arXiv subjects

Jan Valdman

Publications and source records attributed to Jan Valdman.

At least 37 records · Page 2Linked to original sources

arXiv2021

On the application of the SCD semismooth* Newton method to variational inequalities of the second kind

The paper starts with a description of the 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 exhibiting a 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 inequalities of the second kind, where one admits really large numbers of players (firms) and produced commodities.

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arXiv2021

Linearization and Computation for Large-Strain Viscoelasticity

Time-discrete numerical minimization schemes for simple viscoelastic materials in the large strain Kelvin-Voigt rheology are not well-posed due to non-quasiconvexity of the dissipation functional. A possible solution is to resort into non-simple material models with higher-order gradients of deformations. This makes, however, numerical computations much more involved. Here we propose another approach relying on local minimizers of the simple-material model. Computational tests are provided showing a very good agreement between our model and the original one.

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arXiv2021

Minimization of p-Laplacian via the Finite Element Method in MATLAB

Minimization of energy functionals is based on a discretization by the finite element method and optimization by the trust-region method. A key tool is a local evaluation of the approximated gradients together with sparsity of the resulting Hessian matrix. We describe a vectorized MATLAB implementation of the p-Laplace problem in one and two space-dimensions, however it is easily applicable to other energy formulations.

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arXiv2021

On the solution of contact problems with Tresca friction by the semismooth* Newton method

An equilibrium of a linear elastic body subject to loading and satisfying the friction and contact conditions can be described by a variational inequality of the second kind and the respective discrete model attains the form of a generalized equation. To its numerical solution we apply the semismooth* Newton method by Gfrerer and Outrata (2019) in which, in contrast to most available Newton-type methods for inclusions, one approximates not only the single-valued but also the multi-valued part. This is performed on the basis of limiting (Morduchovich) coderivative. In our case of the Tresca friction, the multi-valued part amounts to the subdifferential of a convex function generated by the friction and contact conditions. The full 3D discrete problem is then reduced to the contact boundary. Implementation details of the semismooth* Newton method are provided and numerical tests demonstrate its superlinear convergence and mesh independence.

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arXiv2020

On the application of the semismooth* Newton method to variational inequalities of the second kind

The paper starts with a concise description of the recently developed 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 exhibiting a local superlinear convergence. Thereafter we suggest several globally convergent hybrid algorithms in which one combines the semismooth* Newton method with selected splitting algorithms for the solution of monotone variational inequalities. Their efficiency is documented by extensive numerical experiments.

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arXiv2020

Poincare-Friedrichs Type Constants for Operators Involving grad, curl, and div: Theory and Numerical Experiments

We give some theoretical as well as computational results on Laplace and Maxwell constants. Besides the classical de Rham complex we investigate the complex of elasticity and the complex related to the biharmonic equation and general relativity as well using the general functional analytical concept of Hilbert complexes. We consider mixed boundary conditions and bounded Lipschitz domains of arbitrary topology. Our numerical aspects are presented by examples for the de Rham complex in 2D and 3D which not only confirm our theoretical findings but also indicate some interesting conjectures.

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arXiv2020

MATLAB Implementation of C1 finite elements: Bogner-Fox-Schmit rectangle

Rahman and Valdman (2013) introduced a new vectorized way to assemble finite element matrices. We utilize underlying vectorization concepts and extend MATLAB codes to implementation of Bogner-Fox-Schmit C1 rectangular elements in 2D. Our focus is on the detailed construction of elements and simple computer demonstrations including energies evaluations and their visualizations.

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arXiv2020

MATLAB Implementation of Element-based Solvers

Rahman and Valdman (2013) introduced a vectorized way to assemble finite element stiffness and mass matrices in MATLAB. Local element matrices are computed all at once by array operations and stored in multi-dimentional arrays (matrices). We build some iterative solvers on available multi-dimentional structures completely avoiding the use of a sparse matrix.

