒØÓÒ ÙØÅ ÓÓÖÒØ ÐÑÒØ ÖÐØ ÒÙÑÒÒ ÙØ Ù¼ ؼ ؽ Ø Æµ shows the function call with input ÓÓÖÒØ, ÐÑÒØ, ÖÐØ and ÒÙÑÒÒ together with the vectors ÙØ and Ù¼, which are the start-vector u ϑ of the Newton iteration and the displacement u at time t. The last three variables in the parameter list are the times t, t, and the variable ϑ. Lines 2 7, ±ÁÒØÐ ØÓÒ ¾ ÔÖ ¾ Æ ¾ ƵÉÞÖÓ ¾ Æ ½µÈÞÖÓ ¾ Æ ½µ Рؾ ÐÑÒØ ½ ½ ¾ ¾ µ¹óò Þ ÐÑÒØ ½µ ½µ ½ ¼ ½ ¼ ½ ¼ ± ÑÐÝ ÓÖ ½ Þ ÐÑÒØ ½µ ÄÐ Ø µ Ä Äµ É Äµ ØÑ Ä Äµ É Äµ ÓÓÖÒØ ÐÑÒØ µ µ ÙØ Äµ Ù¼ ĵ µ Ò show the assembling of the stiffness matrix DF and the stiffness vector Q. Line 2 initialises DF, Q, and P, line 3 defines a vector Ð Ø which contains the numbers of degrees of freedom for each element. Lines 4 7 perform the assembling of DF and Q with a call of subroutine ØѺÑ. The element with number j contributes the degrees of freedom ÄÐ Ø µ. The local stiffness matrix and the local stiffness vector of element number j have to be added at the positions LLµ of the global stiffness matrix and Lµ of the global stiffness vector. In the following lines 8 22 ±ÎÓÐÙÑ ÓÖ ÓÖ ½ Þ ÐÑÒØ ½µ ÐÓÓÖÒØ ÓÓÖÒØ ÐÑÒØ µ µöø ½ ½ ½ÐÓÓÖÒØ ³ µ»¾ ½¼ Ö ½¹Øµ ÙÑ ÐÓÓÖÒØ ½µ» ؼµ³» Ø ÙÑ ÐÓÓÖÒØ ½µ» ؽµ³» ½½ È Ð Ø µµè Ð Ø µµ ÖÔÑØ ½µ ½¾ Ò ±ÆÙÑÒÒ ÓÒØÓÒ ½ ÑÔØÝ ÒÙÑÒÒµ ½ ÆРؾ ÒÙÑÒÒ ½ ½ ¾ ¾ µ¹óò Þ ÒÙÑÒÒ ½µ ½µ ÖÔÑØ ½ ¼ ½ ¾µ ½ ÎÓÓÖÒØ ÒÙÑÒÒ ¾µ µ¹óóöòø ÒÙÑÒÒ ½µ µ ½ Ä ÕÖØ ÙÑ Îº Î ¾µµÎκ»Ä Ä ÆÎÎ ¼ ¹½½ ¼ ½ ÓÖ ½ Þ ÒÙÑÒÒ ½µ ½ Ä µ ½¹Øµ ÙÑ ÓÓÖÒØ ÒÙÑÒÒ µ µµ»¾ ÆÎ µ ؼµ³»¾ ººº ½ Ø ÙÑ ÓÓÖÒØ ÒÙÑÒÒ µ µµ»¾ ÆÎ µ ؽµ³»¾µ ¾¼ È ÆÐ Ø µµè ÆÐ Ø µµ ÖÔÑØ ¾ ½µ ¾½ Ò

14 7 C. Carstensen and R. Klose ¾¾ Ò ¾ É¹È the vector F Q P is determined. P contains the work of the volume forces f and the surface forces g at time t ϑ. The values of f are provided by the subroutine ºÑ and evaluated in the centre of gravity x S y S µ of T, to approximate the integral ÊT f ϕ j dx in (4.2), T f ϕ r dx 3 T f k x S y S µ k : mod r 2µ (4.7) The integral Ê E g ϕ k ds in (4.2) that involves the Neumann conditions is approximated with the value of g in the centre x M y M µ of the edge E with length E, E g ϕ r ds 2 Eg k x M y M µ k : mod r 2µ (4.8) The following lines ±ÖÐØ ÓÒØÓÒ ¾ ÒÓ ÙÒÕÙ ÖÐص ¾ ϼ ż Ù ÓÓÖÒØ ÒÓ µ ؼµÏ½ Ž Ù ÓÓÖÒØ ÒÓ µ ؽµ ¾ Ï ½¹Øµ ϼ Ø Ï½Å ½¹Øµ ż Ø Å½ ÔÖ Þ Ï ½µ ¾ Ƶ ¾ ÓÖ ¼½ ¾ ÓÖ ¼½ ¾ ½ ¾ Þ Å ½µ ¾ ÒÓ ¹½ µ Å ½ ¾ Þ Å ½µ ½ µµ ¼ Ò ½ Ò ¾ Ñ Ò ÙÑ µ³µµöæó Ò ÙÑ µµµ Ñ µïï Ñ µòóöñ¼òóöñ ÖÆÓ µµòø¼ take into account Dirichlet conditions whereas gliding boundary conditions are allowed as well. Gliding boundary conditions such as those along symmetry axes, are implemented different from Dirichlet conditions (for all components). The displacement is fixed merely in one specified direction and possibly free in others. A general approach to this type of condition reads, for each node on the Dirichlet boundary, ¼ ½¼ ½ ¼ ½ m U w... (4.9) m d U d w d where m j ¾ Ê d, w j ¾ Ê, and U U d µ are the degrees of freedom for the considered node on the Dirichlet boundary. With the normal vector n along Γ and the canonical basis e j, j d, in Ê d, gliding and classical Dirichlet conditions and the lack of Dirichlet-type conditions on certain parts of Γ can all be included by ¼ ½ n ¼ ½ U.. U d ¼ e T ½¼ ½ U.. ud U d e T d ¼ ½¼ ½ U.. (4.) U d

