arXiv subjects

Jan Valdman

Publications and source records attributed to Jan Valdman.

At least 19 recordsLinked to original sources

arXiv2026

Physics-Informed Neural Network for Diffusion-Reaction Problems with Dead-Core Formation in Catalyst Slabs

This work investigates a nonlinear two-point boundary value problem arising in diffusion--reaction processes in catalyst slabs with power-law kinetics and fractional reaction order. For sufficiently large Thiele modulus, the solution develops a dead-core region separated from the active region by an unknown free boundary. We propose a structured Physics-Informed Neural Network (PINN) framework that incorporates the asymptotic behavior at the dead-core interface into a hard-constrained trial solution and treats the interface location as a trainable parameter. The concentration profile and free boundary are therefore approximated simultaneously without explicit interface tracking or penalty-based enforcement of the interface conditions. The method is validated against the exact solution for power-law kinetics and a high-precision numerical shooting method. Numerical experiments covering near-critical and strongly supercritical regimes demonstrate accurate recovery of both the concentration profile and dead-core location, together with robustness to random initialization and collocation sampling. While classical shooting is more efficient for the present one-dimensional benchmark, the proposed formulation provides a flexible framework for extensions to multidimensional geometries and problems for which analytical solutions are unavailable.

math.NA↗

arXiv2026

Numerical simulations for two-dimensional reaction-diffusion problems with formation of multiple dead zones

The paper deals with dead-core solutions to an isothermal reaction-diffusion problem with power-law kinetics for a single reaction that takes place in a chemical reactor represented by a bounded domain in two dimensions. The model boundary value problem for the stationary non-linear diffusion-reaction equation is solved numerically using an appropriate time-marching method. The spatial discretization is based on the lumped finite element method for piecewise linear functions. The effects of the reaction order and Thiele modulus on the concentration profiles and the size of dead zones are investigated numerically. The paper demonstrates that the formation of multiple dead zones is possible for particular reactor geometries.

math.NA↗

arXiv2026

Sharp mean Hadamard inequalities and polyconvex integrands that give rise to convex functionals

We investigate several instances of the Hadamard inequality in the mean in two dimensions. As a consequence, we prove the uniqueness of minimizers of an integral functional with a polyconvex integrand, subject to mixed Dirichlet and Neumann boundary conditions. The theoretical findings are complemented by computational experiments that illustrate the behavior of the minimizers.

math.AP↗

arXiv2026

Numerical Algorithms for Partially Segregated Elliptic Systems

We develop numerical methods for elliptic systems governed by partial segregation constraints, in which three nonnegative components are required to have a vanishing pointwise product throughout the domain. This constraint enforces that at least one component must be zero at every spatial location, leading to a highly nonconvex admissible set that prevents the use of standard convex optimization techniques. We propose two complementary computational frameworks. The first is a strong-competition penalty method, solved via damped Gauss-Seidel/Picard iterations with a continuation strategy on the penalty parameter, for which we establish compactness results, Lipschitz estimates, and interior exponential improvement in the strong-competition regime. The second is a projected gradient method, together with an accelerated variant, that exploits an explicit pointwise projection onto the three-phase segregation set. Numerical experiments on a suite of benchmark boundary configurations confirm that both algorithms resolve segregated phase patterns.

math.NA↗

arXiv2025

Vectorized 3D mesh refinement and implementation of primal hybrid FEM in MATLAB

In this article, we introduce a Face-to-Tetrahedron connectivity in MATLAB together with a vectorized 3D uniform mesh refinement technique. We introduce a MATLAB vectorized assembly of 3D lowest-order primal hybrid finite element matrices for a second-order elliptic problem. We introduce a parallel solver and a vectorized Schur complement solver to solve the associated linear problem. The numerical results illustrate the software's runtime performance.

math.NA↗

arXiv2025

On Josephy-Halley method for generalized equations

We extend the classical third-order Halley iteration to the setting of generalized equations of the form \[ 0 \in f(x) + F(x), \] where $f\colon X\longrightarrow Y$ is twice continuously Fr\'echet-differentiable on Banach spaces and $F\colon X\tto Y$ is a set-valued mapping with closed graph. Building on predictor-corrector framework, our scheme first solves a partially linearized inclusion to produce a predictor $u_{k+1}$, then incorporates second-order information in a Halley-type corrector step to obtain $x_{k+1}$. Under metric regularity of the linearization at a reference solution and H\"older continuity of $f''$, we prove that the iterates converge locally with order $2+p$ (cubically when $p=1$). Moreover, by constructing a suitable scalar majorant function we derive semilocal Kantorovich-type conditions guaranteeing well-definedness and R-cubic convergence from an explicit neighbourhood of the initial guess. Numerical experiments-including one- and two-dimensional test problems confirm the theoretical convergence rates and illustrate the efficiency of the Josephy-Halley method compared to its Josephy-Newton counterpart.