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arXiv2019

On computation of optimal strategies in oligopolistic markets respecting the cost of change

The paper deals with a class of parametrized equilibrium problems, where the objectives of the players do possess nonsmooth terms. The respective Nash equilibria can be characterized via a parameter-dependent variational inequality of the second kind, whose Lipschitzian stability is thoroughly investigated. This theory is then applied to evolution of a oligopolistic market in which the firms adopt their production strategies to changing input costs, while each change of the production is associated with some "costs of change". We examine both the Cournot-Nash equilibria as well as the two-level case, when one firm decides to take over the role of the Leader (Stackelberg equilibrium). The impact of costs of change is illustrated by academic examples.

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arXiv2019

Numerical approximation of von Kármán viscoelastic plates

We consider metric gradient flows and their discretizations in time and space. We prove an abstract convergence result for time-space discretizations and identify their limits as curves of maximal slope. As an application, we consider a finite element approximation of a quasistatic evolution for viscoelastic von Kármán plates. Computational experiments are provided, too.

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arXiv2019

SpinDoctor: a Matlab toolbox for diffusion MRI simulation

The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses in a heterogeneous medium can be modeled by the multiple compartment Bloch-Torrey partial differential equation (BTPDE). A mathematical model for the time-dependent apparent diffusion coefficient (ADC), called the H-ADC model, was obtained recently using homogenization techniques on the BTPDE. Under the assumption of negligible water exchange between compartments, the H-ADC model produces the ADC of a diffusion medium from the solution of a diffusion equation (DE) subject to a time-dependent Neumann boundary condition. This paper describes a publicly available Matlab toolbox called SpinDoctor that can be used 1) to solve the BTPDE to obtain the dMRI signal (the toolbox provides a way of robustly fitting the dMRI signal to obtain the fitted ADC); 2) to solve the DE of the H-ADC model to obtain the ADC; 3) a short-time approximation formula for the ADC is also included in the toolbox for comparison with the simulated ADC. The PDEs are solved by P 1 finite elements combined with build-in Matlab routines for solving ordinary differential equations. The finite element mesh generation is performed using an external package called Tetgen that is included in the toolbox. SpinDoctor provides built-in options of including 1) spherical cells with a nucleus; 2) cylindrical cells with a myelin layer; 3) an extra-cellular space (ECS) enclosed either a) in a box or b) in a tight wrapping around the cells; 4) deformation of canonical cells by bending and twisting. 5) permeable membranes for the BT-PDE (the H-ADC assumes negligible permeability). Built-in diffusion-encoding pulse sequences include the Pulsed Gradient Spin Echo and the Oscilating Gradient Spin Echo.

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arXiv2019

Interfacial polyconvex energy-enhanced evolutionary model for shape memory alloys

A sharp-interface model describing static equilibrium configurations of shape mory alloys by means of interfacial polyconvex energy density introduced by Šilhavý in 2010 and extended to a quasistatic situation by Knüpfer and Kružík in 2016 is computationally tested. Elastic properties of variants of martensite and the austenite are described by polyconvex energy density functions. Volume fractions of particular variants are modeled by a map of bounded variation. Additionally, energy stored in martensite-martensite and austenite-martensite interfaces is measured by an interface-polyconvex function. It is assumed that transformations between material variants are accompanied by energy dissipation which, in our case, is positively and one-homogeneous giving rise to a rate-independent model. Various two-dimensional computational examples are presented and the used computer code is made available for downloads.

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arXiv2019

Global injectivity in second-gradient Nonlinear Elasticity and its approximation with penalty terms

We present a new penalty term approximating the Ciarlet-Nečas condition (global invertibility of deformations) as a soft constraint for hyperelastic materials. For non-simple materials including a suitable higher order term in the elastic energy, we prove that the penalized functionals converge to the original functional subject to the Ciarlet-Nečas condition. Moreover, the penalization can be chosen in such a way that all low energy deformations, self-interpenetration is completely avoided even for sufficiently small finite values of the penalization parameter. We also present numerical experiments in 2d illustrating our theoretical results.