15 Elastoviscoplastic FE analysis 7 respectively. Line 24 determines all nodes on the Dirichlet boundary and stores them in the variable ÒÓ. Let there be n Dirichlet nodes. Line 26 creates a matrix Å of size 2n 2 and a vector Ï of size 2n. The rows 2 j µ up to 2 j for j n of Å and Ï contain for each Dirichlet node the matrix m m d µ T and the vector w w 2 µ. For the discrete problem, the generalised Dirichlet conditions lead to the linear system BU w (4.) where each row of B ¾ Ê 2n 2N contains the value of some m j at a node on the boundary and w ¾ Ê 2n contains the corresponding values w j at this node. The matrix is calculated in lines 27 3 from the entries of Å. A row m j is not included in B and W, which is realised in line 33. Between lines ÛÐ ÒÓÖÑ ÖÆÓ µµ½¼ ¹½¼µ ½¼ ¹µ ÒÓÖѼµ ² Òؽ¼¼µ ÒØÒØ ½ ± ÐÙÐØÓÒ Ó Ø ÓÐÙØÓÒ ÙØ¹Ï ³ ÔÖ Þ ½µ Þ ½µµ ÜÙØÜ ½¾ ƵÐÜ ¾ Æ ½ Þ Ü ½µµ ± ÑÐÝ ÔÖ ¾ Æ ¾ ƵÉÞÖÓ ¾ Æ ½µ ÓÖ ½ Þ ÐÑÒØ ½µ ¼ ÄÐ Ø µ ½ Ä Äµ É Äµ ØÑ Ä Äµ É Äµ ÓÓÖÒØ ÐÑÒØ µ µ ÙØ Äµ Ù¼ ĵ µ ¾ Ò É¹È Ò Òؽ¼¼µ Ô ³Ð ØÓ ÓÒÚÖ ÛØÒ ½¼¼ ØÖØÓÒ ØÔ ³µ Ò the Newton iteration takes place. Coupling the set of boundary conditions through Lagrange parameters with (4.5) leads to the extended system with DF U k ϑ µ B T B U k ϑ λ b w (4.2) b : DF U k ϑ µuk ϑ f U k ϑ µ (4.3) This system of equations is solved in line 37 by the Matlab solver. Matlab makes use of the properties of a sparse, symmetric matrix for solving the system of equations efficiently. Line 37 determines the parts U k ϑ and λ of the solution. Line 38 to 43 assemble the local stiffness matrix and stiffness vector. Line 32 calculates the set à of free nodes. There are m free nodes z z m µ ¾ Ã. The iteration process only continues in line 34, if the relative residual F U k ϑ z U k ϑ z µ m is greater than a given tolerance and if the maximum number of iterations is less than D Finite Element program ½ ÙÒØÓÒ ÙØÅ ÓÓÖÒØ ÐÑÒØ ÖÐØ ÒÙÑÒÒ ÙØ Ù¼ ؼ ؽ Ø Æµ ±ÁÒØÐ ØÓÒ

16 72 C. Carstensen and R. Klose ¾ ÔÖ ¾ Æ ¾ ƵÉÞÖÓ ¾ Æ ½µÈÞÖÓ ¾ Æ ½µ Рؾ ÐÑÒØ ½ ½ ¾ ¾ µ¹óò Þ ÐÑÒØ ½µ ½µ ½ ¼ ½ ¼ ½ ¼ ± ÑÐÝ ÓÖ ½ Þ ÐÑÒØ ½µ ÄÐ Ø µ Ä Äµ É Äµ ØÑ Ä Äµ É Äµ ÓÓÖÒØ ÐÑÒØ µ µ ÙØ Äµ Ù¼ ĵ µ Ò ±ÎÓÐÙÑ ÓÖ ÓÖ ½ Þ ÐÑÒØ ½µ ÐÓÓÖÒØ ÓÓÖÒØ ÐÑÒØ µ µöø ½ ½ ½ÐÓÓÖÒØ ³ µ»¾ ½¼ Ö ½¹Øµ ÙÑ ÐÓÓÖÒØ ½µ» ؼµ³» Ø ÙÑ ÐÓÓÖÒØ ½µ» ؽµ³» ½½ È Ð Ø µµè Ð Ø µµ ÖÔÑØ ½µ ½¾ Ò ±ÆÙÑÒÒ ÓÒØÓÒ ½ ÑÔØÝ ÒÙÑÒÒµ ½ ÆРؾ ÒÙÑÒÒ ½ ½ ¾ ¾ µ¹óò Þ ÒÙÑÒÒ ½µ ½µ ÖÔÑØ ½ ¼ ½ ¾µ ½ ÎÓÓÖÒØ ÒÙÑÒÒ ¾µ µ¹óóöòø ÒÙÑÒÒ ½µ µ ½ Ä ÕÖØ ÙÑ Îº Î ¾µµÎκ»Ä Ä ÆÎÎ ¼ ¹½½ ¼ ½ ÓÖ ½ Þ ÒÙÑÒÒ ½µ ½ Ä µ ½¹Øµ ÙÑ ÓÓÖÒØ ÒÙÑÒÒ µ µµ»¾ ÆÎ µ ؼµ³»¾ ººº ½ Ø ÙÑ ÓÓÖÒØ ÒÙÑÒÒ µ µµ»¾ ÆÎ µ ؽµ³»¾µ ¾¼ È ÆÐ Ø µµè ÆÐ Ø µµ ÖÔÑØ ¾ ½µ ¾½ Ò ¾¾ Ò ¾ É¹È ±ÖÐØ ÓÒØÓÒ ¾ ÒÓ ÙÒÕÙ ÖÐص ¾ ϼ ż Ù ÓÓÖÒØ ÒÓ µ ؼµÏ½ Ž Ù ÓÓÖÒØ ÒÓ µ ؽµ ¾ Ï ½¹Øµ ϼ Ø Ï½Å ½¹Øµ ż Ø Å½ ÔÖ Þ Ï ½µ ¾ Ƶ ¾ ÓÖ ¼½ ¾ ÓÖ ¼½ ¾ ½ ¾ Þ Å ½µ ¾ ÒÓ ¹½ µ Å ½ ¾ Þ Å ½µ ½ µµ ¼ Ò ½ Ò ¾ Ñ Ò ÙÑ µ³µµöæó Ò ÙÑ µµµ Ñ µïï Ñ µòóöñ¼òóöñ ÖÆÓ µµòø¼ ÛÐ ÒÓÖÑ ÖÆÓ µµ½¼ ¹½¼µ ½¼ ¹µ ÒÓÖѼµ ² Òؽ¼¼µ ÒØÒØ ½ ± ÐÙÐØÓÒ Ó Ø ÓÐÙØÓÒ ÙØ¹Ï ³ ÔÖ Þ ½µ Þ ½µµ ÜÙØÜ ½¾ ƵÐÜ ¾ Æ ½ Þ Ü ½µµ ± ÑÐÝ ÔÖ ¾ Æ ¾ ƵÉÞÖÓ ¾ Æ ½µ ÓÖ ½ Þ ÐÑÒØ ½µ ¼ ÄÐ Ø µ ½ Ä Äµ É Äµ ØÑ Ä Äµ É Äµ ÓÓÖÒØ ÐÑÒØ µ µ ÙØ Äµ Ù¼ ĵ µ ¾ Ò É¹È Ò Òؽ¼¼µ Ô ³Ð ØÓ ÓÒÚÖ ÛØÒ ½¼¼ ØÖØÓÒ ØÔ ³µ Ò D Finite Element program ½ ÙÒØÓÒ ÙØÅ ÓÓÖÒØ ÐÑÒØ ÖÐØ ÒÙÑÒÒ ÙØ Ù¼ ؼ ؽ Ø Æ Æµ ±ÁÒØÐ ØÓÒ ¾ ÔÖ Æ Æµ ÉÞÖÓ Æ ½µÈÞÖÓ Æ ½µ Ð Ø ÐÑÒØ ½ ½ ½ ¾ ¾ ¾ µ¹óò Æ ½µ ¾ ½ ¼ ¾ ½ ¼ ¾ ½ ¼ ¾ ½ ¼ ± ÑÐÝ ÓÖ ½ Þ ÐÑÒØ ½µ ÄÐ Ø µ Ä Äµ É Äµ ØÑ Ä Äµ É Äµ ÓÓÖÒØ ÐÑÒØ µ µ ÙØ Äµ Ù¼ ĵ µ

