arXiv subjects

Lennard Kamenski

Publications and source records attributed to Lennard Kamenski.

18 recordsLinked to original sources

arXiv2025

Adaptive mesh refinement for electromagnetic simulation

We consider problems related to initial meshing and adaptive mesh refinement for the electromagnetic simulation of various structures. The quality of the initial mesh and the performance of the adaptive refinement are of great importance for the finite element solution of the Maxwell equations, since they directly affect the accuracy and the computational time. In this paper, we describe the complete meshing workflow, which allows the simulation of arbitrary structures. Test simulations confirm that the presented approach allows to reach the quality of the industrial simulation software.

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arXiv2022

Non-simplicial Delaunay meshing via approximation by radical partitions

We consider the construction of a polyhedral Delaunay partition as a limit of the sequence of power diagrams (radical partitions). The dual Voronoi diagram is obtained as a limit of the sequence of weighted Delaunay partitions. The problem is reduced to the construction of two dual convex polyhedra, inscribed and superscribed around a circular paraboloid, as a limit of the sequence of pairs of general dual convex polyhedra. The sequence of primal polyhedra should converge to the superscribed polyhedron and the sequence of the dual polyhedra converges to the inscribed polyhedron. We are interested in the case when the vertices of primal polyhedra can move or merge together, i.e., no new faces are allowed for dual polyhedra. These rules define the transformation of the set of initial spheres into the set of Delaunay spheres using radius variation and sphere movement and elimination. Existence theorems are still unavailable but we suggest a functional measuring the deviation of the convex polyhedron from the one inscribed into the paraboloid. It is the discrete Dirichlet functional for the power function which is a linear interpolant of the vertical distance of the dual vertices from the paraboloid. The functional's absolute minimizer is attained on the constant power field, meaning that the inscribed polyhedron can be obtained by a simple translation. This formulation of the functional for the dual surface is not quadratic since the unknowns are the vertices of the primal polyhedron, hence, the transformation of the set of spheres into Delaunay spheres is not unique. We concentrate on the experimental confirmation of the approach viability and put aside mesh quality problems. The zero value of the gradient of the proposed functional defines a manifold describing the evolution of Delaunay spheres. Hence, Delaunay-Voronoi meshes can be optimized using the manifold as a constraint.

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arXiv2021

Sharp bounds on the smallest eigenvalue of finite element equations with arbitrary meshes without regularity assumptions

A proof for the lower bound is provided for the smallest eigenvalue of finite element equations with arbitrary conforming simplicial meshes. The bound has a similar form as the one by Graham and McLean [SIAM J. Numer. Anal., 44 (2006), pp. 1487--1513] but doesn't require any mesh regularity assumptions, neither global nor local. In particular, it is valid for highly adaptive, anisotropic, or non-regular meshes without any restrictions. In three and more dimensions, the bound depends only on the number of degrees of freedom $N$ and the Hölder mean $M_{1-d/2} (\lvert \tildeω \rvert / \lvert ω_i \lvert)$ taken to the power $1-2/d$, $\lvert \tildeω \rvert$ and $\lvert ω_i \rvert$ denoting the average mesh patch volume and the volume of the patch corresponding to the $i^{\text{th}}$ mesh node, respectively. In two dimensions, the bound depends on the number of degrees of freedom $N$ and the logarithmic term $(1 + \lvert \ln (N \lvert ω_{\min} \rvert) \rvert)$, $\lvert ω_{\min} \rvert$ denoting the volume of the smallest patch. Provided numerical examples demonstrate that the bound is more accurate and less dependent on the mesh non-uniformity than the previously available bounds.