math.NA↗

arXiv2025

Finite-strain constitutive model for shape memory alloys formulated in the logarithmic strain space

This work presents a finite-strain version of an established three-dimensional constitutive model for polycrystalline shape memory alloys (SMA) that is able to account for the large deformations and rotations that SMA components may undergo. The model is constructed by applying the logarithmic strain space approach to the original small-strain model, which was formulated within the Generalized Standard Materials framework and features a refined dissipation (rate) function. Additionally, the free energy function is augmented to be more versatile in capturing the transformation kinetics. The model is implemented into finite element software. To demonstrate the model performance and validate the implementation, material parameters are fitted to the experimental data of two SMA, and two computational simulations of SMA components are conducted. The applied approach is highly flexible from the perspective of the future incorporation of other phenomena, e.g., irreversibility associated with plasticity, into the model.

cond-mat.mtrl-sci↗

arXiv2024

New applications of Hadamard-in-the-mean inequalities to incompressible variational problems

Let $\mathbb{D}(u)$ be the Dirichlet energy of a map $u$ belonging to the Sobolev space $H^1_{u_0}(\Omega;\mathbb{R}^2)$ and let $A$ be a subclass of $H^1_{u_0}(\Omega;\mathbb{R}^2)$ whose members are subject to the constraint $\det \nabla u = g$ a.e. for a given $g$, together with some boundary data $u_0$. We develop a technique that, when applicable, enables us to characterize the global minimizer of $\mathbb{D}(u)$ in $A$ as the unique global minimizer of the associated functional $F(u):=\mathbb{D}(u)+ \int_{\Omega} f(x) \, \det \nabla u(x) \, dx$ in the free class $H^1_{u_0}(\Omega;\mathbb{R}^2)$. A key ingredient is the mean coercivity of $F(\varphi)$ on $H^1_0(\Omega;\mathbb{R}^2)$, which condition holds provided the `pressure' $f \in L^{\infty}(\Omega)$ is `tuned' according to the procedure set out in \cite{BKV23}. The explicit examples to which our technique applies can be interpreted as solving the sort of constrained minimization problem that typically arises in incompressible nonlinear elasticity theory.

math.AP↗

arXiv2024

On semismooth$^*$ path-following method and uniformity of strong metric subregularity at/around the reference point

This paper investigates a path-following method inspired by the semismooth$^*$ approach for solving algebraic inclusions, with a primary emphasis on the role of uniform subregularity. Uniform subregularity is crucial for ensuring the robustness and stability of path-following methods, as it provides a framework to uniformly control the distance between the input and the solution set across a continuous path. We explore the problem of finding a mapping $ x: \mathbb{R} \longrightarrow \mathbb{R}^n $ that satisfies $ 0 \in F(t, x(t)) $ for each $ t \in [0, T] $, where $ F $ is a set-valued mapping from $ \mathbb{R} \times \mathbb{R}^n $ to $ \mathbb{R}^n $. The paper discusses two approaches: the first considers mappings with uniform semismooth$^*$ properties along continuous paths, leading to a consistent grid error throughout the interval, while the second examines mappings exhibiting pointwise semismooth$^*$ properties at individual points along the path. The uniform strong subregularity framework is integrated into these approaches to strengthen the stability of solution trajectories and improve algorithmic convergence.

math.OC↗

arXiv2024

Minimization of Nonlinear Energies in Python Using FEM and Automatic Differentiation Tools

This contribution examines the capabilities of the Python ecosystem to solve nonlinear energy minimization problems, with a particular focus on transitioning from traditional MATLAB methods to Python's advanced computational tools, such as automatic differentiation. We demonstrate Python's streamlined approach to minimizing nonlinear energies by analyzing three problem benchmarks - the p-Laplacian, the Ginzburg-Landau model, and the Neo-Hookean hyperelasticity. This approach merely requires the provision of the energy functional itself, making it a simple and efficient way to solve this category of problems. The results show that the implementation is about ten times faster than the MATLAB implementation for large-scale problems. Our findings highlight Python's efficiency and ease of use in scientific computing, establishing it as a preferable choice for implementing sophisticated mathematical models and accelerating the development of numerical simulations.

cs.MS↗

arXiv2024

On a vectorized basic linear algebra package for prototyping codes in MATLAB

When writing high-performance code for numerical computation in a scripting language like MATLAB, it is crucial to have the operations in a large for-loop vectorized. If not, the code becomes too slow to use, even for a moderately large problem. However, in the process of vectorizing, the code often loses its original structure and becomes less readable. This is particularly true in the case of a finite element implementation, even though finite element methods are inherently structured. A basic remedy to this is the separation of the vectorization part from the mathematics part of the code, which is easily achieved through building the code on top of the basic linear algebra subprograms that are already vectorized codes, an idea that has been used in a series of papers over the last fifteen years, developing codes that are fast and still structured and readable. We discuss the vectorized basic linear algebra package and introduce a formalism using multi-linear algebra to explain and define formally the functions in the package, as well as MATLAB pagetime functions. We provide examples from computations of varying complexity, including the computation of normal vectors, volumes, and finite element methods. Benchmarking shows that we also get fast computations. Using the library, we can write codes that closely follow our mathematical thinking, making writing, following, reusing, and extending the code easier.