17 Elastoviscoplastic FE analysis 73 Ò ±ÎÓÐÙÑ ÓÖ ÓÖ ½ Þ ÐÑÒØ ½µ ½¼ ÐÓÓÖÒØ ÓÓÖÒØ ÐÑÒØ µ µúóðø ½ ½ ½ ½ÐÓÓÖÒØ ³ µ» ½½ ÚÓÐ ½¹Øµ ÙÑ ÐÓÓÖÒØ ½µ» ؼµ³» Ø ÙÑ ÐÓÓÖÒØ ½µ» ؽµ³» ½¾ È Ð Ø µµè Ð Ø µµ ÖÔÑØ ½µ ½ Ò ±ÆÙÑÒÒ ÓÒØÓÒ ½ ÑÔØÝ ÒÙÑÒÒµ ½ ÆÐ Ø ÒÙÑÒÒ ½ ½ ½ ¾ ¾ ¾ µ¹óò Þ ÒÙÑÒÒ ½µ ½µ ÖÔÑØ ¾ ½ ¼ ½ µ ½ ÓÖ ½ Þ ÒÙÑÒÒ ½µ ½ ÆÎÖÓ ÓÓÖÒØ ÒÙÑÒÒ ¾µ µ¹óóöòø ÒÙÑÒÒ ½µ µ ººº ½ ÓÓÖÒØ ÒÙÑÒÒ µ µ¹óóöòø ÒÙÑÒÒ ½µ µµ ½ ÖÒÓÖÑ Æε»¾ÆÎÆλÒÓÖÑ Æε ¾¼ Ö ½¹Øµ ÙÑ ÓÓÖÒØ ÒÙÑÒÒ µ µµ»¾ ÆΠؼµ³»¾ ººº ¾½ Ø ÙÑ ÓÓÖÒØ ÒÙÑÒÒ µ µµ»¾ ÆΠؽµ³»¾µ ¾¾ È ÆÐ Ø µµè ÆÐ Ø µµ ÖÔÑØ ½µ ¾ Ò ¾ Ò ¾ É¹È ±ÖÐØ ÓÒØÓÒ ¾ ÒÓ ÙÒÕÙ ÖÐص ¾ ϼ ż Ù ÓÓÖÒØ ÒÓ µ ؼµÏ½ Ž Ù ÓÓÖÒØ ÒÓ µ ؽµ ¾ Ï ½¹Øµ ϼ Ø Ï½Å ½¹Øµ ż Ø Å½ ÔÖ Þ Ï ½µ Ƶ ¾ ÓÖ ¼¾ ¼ ÓÖ ¼¾ ½ ½ Þ Å ½µ ÒÓ ¹¾ µ Å ½ Þ Å ½µ ½ µµ ¾ Ò Ò Ñ Ò ÙÑ µ³µµöæó Ò ÙÑ µµµ Ñ µïï Ñ µòóöñ¼òóöñ ÖÆÓ µµòø¼ ÛÐ ÒÓÖÑ ÖÆÓ µµ½¼ ¹½¼µ ½¼ ¹µ ÒÓÖѼµ ² Òؽ¼¼µ ÒØÒØ ½ ± ÐÙÐØÒ Ø ÓÐÙØÓÒ ÙØ¹Ï ³ ÔÖ Þ ½µ Þ ½µµ ÜÙØÜ ½ ƵÐÜ Æ ½ Þ Ü ½µµ ± ÑÐÝ ¼ ÔÖ Æ ÆµÉÞÖÓ Æ ½µ ½ ÓÖ ½ Þ ÐÑÒØ ½µ ¾ ÄÐ Ø µ Ä Äµ É Äµ ØÑ Ä Äµ É Äµ ÓÓÖÒØ ÐÑÒØ µ µ ÙØ Äµ Ù¼ ĵ µ Ò É¹È Ò Òؽ¼¼µ Ô ³Ð ØÓ ÓÒÚÖ ÛØÒ ½¼¼ ØÖØÓÒ ØÔ ³µ Ò 5. TANGENTIAL LINEAR SYSTEM OF EQUATIONS The tangential stiffness matrix DF in (4.6) as well as the vector F in (4.2) are expressed as a sum over all elements T in Ì. Each part of the sum defines the local stiffness matrix M and the vector of the right-hand side R for the corresponding element. Let k through k K be the numbers of the nodes of an element T. There are dk basis functions with support on T, namely, ϕ π Tµ ϕ k e ϕ π Tdµ ϕ k e d ϕ π T2K µ ϕ k K e ϕ π TdKµ ϕ k K e d, where e j is the j-th unit vector. The function ϕ k j is the local scalar hat function of node k j. The function π maps the indices dk of the local numeration with respect to T to the global numeration. In the sequel we want to compute the local stiffness matrix M and a local vector R for element T. For simplicity we write ϕ ϕ dk instead of ϕ π Tµ ϕ π TdKµ.

18 74 C. Carstensen and R. Klose The local vector R is defined as R r : T σ ε U ϑ U µ σ µ : ε ϕ r µdx r dk (5.) The local stiffness matrix for element T is defined, for rs dk, by R r M rs U k ϑ s U k ϑ s T 5.. Notation for triangular elements dk σ ε U kϑ ϕ U σ : ε ϕ r µdx (5.2) The Matlab implementation of M and R can be done in a compact way by defining a matrix Ô ÐÓÒ by Ô ÐÓÒ : ε ϕ j µ ε 2 ϕ j µ ε 2 ϕ j µ ε 22 ϕ µ 6 j ؽ ؾµ 2 2 Dϕ jµ Dϕ j µ 2 Dϕ j µ 2 Dϕ j µ 22 6 j Dϕ j µ Dϕ j µ 2 Dϕ j µ 2 Dϕ j µ 22 6 j µ (5.3) with ¾ ϕ k ϕ k ϕ k ϕ k ϕ k2 ϕ k2 ؽ ϕ k2 ϕ k2 ϕ k3 ϕ k3 ϕ k3 ϕ k3 ¾ ؾ ϕ k ϕ k ϕ k ϕ k ϕ k2 ϕ k2 ϕ k2 ϕ k2 ϕ k3 ϕ k3 ϕ k3 ϕ k3 (5.4) With the coordinates x y µ, x 2 y 2 µ, x 3 y 3 µ of the vertices of the element T, the entries of ؽ and ؾ are stored in the matrix Ô, ¾ Ô ϕ k ϕ k2 ϕ k3 ϕ k ϕ k2 ϕ k3 ¾ ¾ x x 2 x 3 (5.5) y y 2 y 3 With Ô from (5.5) the following Matlab lines calculate the matrices ؽ and ؾ. ؽÞÖÓ µø½ ½¾ ½¾µÔؽ ¾¾ µô ؾÞÖÓ µø¾ ½¾ ½ µôø¾ ¾¾ ¾ µô