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arXiv2019

Conditioning of implicit Runge-Kutta integration for finite element approximation of linear diffusion equations on anisotropic meshes

The conditioning of implicit Runge-Kutta (RK) integration for linear finite element approximation of diffusion equations on general anisotropic meshes is investigated. Bounds are established for the condition number of the resulting linear system with and without diagonal preconditioning for the implicit Euler and general implicit RK methods. Two solution strategies are considered for the linear system resulting from general implicit RK integration: the simultaneous solution (the system is solved as a whole) and a successive solution which follows the commonly used implementation of implicit RK methods to first transform the system into smaller systems using the Jordan normal form of the RK matrix and then solve them successively. For the simultaneous solution in case of a positive semidefinite symmetric part of the RK coefficient matrix and for the successive solution it is shown that . If the smallest eigenvalue of the symmetric part of the RK coefficient matrix is negative and the simultaneous solution strategy is used, an upper bound on the time step is given so that the system matrix is positive definite. The obtained bounds for the condition number have explicit geometric interpretations and take the interplay between the diffusion matrix and the mesh geometry into full consideration. They show that there are three mesh-dependent factors that can affect the conditioning: the number of elements, the mesh nonuniformity measured in the Euclidean metric, and the mesh nonuniformity with respect to the inverse of the diffusion matrix. They also reveal that the preconditioning using the diagonal of the system matrix, the mass matrix, or the lumped mass matrix can effectively eliminate the effects of the mesh nonuniformity measured in the Euclidean metric. Numerical examples are given.

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arXiv2019

Stability of explicit Runge-Kutta methods for high order finite element approximation of linear parabolic equations

We study the stability of explicit Runge-Kutta methods for high order Lagrangian finite element approximation of linear parabolic equations and establish bounds on the largest eigenvalue of the system matrix which determines the largest permissible time step. A bound expressed in terms of the ratio of the diagonal entries of the stiffness and mass matrices is shown to be tight within a small factor which depends only on the dimension and the choice of the reference element and basis functions but is independent of the mesh or the coefficients of the initial-boundary value problem under consideration. Another bound, which is less tight and expressed in terms of mesh geometry, depends only on the number of mesh elements and the alignment of the mesh with the diffusion matrix. The results provide an insight into how the interplay between the mesh geometry and the diffusion matrix affects the stability of explicit integration schemes when applied to a high order finite element approximation of linear parabolic equations on general nonuniform meshes.

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arXiv2019

A new anisotropic mesh adaptation method based upon hierarchical a posteriori error estimates

A new anisotropic mesh adaptation strategy for finite element solution of elliptic differential equations is presented. It generates anisotropic adaptive meshes as quasi-uniform ones in some metric space, with the metric tensor being computed based on hierarchical a posteriori error estimates. A global hierarchical error estimate is employed in this study to obtain reliable directional information of the solution. Instead of solving the global error problem exactly, which is costly in general, we solve it iteratively using the symmetric Gauss--Seidel (GS) method. Numerical results show that a few GS iterations are sufficient for obtaining a reasonably good approximation to the error for use in anisotropic mesh adaptation. The new method is compared with several strategies using local error estimators or recovered Hessians. Numerical results are presented for a selection of test examples and a mathematical model for heat conduction in a thermal battery with large orthotropic jumps in the material coefficients.

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arXiv2019

Why do we need Voronoi cells and Delaunay meshes? Essential properties of the Voronoi finite volume method

Unlike other schemes that locally violate the essential stability properties of the analytic parabolic and elliptic problems, Voronoi finite volume methods (FVM) and boundary conforming Delaunay meshes provide good approximation of the geometry of a problem and are able to preserve the essential qualitative properties of the solution for any given resolution in space and time as well as changes in time scales of multiple orders of magnitude. This work provides a brief description of the essential and useful properties of the Voronoi FVM, application examples, and a motivation why Voronoi FVM deserve to be used more often in practice than they are currently.

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arXiv2017

Tetrahedral mesh improvement using moving mesh smoothing, lazy searching flips, and RBF surface reconstruction

Given a tetrahedral mesh and objective functionals measuring the mesh quality which take into account the shape, size, and orientation of the mesh elements, our aim is to improve the mesh quality as much as possible. In this paper, we combine the moving mesh smoothing, based on the integration of an ordinary differential equation coming from a given functional, with the lazy flip technique, a reversible edge removal algorithm to modify the mesh connectivity. Moreover, we utilize radial basis function (RBF) surface reconstruction to improve tetrahedral meshes with curved boundary surfaces. Numerical tests show that the combination of these techniques into a mesh improvement framework achieves results which are comparable and even better than the previously reported ones.