cs.MS↗

arXiv2024

Hadamard's inequality in the mean

Let $Q$ be a Lipschitz domain in $\mathbb{R}^n$ and let $f \in L^{\infty}(Q)$. We investigate conditions under which the functional $$I_n(φ)=\int_Q |\nabla φ|^n+ f(x)\,\mathrm{det} \nabla φ\, \mathrm{d}x $$ obeys $I_n \geq 0$ for all $φ\in W_0^{1,n}(Q,\mathbb{R}^n)$, an inequality that we refer to as Hadamard-in-the-mean, or (HIM). We prove that there are piecewise constant $f$ such that (HIM) holds and is strictly stronger than the best possible inequality that can be derived using the Hadamard inequality $n^{\frac{n}{2}}|\det A|\leq |A|^n$ alone. When $f$ takes just two values, we find that (HIM) holds if and only if the variation of $f$ in $Q$ is at most $2n^{\frac{n}{2}}$. For more general $f$, we show that (i) it is both the geometry of the `jump sets' as well as the sizes of the `jumps' that determine whether (HIM) holds and (ii) the variation of $f$ can be made to exceed $2n^{\frac{n}{2}}$, provided $f$ is suitably chosen. Specifically, in the planar case $n=2$ we divide $Q$ into three regions $\{f=0\}$ and $\{f=\pm c\}$, and prove that as long as $\{f=0\}$ `insulates' $\{f= c\}$ from $\{f= -c\}$ sufficiently, there is $c>2$ such that (HIM) holds. Perhaps surprisingly, (HIM) can hold even when the insulation region $\{f=0\}$ enables the sets $\{f=\pm c\}$ to meet in a point. As part of our analysis, and in the spirit of the work of Mielke and Sprenger (1998), we give new examples of functions that are quasiconvex at the boundary.

math.AP↗

arXiv2024

Vectorized implementation of primal hybrid FEM in MATLAB

We present efficient MATLAB implementations of the lowest-order primal hybrid finite element method (FEM) for linear second-order elliptic and parabolic problems with mixed boundary conditions in two spatial dimensions. We employ the Crank-Nicolson finite difference scheme for the complete discrete setup of the parabolic problem. All the codes presented are fully vectorized using matrix-wise array operations. Numerical experiments are conducted to show the performance of the software.

math.NA↗

arXiv2023

Minimization of energy functionals via FEM: implementation of hp-FEM

Many problems in science and engineering can be rigorously recast into minimizing a suitable energy functional. We have been developing efficient and flexible solution strategies to tackle various minimization problems by employing finite element discretization with P1 triangular elements [1,2]. An extension to rectangular hp-finite elements in 2D is introduced in this contribution.

cs.CE↗

arXiv2023

Surface penalization of self-interpenetration in linear and nonlinear elasticity

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.

math.AP↗

arXiv2022

Elastoplastic deformations of layered structures

We formulate a large-strain model of single-slip crystal elastoplasticity in the framework of energetic solutions. Numerical performance of the model is compared with lab experiments on the compression of a stack of note papers.

math.AP↗

arXiv2022

MATLAB implementation of hp finite elements on rectangles

A simple MATLAB implementation of hierarchical shape functions on 2D rectangles is explained and available for download. Global shape functions are ordered for a given polynomial degree according to the indices of the nodes, edges, or elements to which they belong. For a uniform p-refinement, the hierarchical structure enables an effective assembly of mass and stiffness matrices. A solution of a boundary value problem is approximated for various levels of uniform h and p refinements.

cs.MS↗

arXiv2022

Fast MATLAB evaluation of nonlinear energies using FEM in 2D and 3D: nodal elements

Nonlinear energy functionals appearing in the calculus of variations can be discretized by the finite element (FE) method and formulated as a sum of energy contributions from local elements. A fast evaluation of energy functionals containing the first order gradient terms is a central part of this contribution. We describe a vectorized implementation using the simplest linear nodal (P1) elements in which all energy contributions are evaluated all at once without the loop over triangular or tetrahedral elements. Furthermore, in connection to the first-order optimization methods, the discrete gradient of energy functional is assembled in a way that the gradient components are evaluated over all degrees of freedom all at once. The key ingredient is the vectorization of exact or approximate energy gradients over nodal patches. It leads to a time-efficient implementation at higher memory-cost. Provided codes in MATLAB related to 2D/3D hyperelasticity and 2D p-Laplacian problem are available for download and structured in a way it can be easily extended to other types of vector or scalar forms of energies.

cs.MS↗