19 Elastoviscoplastic FE analysis Notation for tetrahedral elements In the three-dimensional case the matrix Ô ÐÓÒ is defined by Ô ÐÓÒ : ε ϕ j µ ε 2 ϕ j µ ε 3 ϕ j µ ε 2 ϕ j µ ε 22 ϕ j µ ε 23 ϕ j µ ε 3 ϕ j µ ε 32 ϕ j µ ε 33 ϕ j µ 2 j with ؽ ؾµ (5.6) 2 ؽ Dϕ j µ Dϕ j µ 2 Dϕ j µ 3 Dϕ j µ 2 Dϕ j µ 22 Dϕ j µ 23 Dϕ j µ 3 Dϕ j µ 32 Dϕ j µ 33 2 j ¾ and ϕ k ϕ k ϕ k z ϕ k ϕ k ϕ k z. ϕ k4 ϕ k4 ϕ k4 ϕ k ϕ k ϕ k z z ϕ k4 ϕ k4 ϕ k4 z ϕ k4 ϕ k4. ϕ k4 z (5.7) ؾ Dϕ j µ Dϕ j µ 2 Dϕ j µ 3 Dϕ j µ 2 Dϕ j µ 22 Dϕ j µ 32 Dϕ j µ 3 Dϕ j µ 23 Dϕ j µ 33 2 j ¾ ϕ k ϕ k ϕ k z ϕ k ϕ k ϕ k z ϕ k ϕ k ϕ k z.. (5.8) ϕ k4 ϕ k4 ϕ k4 z ϕ k4 ϕ k4 ϕ k4 z ϕ k4 ϕ k4 ϕ k4 z The entries of ؽ and ؾ are stored in the matrix Ô, ¾ ϕ k ϕ k ϕ k z ¾ ¾ ϕ k2 ϕ k2 ϕ k2 z x x 2 x 3 x 4 Ô ϕ k3 ϕ k3 ϕ k3 (5.9) y y 2 y 3 y 4 z z z 2 z 3 z 4 ϕ k4 ϕ k4 ϕ k4 z where x y z µ, x 2 y 2 z 2 µ, x 3 y 3 z 3 µ, x 4 y 4 z 4 µ are the coordinates of the vertices of the element T. The following Matlab lines calculate the matrices ؽ and ؾ.

20 76 C. Carstensen and R. Klose ؽÞÖÓ ½¾ µø¾þöó ½¾ µ ؽ ½ ½¼ ½ µôø½ ¾ ½½ µôø½ ½¾ µô ؾ ½ ½¼ ½ µôø¾ ¾ ½½ ¾ µôø¾ ½¾ µô 5.3. Local stiffness matrix M and local vector R The subsequent algorithms perform a fall differentiation between the elastic and the plastic phase. In the plastic phase the local stiffness matrix M and the local vector R for element T are obtained by evaluating (5.2) and (5.) for the different cases of hardening. In the elastic phase M and R are well known, c.f. [3]. The Matlab routine ØÑºÑ calculates the local stiffness matrix M according to (5.2) and the first summand of (5.). With M and R a step of the assembling procedure is performed in ØѺÑ. Example 5. (Perfect viscoplasticity for 2D and 3D). The Matlab program calculating the local stiffness matrix and the vector R depending on the element T is listed at the end of this example for the 2D and 3D case, together with a special implementation of the deviator and trace function, called Ú¾ and ØÖ¾ in the 2D case and Ú and ØÖ in the 3D case. The 2D program is described below. In line 2 global variables are transfered to the routine, C : λ 2µd C 2 : ν βν ϑkµ C 3 : ϑkσ y βν ϑkµ C 4 : 2µ In lines 4 7 the matrix Ô ÐÓÒ is calculated as described in the previous section. In line 7 we define the matrix v : ε U ϑ U µ σ and store it row wise. In line 8 the area of element T is determined. In line 9 a fall differentiation is performed. If dev vµ σ y 2µµ the plastic phase occurs, otherwise the elastic phase. Depending on the result of this distinction, a constant C 5 and a vector C 6 are defined by C 5 : C2 C 3 dev vµ if dev vµ σ y 2µµ 2µ else (5.) C 3 dev vµ 3 Ú¾ ε ϕ j µµ : Ú¾ vµ 6 j if dev vµ σ y 2µµ C 6 : T else. (5.) In lines 4 and 5 an update of the local stiffness matrix and the first summand of the vector R is performed. The components of the local stiffness matrix read M jk T C tr ε ϕ j µµtr ε ϕ k µµ C 5 dev ε ϕ j µµ : ε ϕ k µ C 6 µ j dev vµ : ε ϕ k µµ (5.2) The components of the first summand of the local vector R read R j T C tr vµtr ε ϕ j µµ C 5 dev vµ : ε ϕ j µµ (5.3) The Matlab routines ØѺÑ, ØÖ¾ºÑ and Ú¾ºÑ for the 2D case are listed below.