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arXiv2017

On the mesh nonsingularity of the moving mesh PDE method

The moving mesh PDE (MMPDE) method for variational mesh generation and adaptation is studied theoretically at the discrete level, in particular the nonsingularity of the obtained meshes. Meshing functionals are discretized geometrically and the MMPDE is formulated as a modified gradient system of the corresponding discrete functionals for the location of mesh vertices. It is shown that if the meshing functional satisfies a coercivity condition, then the mesh of the semi-discrete MMPDE is nonsingular for all time if it is nonsingular initially. Moreover, the altitudes and volumes of its elements are bounded below by positive numbers depending only on the number of elements, the metric tensor, and the initial mesh. Furthermore, the value of the discrete meshing functional is convergent as time increases, which can be used as a stopping criterion in computation. Finally, the mesh trajectory has limiting meshes which are critical points of the discrete functional. The convergence of the mesh trajectory can be guaranteed when a stronger condition is placed on the meshing functional. Two meshing functionals based on alignment and equidistribution are known to satisfy the coercivity condition. The results also hold for fully discrete systems of the MMPDE provided that the time step is sufficiently small and a numerical scheme preserving the property of monotonically decreasing energy is used for the temporal discretization of the semi-discrete MMPDE. Numerical examples are presented.

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arXiv2016

Stability of explicit one-step methods for P1-finite element approximation of linear diffusion equations on anisotropic meshes

We study the stability of explicit one-step integration schemes for the linear finite element approximation of linear parabolic equations. The derived bound on the largest permissible time step is tight for any mesh and any diffusion matrix within a factor of $2(d+1)$, where $d$ is the spatial dimension. Both full mass matrix and mass lumping are considered. The bound reveals that the stability condition is affected by two factors. The first one depends on the number of mesh elements and corresponds to the classic bound for the Laplace operator on a uniform mesh. The other factor reflects the effects of the interplay of the mesh geometry and the diffusion matrix. It is shown that it is not the mesh geometry itself but the mesh geometry in relation to the diffusion matrix that is crucial to the stability of explicit methods. When the mesh is uniform in the metric specified by the inverse of the diffusion matrix, the stability condition is comparable to the situation with the Laplace operator on a uniform mesh. Numerical results are presented to verify the theoretical findings.

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arXiv2015

A geometric discretization and a simple implementation for variational mesh generation and adaptation

We present a simple direct discretization for functionals used in the variational mesh generation and adaptation. Meshing functionals are discretized on simplicial meshes and the Jacobian matrix of the continuous coordinate transformation is approximated by the Jacobian matrices of affine mappings between elements. The advantage of this direct geometric discretization is that it preserves the basic geometric structure of the continuous functional, which is useful in preventing strong decoupling or loss of integral constraints satisfied by the functional. Moreover, the discretized functional is a function of the coordinates of mesh vertices and its derivatives have a simple analytical form, which allows a simple implementation of variational mesh generation and adaptation on computer. Since the variational mesh adaptation is the base for a number of adaptive moving mesh and mesh smoothing methods, the result in this work can be used to develop simple implementations of those methods. Numerical examples are given.

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arXiv2015

A comparative numerical study of meshing functionals for variational mesh adaptation

We present a comparative numerical study for three functionals used for variational mesh adaptation. One of them is a generalisation of Winslow's variable diffusion functional while the others are based on equidistribution and alignment. These functionals are known to have nice theoretical properties and work well for most mesh adaptation problems either as a stand-alone variational method or combined within the moving mesh framework. Their performance is investigated numerically in terms of equidistribution and alignment mesh quality measures. Numerical results in 2D and 3D are presented.

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arXiv2014

How a nonconvergent recovered Hessian works in mesh adaptation

Hessian recovery has been commonly used in mesh adaptation for obtaining the required magnitude and direction information of the solution error. Unfortunately, a recovered Hessian from a linear finite element approximation is nonconvergent in general as the mesh is refined. It has been observed numerically that adaptive meshes based on such a nonconvergent recovered Hessian can nevertheless lead to an optimal error in the finite element approximation. This also explains why Hessian recovery is still widely used despite its nonconvergence. In this paper we develop an error bound for the linear finite element solution of a general boundary value problem under a mild assumption on the closeness of the recovered Hessian to the exact one. Numerical results show that this closeness assumption is satisfied by the recovered Hessian obtained with commonly used Hessian recovery methods. Moreover, it is shown that the finite element error changes gradually with the closeness of the recovered Hessian. This provides an explanation on how a nonconvergent recovered Hessian works in mesh adaptation.