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arXiv2018

Efficient and flexible MATLAB implementation of 2D and 3D elastoplastic problems

We propose an effective and flexible way to implement 2D and 3D elastoplastic problems in MATLAB using fully vectorized codes. Our technique is applied to a broad class of the problems including perfect plasticity or plasticity with hardening and several yield criteria. The problems are formulated in terms of displacements, discretized by the implicit Euler method in time and the finite element method in space, and solved by the semismooth Newton method. We discuss in detail selected models with the von Mises and Prager-Drucker yield criteria and four types of finite elements. The related codes are available for download. A particular interest is devoted to the assembling of tangential stiffness matrices. Since these matrices are repeatedly constructed in each Newton iteration and in each time step, we propose another vectorized assembling than current ones known for the elastic stiffness matrices. The main idea is based on a construction of two large and sparse matrices representing the strain-displacement and tangent operators, respectively, where the former matrix remains fixed and the latter one is updated only at some integration points. Comparisons with other available MATLAB codes show that our technique is also efficient for purely elastic problems. In elastoplasticity, the assembly times are linearly proportional to the number of integration points in a plastic phase and additional times due to plasticity never exceed assembly time of the elastic stiffness matrix.

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arXiv2017

Verifications of primal energy identities for variational problems with obstacles

We discuss error identities for two classes of free boundary problems generated by obstacles. The identities suggest true forms of the respective error measures which consist of two parts: standard energy norm and a certain nonlinear measure. The latter measure controls (in a weak sense) approximation of free boundaries. Numerical tests confirm sharpness of error identities and show that in different examples one or another part of the error measure may be dominant.

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arXiv2017

Error identities for variational problems with obstacles

The paper is concerned with a class of nonlinear free boundary problems, which are usually solved by variational methods based on primal (or primal-dual) variational settings. We deduce and investigate special relations (error identities). They show that a certain nonlinear measure of the distance to the exact solution (specific for each problem) is equivalent to the respective duality gap, which minimization is a keystone of all variational numerical methods. Therefore, the identity defines the measure that contains maximal quantitative information on the quality of a numerical solution available through these methods. The measure has quadratic terms generated by the linear part of the differential operator and nonlinear terms associated with free boundaries. We obtain fully computable two sided bounds of this measure and show that they provide efficient estimates of the distance between the minimizer and any function from the corresponding energy space. Several examples show that for different minimization sequence the balance between different components of the overall error measure may be different and domination of nonlinear terms may indicate that coincidence sets are approximated incorrectly.

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arXiv2016

Computational modeling of magnetic hysteresis with thermal effects

We study computational behavior of a mesoscopic model describing temperature/external magnetic field-driven evolution of magnetization. Due to nonconvex anisotropy energy describing magnetic properties of a body, magnetization can develop fast spatial oscillations creating complicated microstructures. These microstructures are encoded in Young measures, their first moments then identify macroscopic magnetization. Our model assumes that changes of magnetization can contribute to dissipation and, consequently, to variations of the body temperature affecting the length of magnetization vectors. In the ferromagnetic state, minima of the anisotropic energy density depend on temperature and they tend to zero as we approach the so-called Curie temperature. This brings the specimen to a paramagnetic state. Such a thermo-magnetic model is fully discretized and tested on two-dimensional examples. Computational results qualitatively agree with experimental observations. The own MATLAB code used in our simulations is available for download.

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arXiv2016

A FEM approximation of a two-phase obstacle problem and its a posteriori error estimate

This paper is concerned with the two--phase obstacle problem, a type of a variational free boundary problem. We recall the basic estimates of Repin and Valdman (2015) and verify them numerically on two examples in two space dimensions. A solution algorithm is proposed for the construction of the finite element approximation to the two--phase obstacle problem. The algorithm is not based on the primal (convex and nondifferentiable) energy minimization problem but on a dual maximization problem formulated for Lagrange multipliers. The dual problem is equivalent to a quadratic programming problem with box constraints. The quality of approximations is measured by a functional a posteriori error estimate which provides a guaranteed upper bound of the difference of approximated and exact energies of the primal minimization problem. The majorant functional in the upper bound contains auxiliary variables and it is optimized with respect to them to provide a sharp upper bound. A space density of the nonlinear related part of the majorant functional serves as an indicator of the free boundary.

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