21 Elastoviscoplastic FE analysis 77 ½ ÙÒØÓÒ Å ØÑ Å¼ ¼ ÐÓÓÖÒØ ÙØ Ù¼ µ ¾ ÐÓÐ ÐÑ ÑÙ ÑÝ ½ ¾ ¼ ÔÒÚ ½ ½ ½ÐÓÓÖÒØ ³ µ ¼ ¼Ý ¾µ Ô ÞÖÓ µø½þöó µø¾þöó µ Ø½ ½¾ ½¾µÔؽ ¾¾ µô ؾ ½¾ ½ µôø¾ ¾¾ ¾ µô Ô Ø½ ؾµ»¾Ú Ùعټµ³ Ô ¼ µ ÌØ ½ ½ ½ÐÓÓÖÒØ ³ µ»¾ ÒÓÖÑ Ú¾ Úµµ¹ ÑÝ» ¾ ÑÙµ¼ ½¼ ¾»ÒÓÖÑ Ú¾ Úµµ»ÒÓÖÑ Ú¾ Úµµ Ú¾ Ô µ Ú¾ Úµ³µ ½½ Ð ½¾ ¾ ÑÙÞÖÓ ½µ ½ Ò ½ Åż Ì ½ ØÖ¾ Ô µ ØÖ¾ Ô µ³ Ú¾ Ô µ Ô ³¹ Ú¾ Úµ Ô ³µ ½ ¼ Ì ½ ØÖ¾ Úµ ØÖ¾ Ô µ Ô Ú¾ Úµ³µ ÙÒØÓÒ ØÖ¾ØÖ¾ µ ØÖ¾ ½µ µ ÙÒØÓÒ Ú¾Ú¾ µ Ú¾ ½µ¹ µµ»¾ ¾µ µ µ¹ ½µµ»¾ The Matlab routines ØѺÑ, ØÖ ºÑ and Ú ºÑ for the 3D case are listed below. ½ ÙÒØÓÒ Å ØÑ Å¼ ¼ ÐÓÓÓÖÒØ Ù½ Ù¼ µ ¾ ÐÓÐ ÑÙ ÑÝ ½ ¾ ¼ ÔÒÚ ½ ½ ½ ½ÐÓÓÓÖÒØ ³ µ ÞÖÓ ½ µý µ Ô ÞÖÓ ½¾ µø½þöó ½¾ µø¾þöó ½¾ µ ؽ ½ ½¼ ½ µôø½ ¾ ½½ µôø½ ½¾ µô ؾ ½ ½¼ ½ µôø¾ ¾ ½½ ¾ µôø¾ ½¾ µô Ô Ø½ ؾµ»¾Ú Ù½¹Ù¼µ³ Ô ¼ µ ÌØ ½ ½ ½ ½ÐÓÓÓÖÒØ ³ µ» ÒÓÖÑ Ú Úµµ¹ ÑÝ» ¾ ÑÙµ¼ ½¼ ¾»ÒÓÖÑ Ú Úµµ»ÒÓÖÑ Ú Úµµ Ú Ô µ Ú Úµ³µ ½½ Ð ½¾ ÞÖÓ ½¾ ½µ ½ Ò ½ Åż Ì ½ ØÖ Ô µ ØÖ Ô µ³ Ú Ô µ Ô ³¹ Ú Úµ Ô ³µ ½ ¼ Ì ½ ØÖ Úµ ØÖ Ô µ Ô Ú Úµ³µ ÙÒØÓÒ ØÖØÖ µ ØÖ ½µ µ µ ÙÒØÓÒ ÚÚ µ Ú¹ØÖ µ ½ ¼ ¼ ¼ ½ ¼ ¼ ¼ ½ Example 5.2 (Isotropic hardening in 2D). The Matlab program calculating the local stiffness matrix and the local vector R is listed at the end of this example. In line 2 global variables C up to C 6 are transfered to the routine, C : λ µ C 2 : ν 2µ H2 σ 2 y µ ϑk 2µ H H 2 σ 2 y C 3 : ϑkσ y α Hµ C 4 : H H 2 ϑkσ 2 y ν H2 σ 2 y µ C 5 : 2µ C 6 : 2µ α Hµσ y Lines 3 8 are analog to the last example and perform the calculation of Ô ÐÓÒ,

22 78 C. Carstensen and R. Klose define the vector v and determine the element area T. In line 9 the fall differentiation is performed. If dev vµ C 6 the plastic phase occurs, otherwise the elastic phase. Depending on the result of this distinction, constants C 7 and a vector C 8 are defined by C 7 : C 5 C3 C 2 dev vµµ C 4 C 2 if dev vµ C 6 else (5.4) C 8 µ j : C3 C 2 dev vµ 3 µú¾ ε ϕ j µµ : Ú¾ vµ if dev vµ C 6 T else. (5.5) In lines 4 and 5 an update of the local stiffness matrix and the first summand of the local vector R is performed. The components of the local stiffness matrix read M jk T C tr ε ϕ j µµtr ε ϕ k µµ C 7 dev ε ϕ j µµ : ε ϕ k µ C 8 µ j dev vµ : ε ϕ k µµ (5.6) The components of the first summand of the vector R read R j T C tr vµtr ε ϕ j µµ C 7 dev vµ : ε ϕ j µµ (5.7) The Matlab routine ØÑºÑ is listed below. ½ ÙÒØÓÒ Å Ê ØÑ Å¼ ʼ ÐÓÓÓÖÒØ Ù½ Ù¼ µ ¾ ÐÓÐ ½ ¾ ¼ ÔÒÚ ½ ½ ½ÐÓÓÓÖÒØ ³ µ ¼ ¼Ý ¾µ Ô ÞÖÓ µø½þöó µø¾þöó µ Ø½ ½¾ ½¾µÔؽ ¾¾ µô ؾ ½¾ ½ µôø¾ ¾¾ ¾ µô Ô Ø½ ؾµ»¾Ú Ù½¹Ù¼µ³ Ô ¼ µ ÌØ ½ ½ ½ÐÓÓÓÖÒØ ³ µ»¾ ÒÓÖÑ Ú¾ Úµµ¹ µ¼ ½¼ µ» ¾ ÒÓÖÑ Ú¾ Úµµµ»¾ µ» ¾ ÒÓÖÑ Ú¾ Úµµ µ Ú¾ Ô µ Ú¾ Úµ³ ½½ Ð ½¾ ÞÖÓ ½µ ½ Ò ½ Åż Ì ½ ØÖ¾ Ô µ ØÖ¾ Ô µ³ Ú¾ Ô µ Ô ³ Ú¾ Úµ Ô ³µ ½ Êʼ Ì ½ ØÖ¾ Úµ ØÖ¾ Ô µ Ô Ú¾ Úµ³µ Example 5.3 (Kinematic hardening in 2D). The Matlab program calculating the local stiffness matrix and the local vector R is listed at the end of this example. In line 2 global variables C up to C 4 are transfered to the routine, C : λ µ C 2 : C 3 : ϑkk 2ν ϑk ϑkk 2µµ νµ ϑkσ y ϑk ϑkk 2µµ νµ C 4 : 2µ Lines 3 8 are analog to the last example. The strain tensor Ô ÐÓÒ is calculated, and matrices v and v are defined, with v : ε U ϑ U µ σ, v :