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arXiv2013

A study on the conditioning of finite element equations with arbitrary anisotropic meshes via a density function approach

The linear finite element approximation of a general linear diffusion problem with arbitrary anisotropic meshes is considered. The conditioning of the resultant stiffness matrix and the Jacobi preconditioned stiffness matrix is investigated using a density function approach proposed by Fried in 1973. It is shown that the approach can be made mathematically rigorous for general domains and used to develop bounds on the smallest eigenvalue and the condition number that are sharper than existing estimates in one and two dimensions and comparable in three and higher dimensions. The new results reveal that the mesh concentration near the boundary has less influence on the condition number than the mesh concentration in the interior of the domain. This is especially true for the Jacobi preconditioned system where the former has little or almost no influence on the condition number. Numerical examples are presented.

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arXiv2012

Conditioning of Finite Element Equations with Arbitrary Anisotropic Meshes

Bounds are developed for the condition number of the linear finite element equations of an anisotropic diffusion problem with arbitrary meshes. They depend on three factors. The first, factor proportional to a power of the number of mesh elements, represents the condition number of the linear finite element equations for the Laplacian operator on a uniform mesh. The other two factors arise from the mesh nonuniformity viewed in the Euclidean metric and in the metric defined by the diffusion matrix. The new bounds reveal that the conditioning of the finite element equations with adaptive anisotropic meshes is much better than what is commonly feared. Diagonal scaling for the linear system and its effects on the conditioning are also studied. It is shown that the Jacobi preconditioning, which is an optimal diagonal scaling for a symmetric positive definite sparse matrix, can eliminate the effects of mesh nonuniformity viewed in the Euclidean metric and reduce those effects of the mesh viewed in the metric defined by the diffusion matrix. Tight bounds on the extreme eigenvalues of the stiffness and mass matrices are obtained. Numerical examples are given.

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arXiv2012

Adaptive finite elements with anisotropic meshes

The paper presents a numerical study for the finite element method with anisotropic meshes. We compare the accuracy of the numerical solutions on quasi-uniform, isotropic, and anisotropic meshes for a test problem which combines several difficulties of a corner singularity, a peak, a boundary layer, and a wavefront. Numerical experiment clearly shows the advantage of anisotropic mesh adaptation. The conditioning of the resulting linear equation system is addressed as well. In particular, it is shown that the conditioning with adaptive anisotropic meshes is not as bad as generally assumed.

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arXiv2012

A Study on Using Hierarchical Basis Error Estimates in Anisotropic Mesh Adaptation for the Finite Element Method

A common approach for generating an anisotropic mesh is the M-uniform mesh approach where an adaptive mesh is generated as a uniform one in the metric specified by a given tensor M. A key component is the determination of an appropriate metric which is often based on some type of Hessian recovery. Recently, the use of a global hierarchical basis error estimator was proposed for the development of an anisotropic metric tensor for the adaptive finite element solution. This study discusses the use of this method for a selection of different applications. Numerical results show that the method performs well and is comparable with existing metric tensors based on Hessian recovery. Also, it can provide even better adaptation to the solution if applied to problems with gradient jumps and steep boundary layers. For the Poisson problem in a domain with a corner singularity, the new method provides meshes that are fully comparable to the theoretically optimal meshes.

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arXiv2010

Anisotropic Mesh Adaptation for Variational Problems Using Error Estimation Based on Hierarchical Bases

Anisotropic mesh adaptation has been successfully applied to the numerical solution of partial differential equations but little considered for variational problems. In this paper, we investigate the use of a global hierarchical basis error estimator for the development of an anisotropic metric tensor needed for the adaptive finite element solution of variational problems. The new metric tensor is completely a~posteriori and based on residual, edge jumps and the hierarchical basis error estimator. Numerical results show that it performs comparable with existing metric tensors based on Hessian recovery. A few sweeps of the symmetric Gauß-Seidel iteration for solving the global error problem prove sufficient to provide directional information necessary for successful mesh adaptation. .

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