Computation in finite-strain viscoelasticity: finite elements based on the interpretation as differential–algebraic equations

Hartmann, Stefan

doi:10.1016/s0045-7825(01)00332-2

Cited by 77 publications

(58 citation statements)

References 33 publications

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“…), see for different applications [28,30,31]. Thereby, u ∈ R nu are the unknown nodal displacements and q ∈ R nQ the internal variables at all spatial integration points.…”

Section: F(t Y(t)ẏ(t)) := G(t U(t) Q(t)) Q(t) − R(t U(t)u(t) Qmentioning

confidence: 99%

“…The treatment of constitutive models of evolutionary-type within implicit finite elements leads after the spatial discretization to a system of differential-algebraic equations (DAE-system) where the algebraic part results from the discretized weak formulation of the equilibrium conditions and the differential part is the outcome of the assemblage of all constitutive model's evolutionary equations at all spatial integration points -Gauss-points (see, for example, [21,28]). Usually, this is solved by a Backward-Euler method.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Iterative solvers within sequences of large linear systems in non‐linear structural mechanics

Hartmann¹,

Tebbens

Quint

et al. 2009

Z Angew Math Mech

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This article treats the computation of discretized constitutive models of evolutionary-type (like models of viscoelasticity, plasticity, and viscoplasticity) with quasi-static finite elements using diagonally implicit Runge-Kutta methods (DIRK) combined with the Multilevel-Newton algorithm (MLNA). The main emphasis is on promoting iterative methods, as opposed to the more traditional direct methods, for solving the non-symmetric systems which occur within the DIRK/MLNA. It is shown that iterative solution of the arising sequences of linear systems can be substantially accelerated by various techniques that aim at sharing part of the computational effort throughout the sequence. In this way iterative solution becomes attractive at clearly lower dimensions than the dimensions where direct solvers start to fail for memory reasons. The applications are related to small strain viscoplasticity of a smooth constitutive model for plastics and a finite strain plasticity model with non-linear kinematic hardening developed for metals.

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“…), see for different applications [28,30,31]. Thereby, u ∈ R nu are the unknown nodal displacements and q ∈ R nQ the internal variables at all spatial integration points.…”

Section: F(t Y(t)ẏ(t)) := G(t U(t) Q(t)) Q(t) − R(t U(t)u(t) Qmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Iterative solvers within sequences of large linear systems in non‐linear structural mechanics

Hartmann¹,

Tebbens

Quint

et al. 2009

Z Angew Math Mech

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“…In this investigation an overstress-type model of finite strain viscoelasticity with an extention to a poly-convex hyperelasticity relation is applied, see [3] and [4]. The model is based on the multiplicative decomposition of the deformation gradient F into an elastic and a viscous part, F = F e F v .…”

Section: Constitutive Modelmentioning

confidence: 99%

Application of Rosenbrock‐type methods to a finite element formulation based on large strain viscoelasticity

Hamkar

Hartmann

2008

Proc Appl Math and Mech

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The present paper is focused on the solution of differential-algebraic equation systems (DAE), which arise in large viscoelastic deformations within the quasi-static finite element context. For this purpose linearly implicit methods of Rosenbrock-type are used, which avoid completely the solution of non-linear equations. This article investigates a possible treatment of the new global approach with respect to expense and achievable accuracy. Constitutive ModelIn this investigation an overstress-type model of finite strain viscoelasticity with an extention to a poly-convex hyperelasticity relation is applied, see [3] and [4]. The model is based on the multiplicative decomposition of the deformation gradient F into an elastic and a viscous part, F = F e F v . The total deformation, represented by the right Cauchy-Green tensor C = F T F, defines the equilibrium stress partT eq = 2dψ(J,C) dC , which is carried out by an additive decomposition of the strain-energy function ψ(J, C) = U (J) + w eq (C) into a volume-changing and a volume-preserving part, where the volume-changing part depends on J = det F and the volume-preserving part on the invariants of the unimodular right Cauchy-Green tensor C = (det C) −1/3 C. The over-stresses are given by a Neo-Hooke type model w ov (C e ) = µ(I Ce − 3) which are formulated by quantities relative to the inelastic intermediate configuration:C e = (detC e ) −1/3 C e defines the unimodular elastic right Cauchy-Green tensor withstands for the 2nd Piola-Kirchhoff tensor, where T is the Cauchy-stress tensor. In the case of viscous (inelastic) deformations the internal variables develop according to the associated flow ruleĊ v = 4µ η, described by a process-dependent viscosity η = η 0 exp(−s(CT ov ·T ov C) 1/2 ) and the viscous right Cauchy-Green tensor Application of Rosenbrock-type methodsThe numerical treatment of quasi-static problems with constitutive equations of evolutionary type, is currently explained as a solution procedure of DAEs related to the consistent application of the vertical method of lines , see [1] and the literature cited therein. In view of this global DAE-approach the calculation of the reaction forces are involved using the method of the Lagrange multipliers, see [5]. This extends the DAE-system with a constraint equation to the form F(t, y(t),ẏ(t))where λ ∈ R n p are the Lagrange multipliers interpreted as the reaction forces. In contrast to the classical "displacementdriven" formulation the discretized weak form g a = {g,ĝ} T depends on all nodal displacements u a ∈ R n dof . Here the decomposition u a = {u,û} T , into displacements u ∈ R n u at those nodes where no constraints are given and the displacementŝ u ∈ R n p where the prescribed displacements u ∈ R n p apply, is used. q ∈ R n Q is the vector of all internal variables occurring at all Gauss-points. In [5] diagonally implicit Runge-Kutta methods (DIRK), which embraces the classical Backward-Euler method as a particular case, are applied to (1). The integration procedure yield a coupled system of no...

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“…Transforming the coupled ODE-system of second order of Eq. (1) in an equivalent ODE-system of first order and applying the DIRK approach, see [1,7,8], yields in each stage T ni = t n + c i ∆t n , i = 1, . .…”

Section: Introduction and Space-time Discretizationmentioning

confidence: 99%

Time‐adaptive non‐linear finite‐element computations for thermo‐mechanically coupled analysis in inelastic dynamical systems

Grafenhorst

Hartmann

Rang

2016

Proc Appl Math and Mech

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In structural dynamics the initial boundary-value problem has to be solved in the space and time domain. The spatial discretization is done using finite elements yielding systems of differential equations of second-order. Incorporating inelastic material properties, thermo-mechanical coupling and particular Dirichlet boundary conditions essentially changes the underlying mathematical problem. The main goal is to provide higher-order time integration schemes using diagonally-implicit RungeKutta methods (DIRK) and the Generalized α-method consistently applied to the semi-discretized ODE-system comprising the semi-discretized equations of motion, the unsteady semi-discrete heat equation and constitutive equations of evolutionarytype. Furthermore, we want to show the applicability of time-adaptivity by embedded schemes so that step-sizes are chosen automatically. The inelastic constitutive equations are given by a thermo-viscoplasticity model of Perzyna/Chaboche-type with non-linear kinematic hardening.

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Computation in finite-strain viscoelasticity: finite elements based on the interpretation as differential–algebraic equations

Cited by 77 publications

References 33 publications

Iterative solvers within sequences of large linear systems in non‐linear structural mechanics

Iterative solvers within sequences of large linear systems in non‐linear structural mechanics

Application of Rosenbrock‐type methods to a finite element formulation based on large strain viscoelasticity

Time‐adaptive non‐linear finite‐element computations for thermo‐mechanically coupled analysis in inelastic dynamical systems

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