23 Elastoviscoplastic FE analysis 79 2µµα. In line 9 the fall differentiation is performed. If dev vµ σ y 2µµ the plastic phase occurs, otherwise the elastic phase. Depending on the result of this distinction, a constant C 5, a vector C 6 and a Kronecker delta d are defined by C 5 : C 3 dev vµµ C 2 if dev vµ σ y 2µµ C 4 else (5.8) C C 3 dev vµ 3 Ú¾ ε ϕ j µµ : Ú¾ vµ if dev vµ σ y 2µµ 6 µ j : T else (5.9) if dev vµ σ d y 2µµ : (5.2) else. In lines 4 and 5 an update of the local stiffness matrix and the first summand of the local vector R is performed. The components of the local stiffness matrix read M jk T C tr ε ϕ j µµtr ε ϕ k µµ C 5 dev ε ϕ j µµ : ε ϕ k µ C 6 µ j dev vµ : ε ϕ k µµ (5.2) The components of the first summand of the vector R read R j T C tr vµtr ε ϕ j µµ C 5 dev vµ : ε ϕ j µ d dev α µ : ε ϕ j µµ (5.22) The Matlab routine ØÑºÑ is listed below. ½ ÙÒØÓÒ Å Ê ØÑ Å¼ ʼ ÐÓÓÓÖÒØ Ù½ Ù¼ µ ¾ ÐÓÐ ÑÙ ÑÝ ½ ¾ ¼ м ÔÒÚ ½ ½ ½ÐÓÓÓÖÒØ ³ µ ¼ ¼Ý ¾µ Ô ÞÖÓ µø½þöó µø¾þöó µ Ø½ ½¾ ½¾µÔؽ ¾¾ µô ؾ ½¾ ½ µôø¾ ¾¾ ¾ µô Ô Ø½ ؾµ»¾Ú¼ Ù½¹Ù¼µ³ Ô ¼ µú½¹½» ¾ ÑÙµ м µúú¼ Ú½ ÌØ ½ ½ ½ÐÓÓÓÖÒØ ³ µ»¾ ÒÓÖÑ Ú¾ Úµµ¹ ÑÝ» ¾ ÑÙµ¼ ½¼ ¾»ÒÓÖÑ Ú¾ Úµµ»ÒÓÖÑ Ú¾ Úµµ µ Ú¾ Ô µ Ú¾ Úµ³½ ½½ Ð ½¾ ÞÖÓ ½µ¼ ½ Ò ½ Åż Ì ½ ØÖ¾ Ô µ ØÖ¾ Ô µ³ Ú¾ Ô µ Ô ³¹ Ú¾ Úµ Ô ³µ ½ Êʼ Ì ½ ØÖ¾ Ú¼µ ØÖ¾ Ô µ Ô Ú¾ Ú Ú½µ³ Ô Ú¾ м µµ³µ 6. POSTPROCESSING 6.. Displaying the solution Our two dimensional problems model the plain stress condition. In that case, the complete stress tensor σ ¾ Ê 3 3 sym has the form ¼ σ σ 2 ½ σ σ 2 σ 22

24 8 C. Carstensen and R. Klose with σ 33 2 λ µ λ µ σ σ 22 µ. It then follows for dev σ 2, where deva : A 3 traid 3 3 and A : n jk A2 jk 2 is the Frobenius norm, that devσ 2 µ 2 6 µ λ µ 2 σ σ 22 µ 2 2 σ2 2 σ σ 22 µ 2 The function ÓÛºÑ, listed at the end of this subsection, calculates the variable ÚË in lines 2 9, which stores the nodal values of σ h. Here σ h is the stress calculated by the finite element method and σ h is a smoother approximation of the discrete stress σ h, which may be used for error control [8]. Line determines the shear energy density devσ h 2 4µµ and stores it in the variable Ú. In line 3 the Matlab routine ØÖ ÙÖ is used to draw the deformed mesh with a magnification of the displacement, which is fixed in line. Grey tones are attached to the displayed meshes which are proportional to the elastic shear energy density devσ h 2 4µµ. ½ ÙÒØÓÒ ÓÛ ÓÓÖÒØ ÐÑÒØ ËÑ Ù ÐÑ ÑÙµ ¾ ÖÇÑÞÖÓ Þ ÓÓÖÒØ ½µ ½µ ÚËÞÖÓ Þ ÓÓÖÒØ ½µ µ ÓÖ ½ Þ ÐÑÒØ ½µ ÖØ ½ ½ ½ÓÓÖÒØ ÐÑÒØ µ µ³ µ»¾ ÖÇÑ ÐÑÒØ µµöçñ ÐÑÒØ µµ Ö ÚË ÐÑÒØ µ µúë ÐÑÒØ µ µ Ö ½½½ ËÑ µ Ò ÚËÚ˺» ÖÇÑ ½ ½ ½ ½ µ ½¼ Ú ÑÙ» ¾ ÑÙ Ðѵ¾µ ½» ÑÙµµ ÚË ½µ ººº ÚË µµº¾ ½» ¾ ÑÙµ ÚË ¾µº¾¹ÚË ½µº ÚË µµ ½½ ÑÒݾ¼ ½¾ ÓÐÓÖÑÔ ½¹Öݵ ½ ØÖ ÙÖ ÐÑÒØ ÑÒÝ Ù ½¾ Þ Ù ½µµ ÓÓÖÒØ ½µ ººº ÑÒÝ Ù ¾¾ Þ Ù ½µµ ÓÓÖÒØ ¾µ ººº ½ ÞÖÓ Þ ÓÓÖÒØ ½µ ½µ Ú ³ÓÐÓÖ³ ³ÒØÖÔ³µ ½ ÚÛ ¼ ¼µ ½ ÓÐÓÖÖ ³ÚÖسµ The Matlab routine ØÖ ÙÖ ÐÑÒØ Ü Ý Þ Ú ³ÓÐÓÖ³ ³ÒØÖÔ³µ is used to draw triangulations for equal types of elements. Every row of the matrix ÐÑÒØ determines one polygon where the x-, y- and z-coordinate of each corner of this polygon is given by the corresponding entry in Ü, Ý and Þ. For a 2D-problem, we use for the z-coordinate the same value (zero) in all mesh points. The values Ú together with the options ³ÓÐÓÖ³ ³ÒØÖÔ³ determine the grey tone of the area. The grey tone of each node is determined by Ú, that of the rest of the area is linearly interpolated A posteriori error indication The averaged stress field σ h, allows an a posteriori error estimation by comparing it to the discrete (discontinuous) stress σ h. For one time step, the computable quantity η h σ h σ h L 2 Ωµ (6.)

25 Elastoviscoplastic FE analysis 8 defines the error estimate [8] (for pure Dirichlet conditions Neumann boundary conditions require modifications [9]) σ σ h L 2 Ωµ Cη h (6.2) Here, σ h may be any piecewise affine and globally continuous stress approximation. It is emphasized that the error bound (6.2) holds only if the time-discretization error is neglected. It is generally believed that the accumulation error (in time) cannot be captured by averaging (in space). However the quantity η h may monitor the local spatial approximation error and a large size of η h indicates a larger spatial error and motivates refinements. Theorem 6. (Efficeny). Assume that the stress field σ at time t is smooth, i.e. σ ¾ H Ω; Ê d d sym µ, then η h is an efficient error estimator up to higher order terms in the sense that η h 4σ σ h L 2 Ωµ hot (6.3) where generically hot O h 2 Dσ L 2 Ωµ µ σ σ h L 2 Ωµ. Proof. The triangle inequality provides, for all τ h which are globally continous and Ì -piecewise affine (like σ h ), that σ h τ h L 2 Ωµ σ σ h L 2 Ωµ σ τ h L 2 Ωµ σ σ h L 2 Ωµ hot (6.4) for the nodal interpolation τ h Iσ of σ [5]. One can prove that (with τ h globaly continous and Ì -piecewise affine but arbitrary otherwise) η M : minσ h τ h L 2 Ωµ σ h σ τ h L 2 Ωµ 4η M h This is shown for ab arbitrary constant in [6,4,6] and for the constant 4 and quite general boundary conditions in [7]. The main argument is a (local) inverse estimate, caused by equivalence of norms on finite dimensional vector spaces. The combination of the two inequalities leads to η h σ σ h L 2 Ωµ 4η M 4σ σ h L 2 Ωµ 4σ τ h L 2 Ωµ for any τ h, in particular for τ h Iσ. It is emphasized once more that the reliability (6.2) is problematic and holds only for one time step and for large hardening and large viscosity [8]. The Matlab routine at the end of this subsection calculates ؽ at time t t. Lines 3 4 initialise ω z, the patch of the hat function at node z, which is stored in ÖÇÑ and σ h, which is stored in ÚË. Line 5 determines weights and barycentric coordinates Ê by a two dimensional quadrature rule. In our algorithm we have σ h zµ ω z σ h dxω z. Lines 6 calculate the area of the elements (line 7), the

