On the SCD semismooth* Newton method for generalized equations with application to a class of static contact problems with Coulomb friction
Authors
Gfrerer, H.; Mandlmayr, M.; Outrata, J.; Valdman, J.
Year
2023
Published
Computational Optimization and Applications. 2023, 86(3), 1159-1191. ISSN 0926-6003.
Type
Article
Departments
Annotation
In the paper, a variant of the semismooth* Newton method is developed for the numerical solution of generalized equations, in which the multi-valued part is a so-called SCD (subspace containing derivative) mapping. Under a rather mild regularity requirement, the method exhibits (locally) superlinear convergence behavior. From the main conceptual algorithm, two implementable variants are derived whose efficiency is tested via a generalized equation modeling a discretized static contact problem with Coulomb friction.
Surface penalization of self-interpenetration in linear and nonlinear elasticity
Authors
Krömer, S.; Valdman, J.
Year
2023
Published
Applied Mathematical Modelling. 2023, 122 641-664. ISSN 0307-904X.
Type
Article
Departments
Annotation
We analyze a term penalizing surface self-penetration, as a soft constraint for models of hyperelastic materials to approximate the Ciarlet-Nečas condition (almost everywhere global invertibility of deformations). For a linear elastic energy subject to an additional local invertibility constraint, we prove that the penalized elastic functionals converge to the original functional subject to the Ciarlet-Nečas condition. The approach also works for nonlinear models of non-simple materials including a suitable higher order term in the elastic energy, without artificial local constraints. Numerical experiments illustrate our results for a self-contact problem in 3d. © 2023 Elsevier Inc.
Elastoplastic Deformations of Layered Structures
Authors
Drozdenko, D.; Knapek, M.; Kružík, M.; Mathis, K.; Švadlenka, K.; Valdman, J.
Year
2022
Published
Milan Journal of Mathematics. 2022, 90(2), 691-706. ISSN 1424-9286.
Type
Article
Departments
Annotation
We formulate a large-strain model of single-slip crystal elastoplasticity in the framework of energetic solutions. The numerical performance of the model is compared with laboratory experiments on the compression of a stack of papers.
On the Application of the SCD Semismooth* Newton Method to Variational Inequalities of the Second Kind
Authors
Gfrerer, H.; Outrata, J.; Valdman, J.
Year
2022
Published
Set-Valued and Variational Analysis. 2022, 30(4), 1453-1484. ISSN 1877-0533.
Type
Article
Departments
Annotation
The paper starts with a description of SCD (subspace containing derivative) mappings and the SCD semismooth* Newton method for the solution of general inclusions. This method is then applied to a class of variational inequalities of the second kind. As a result, one obtains an implementable algorithm which exhibits locally superlinear convergence. Thereafter we suggest several globally convergent hybrid algorithms in which one combines the SCD semismooth* Newton method with selected splitting algorithms for the solution of monotone variational inequalities. Finally, we demonstrate the efficiency of one of these methods via a Cournot-Nash equilibrium, modeled as a variational inequality of the second kind, where one admits really large numbers of players (firms) and produced commodities.
Vectorized MATLAB Implementation of the Incremental Minimization Principle for Rate-Independent Dissipative Solids Using FEM: A Constitutive Model of Shape Memory Alloys
Type
Article
Departments
Annotation
The incremental energy minimization principle provides a compact variational formulation for evolutionary boundary problems based on constitutive models of rate-independent dissipative solids. In this work, we develop and implement a versatile computational tool for the resolution of these problems via the finite element method (FEM). The implementation is coded in the MATLAB programming language and benefits from vector operations, allowing all local energy contributions to be evaluated over all degrees of freedom at once. The monolithic solution scheme combined with gradient-based optimization methods is applied to the inherently nonlinear, non-smooth convex minimization problem. An advanced constitutive model for shape memory alloys, which features a strongly coupled rate-independent dissipation function and several constraints on internal variables, is implemented as a benchmark example. Numerical simulations demonstrate the capabilities of the computational tool, which is suited for the rapid development and testing of advanced constitutive laws of rate-independent dissipative solids.