26 82 C. Carstensen and R. Klose patch size ω z (line 8) and Ê ω z σ h dx (line 9) by a loop over all elements. In line, σ h is obtained by dividing ω z σ h dx through the patch size. Lines 2 6 compute ؽ, where for σ h four interpolation points are used on each element T. ½ ÙÒØÓÒ ØÈÓ ØÖÓÖ ÓÓÖÒØ ÐÑÒØ ÒÙÑÒÒ Ñ Øµ ¾ ؼ ÖÇÑ ÞÖÓ Þ ÓÓÖÒØ ½µ ½µ ÚË ÞÖÓ Þ ÓÓÖÒØ ½µ µ ÛØ ÖÝ ÕÙÖØÙÖ¾ ¾µ ÓÖ ½ Þ ÐÑÒØ ½µ Ö µø ½ ½ ½ÓÓÖÒØ ÐÑÒØ µ µ³ µ»¾ ÖÇÑ ÐÑÒØ µµöçñ ÐÑÒØ µµ Ö µ ÚË ÐÑÒØ µ µúë ÐÑÒØ µ µ Ö µ ½½½ Ñ µ ½¼ Ò ½½ ÚËÚ˺» ÖÇÑ ½ ½ ½ ½ µ ½¾ ÓÖ ½ Þ ÐÑÒØ ½µ ½ ØØ Ö µ ÛØ ÙÑ ÓÒ Þ ÛØ ¾µ ½µ Ñ µ¹ººº ½ Öݳ ÚË ÐÑÒØ µ µµº¾ ¾µ ½ Ò ½ ØØ ½»¾µ 7. NUMERICAL EXAMPLES 7.. On a plate with circular hole A two dimensional squared plate with a hole, Ω 22µ 2 Ò B µ is submitted to time dependent surface forces g tµ 6tµn at the top (y 2) and the bottom ( y 2), where n denotes the outer normal to Ω. The rest of the boundary is traction free. As the problem is symmetric only a quarter of Ω is discretised. The boundary conditions are specified in files Ù ºÑ, ºÑ, and ºÑ : ÙÒØÓÒ Ï Å Ù Ü Øµ ÅÞÖÓ ¾ Þ Ü ½µ ¾µ ÏÞÖÓ ¾ Þ Ü ½µ ½µ ± ÝÑÑØÖÝ ÓÒØÓÒ ÓÒ Ø Ü¹Ü ØÑÔÒ Ü ½µ¼ ² Ü ¾µ¼µ Å ¾ ØÑÔ¹½ ½¾µÓÒ Þ ØÑÔ ½µ ½µ ¼ Ï ¾ ØÑÔ¹½ ½µÞÖÓ Þ ØÑÔ ½µ ½µ ± ÝÑÑØÖÝ ÓÒØÓÒ ÓÒ Ø Ý¹Ü ØÑÔÒ Ü ¾µ¼ ² Ü ½µ¼µ Å ¾ ØÑÔ¹½ ½¾µÓÒ Þ ØÑÔ ½µ ½µ ½ Ï ¾ ØÑÔ¹½ ½µÞÖÓ Þ ØÑÔ ½µ ½µ ÙÒØÓÒ ÚÓÐÓÖ Üµ ÚÓÐÓÖÞÖÓ Þ Ü ½µ ¾µ ½ ¼ ÙÒØÓÒ ÓÖ Ü Ò Øµ ÓÖÞÖÓ Þ Ü ½µ ¾µ Ò ¾µ½µ ÓÖ ¾µ¼¼ Ø Ò

27 Elastoviscoplastic FE analysis Γ N Γ D Γ N Γ N.2 Γ D Figure 3. Deformed mesh for membrane with hole in the example of Subsection 7. calculated with 3474 degrees of freedom. The grey tone visualises devσ h 2 4µµ as described in Subsection 6.. This numerical example from [7] models perfect viscoplasticity with Young s modulus E 269, Poisson s ratio ν 29, yield stress σ y 45 and vanishing initial data for the stress tensor σ. The solution is calculated with 3474 degrees of freedom in the time interval from t to t 4 by the implicit Euler method in 8 uniform steps of length k 2. The material remains elastic between t and t 2. Figure 3 shows the deformed mesh at t 2, Table shows the number of iterations in Newton s method and the estimated error for each time step. Table. Number of iterations and indicated error for membrane with hole in the example of Subsection 7. for various time steps. step iterations η σ h σ h 2σ h

28 84 C. Carstensen and R. Klose 7.2. Numerical examples for Cooks membrane problem In the second numerical example we solve the viscoplastic problem with isotropic hardening for a two-dimensional model of a tapered panel of plexi glass described by Ω conv µ, 4844µ, 486µ, 44µ. The panel is clamped at one end (x ) and subjected to a shearing load g tµ on the opposite end (x 48) with vanishing volume force f. The boundary conditions are specified in files Ù ºÑ, ºÑ, and ºÑ : ÙÒØÓÒ Ï Å Ù Ü Øµ Å ÞÖÓ ¾ Þ Ü ½µ ¾µ Ï ÞÖÓ ¾ Þ Ü ½µ ½µ Å ½¾¾ Þ Ü ½µ¹½ ½µÓÒ Þ Ü ½µ ½µ Å ¾¾¾ Þ Ü ½µ ¾µÓÒ Þ Ü ½µ ½µ ÚÐÙÞÖÓ Þ Ü ½µ ½µ Ï ½¾¾ Þ Ù ½µ¹½ ½µÚÐÙ ½µ Ï ¾¾¾ Þ Ù ½µ ½µÚÐÙ ¾µ ÙÒØÓÒ ÚÓÐÓÖ Üµ ÚÓÐÓÖ ÞÖÓ Þ Ü ½µ ¾µ ÙÒØÓÒ ÓÖ Ù Ò Øµ ÓÖÞÖÓ Þ Ü ½µ ¾µ ÓÖ Ò Ò ¾µ½µ ¾µÜÔ ¾ ص For plexi glass we set E 29 and ν 4. This example is often referred to as the Cooks membrane problem and constitutes a standard test for bending dominated response. We calculate the solution with 845 degrees of freedom in the time interval [,.] with vanishing initial data for the displacement u, the stress tensor σ and the hardening parameter α. The time step in the implicit Euler method is k. The plastic parameters are H, H and σ y. The problem becomes plastic at t 2. Figure 4 shows the deformed mesh at t for 278 degrees of freedom, while Table 2 shows the number of iterations in Newton s method and the estimated error for each time step. Table 2. Number of iterations and indicated error for Cook s membrane problem in the example of Subsection 7.2 for various time steps. step iterations η σ h σ h 2σ h

29 Elastoviscoplastic FE analysis 85 7 x Γ N Γ N Γ D Γ N Figure 4. Deformed mesh for Cooks membrane problem in the example of Subsection 7.2 calculated with 278 degrees of freedom. The grey tone visualises devσ h 2 4µµ as described in Subsection Numerical example for axissymmetric ring The third numerical example is taken from [4]. We solve a viscoplastic model with kinematic hardening. The geometry is a two-dimensional section of a long tube with inner radius and outer radius 2 (Fig. 6). We have no volume force, f but time depending surface forces g rϕtµ te r and g 2 rϕtµ t4e r. The system is required to keep centered at the origin and rotation is prohibited. The boundary conditions are specified in files Ù ºÑ, ºÑ, and ºÑ : ÙÒØÓÒ Ï Å Ù Ü Øµ Å ÞÖÓ ¾ Þ Ü ½µ ¾µ Ï ÞÖÓ ¾ Þ Ü ½µ ½µ ± ÝÑÑØÖÝ ÓÒØÓÒ ÓÒ Ø Ü¹Ü ØÑÔ Ò Ü ½µ¼ ² Ü ¾µ¼µ Å ¾ ØÑÔ¹½ ½¾µ ÓÒ Þ ØÑÔ ½µ ½µ ¼ Ï ¾ ØÑÔ¹½ ½µ ÞÖÓ Þ ØÑÔ ½µ ½µ ± ÝÑÑØÖÝ ÓÒØÓÒ ÓÒ Ø Ý¹Ü ØÑÔ Ò Ü ¾µ¼ ² Ü ½µ¼µ Å ¾ ØÑÔ¹½ ½¾µ ÓÒ Þ ØÑÔ ½µ ½µ ½ Ï ¾ ØÑÔ¹½ ½µ ÞÖÓ Þ ØÑÔ ½µ ½µ ½ ¼ ÙÒØÓÒ ÚÓÐÓÖ Ü Øµ ÚÓÐÓÖÞÖÓ Þ Ü ½µ ¾µ

30 86 C. Carstensen and R. Klose ÙÒØÓÒ ÓÖ Ü Ò Øµ Ô Ö ÖؾÔÓÐ Ü ½µ Ü ¾µµ ÔÆÒ Ô¼µ Ô ÔÆµÔ ÔƵººº ¾ Ô ÓÒ Þ ÔƵµ ÓÖ ½ Þ Ü ½µ Ö µº ² Ö µ½º µø Ó Ô µµ Ò Ô µµ Ð Ö µ½º ² Ö µ¾º µ¹ø» Ó Ô µµ Ò Ô µµ Ò Ü ½µ¼ Ü ¾µ¼ µ¼ ¼ Ò Ò The exact solution is u rϕtµ u r rtµe r σ rϕtµ σ r rtµe r Å e r σ ϕ e ϕ Å e ϕ p rϕtµ P r rtµ e r Å e r e ϕ Å e ϕ µ (7.) with e r cosϕsinϕµ, e ϕ sinϕcosϕµ and with a µ λ, ß 2µ 2µ λ µ, t 2µr 3 ßI R tµµ r 4a r R tµ µr u r rtµ t 2µr 3 ßI R tµµ 4r 4a (7.2) ßI rµr r R tµ µr t 8 r σ 2 3 aßi R tµµ 4 r 2 r R tµ r rtµ (7.3) t 8 r 2 3 aßi R tµµ r 2 2aßI rµ r R tµ σ ϕ rtµ rσ rµ r P r rtµ I rµ r R tµ σ y Ô 2 aß k µ R 2 r 2 µ r R tµ (7.4) (7.5) σ y Ô 2 aß k µ lnr 2 R 2 r 2 R 2 µµ (7.6) With α 4aß 3 aß k µµ, the radius of the plastic boundary R tµ is the positive root of Ô 2 f Rµ 2α lnr α µr 2 α t (7.7) σ y

31 Elastoviscoplastic FE analysis 87 Table 3. Number of iterations, indicated error, and exact error for axissymmetric ring in the example of Subsection 7.3. This numerical example validates our claim that η can be an accurate error guess. step iterations e σ σ h 2σ 2 η σ h σ h 2σ h 2 eη The material parameters are E 7 for Young s modulus, ν 33 for Poisson s ratio and the yield stress is σ y 2. The solution is calculated in the time interval [,.2] with 322 degrees of freedom and vanishing initial data for the displacement u, the stress tensor σ and the hardening parameter α. The time step in the implicit Euler method is k. The hardening parameter is k. The material becomes plastic at t 6. Table 3 shows the error and the estimated error for the different time steps together with the number of iterations in Newton s method. Figure 5 shows the deformed mesh at t 8, displacements magnified by the factor Numerical example in three dimensions As a three-dimensional example we solve a perfect viscoplastic problem on the cube 3 Ò 5 3. The face with x is Dirichlet boundary, the rest is Neumann boundary. The volume force f is zero while the surface force is g tn, if the normal vector n points in y-direction. The boundary conditions are specified in files Ù ºÑ, ºÑ, and ºÑ : ÙÒØÓÒ Ï Å Ù Ü Øµ Å ÞÖÓ Þ Ü ½µ µ Ï ÞÖÓ Þ Ü ½µ ½µ Å ½ Þ Ü ½µ¹¾ ½µ ÞÖÓ Þ Ü ½µ ½µ Å ¾ Þ Ü ½µ¹½ ¾µ ÞÖÓ Þ Ü ½µ ½µ Å Þ Ü ½µ µþöó Þ Ü ½µ ½µ ÚÐÙÞÖÓ Þ Ü ½µ µ Ï ½ Þ Ü ½µ¹¾ ½µÚÐÙ ½µ Ï ¾ Þ Ü ½µ¹½ ½µÚÐÙ ¾µ Ï Þ Ü ½µ ½µÚÐÙ µ ÙÒØÓÒ ÚÓÐÓÖ Üµ ÚÓÐÓÖ ÞÖÓ Þ Ü ½µ ¾µ ÙÒØÓÒ ÓÖ Ù Ò Øµ ÓÖÞÖÓ Þ Ü ½µ ¾µ ÓÖ Ò Ò ¾µ½µ ¾µØ

32 88 C. Carstensen and R. Klose 2.5 x Figure 5. Deformed mesh for axissymmetric ring in the example of Subsection 7.3 calculated with 322 degrees of freedom. The grey tone visualises devσ h 2 4µµ as described in Subsection 6.. g 2 f= g Ω 2 E 7 ν 33 σ y 2 k g te r g 2 t4e r Figure 6. Axissymmetric ring in the example of Subsection 7.3 with initial triangulation of a quarter of the ring endowed with symmetrie boundary conditions.