arXiv subjects

René R. Hiemstra

Publications and source records attributed to René R. Hiemstra.

10 recordsLinked to original sources

arXiv2026

Wigner-Eckart Factorization of the Polyatomic Boltzmann Collision Operator

We extend the Wigner-Eckart factorization of the spectral Boltzmann collision operator to polyatomic gases with continuous internal energy. Because internal energies are invariant under spatial rotations, the SO(3) reduction survives the Borgnakke-Larsen energy exchange, and the twelve-dimensional collision integral collapses onto a nine-dimensional kinematic core. The core splits into a sparse geometric tensor, evaluated exactly, and a dense physical tensor, integrated by singularity-resolving Gauss rules with an auxiliary Laplace representation of the fractional energy couplings. The quadrature attains near machine precision at the fractional exponents of real gases. The collision invariants are embedded exactly, preserving the translational-internal energy exchange. The factorization compresses the operator by three to nearly four orders of magnitude and accelerates its evaluation 40-fold over dense formulations. The method is validated against the exact monatomic limit, Landau-Teller relaxation, and an analytic frozen-channel Prandtl number, and it matches a published calibration of the same kernel for N2, CO, and H2.

math.NA↗

arXiv2026

Isogeometric Analysis for Explicit Wave Propagation in Poroelastic Media

For higher-order discretizations of explicit dynamics problems, Isogeometric Analysis (IGA) has several favorable properties as compared to classical Finite Element Analysis (FEA). While FEA produces spurious modes at orders beyond linear, this is not the case for IGA. Consequently, fewer degrees of freedom are required for comparable accuracy, larger timesteps can be taken, and the method is more robust for nonlinear problems. If outlier modes are removed, the timestep even becomes virtually independent of the order. In this paper, we investigate how these advantages apply to the poroelastic continuum model. We consider both a primal formulation, wherein our variables are the displacement of the matrix material, the fluid displacement, and the pressure, as well as a reduced form wherein the pressure is eliminated. For our discretizations, we employ divergence-conforming spline spaces. Conforming spline spaces for the fluid displacement ensure inf-sup stability for the primal form, as well as a correct null space in the reduced form. Furthermore, we prove and demonstrate that the two formulations coincide when both displacements are discretized with conforming spline spaces. Through spectral analysis, we find that the aforementioned benefits of IGA do carry over directly to the context of poroelasticity. In 1D, we split the discrete spectrum into fast and slow waves. When normalized against an analytical solution, each of these sub-spectra closely resembles results known in elasticity. Consequently, when poroelasticity is discretized with outlier-free IGA, the timestep is essentially independent of the order. We show this timestep scaling in 2D as well.

math.NA↗

arXiv2026

Wigner-Eckart Factorization of the Spectral Boltzmann Collision Operator

We reduce the eight-dimensional weak form of the bilinear Boltzmann collision operator to a five-dimensional kinematic core by rigidly rotating the laboratory frame to align with the colliding pair and integrating over the $\mathrm{SO}(3)$ rotation group. This reduction yields an exact Wigner--Eckart factorization within a spectral Galerkin framework of associated Laguerre polynomials and spherical harmonics. The decomposition decouples the angular geometry from the scattering physics. The former, represented by Clebsch--Gordan coefficients, is evaluated exactly, while the latter is evaluated to machine precision by a spectrally convergent singular quadrature strategy. By explicitly zeroing specific entries, the macroscopic collision invariants are embedded without approximation. Cache-optimized contractions deliver up to a 37-fold single-core speedup and a 1000-fold memory reduction over standard dense Cartesian formulations. The approach is validated against analytical solutions for Maxwell molecules and infinite-order Chapman--Enskog viscosity coefficients for hard spheres.

math.NA↗

arXiv2025

A locking-free isogeometric thin shell formulation based on higher order accurate diagonalized strain projection via approximate dual splines

We present a novel isogeometric discretization approach for the Kirchhoff-Love shell formulation based on the Hellinger-Reissner variational principle. For mitigating membrane locking, we discretize the independent strains with spline basis functions that are one degree lower than those used for the displacements. To enable computationally efficient condensation of the independent strains, we first discretize the variations of the independent strains with approximate dual splines to obtain a projection matrix that is close to a diagonal matrix. We then diagonalize this strain projection matrix via row-sum lumping. Due to this diagonalization, the static condensation of the independent strain fields becomes computationally inexpensive, as no matrix needs to be inverted. At the same time, our approach maintains higher-order accuracy at optimal rates of convergence. We illustrate the numerical properties and the performance of our approach through numerical benchmarks, including a curved Euler-Bernoulli beam and the examples of the shell obstacle course.

cs.CE↗

arXiv2024

Nonlinear dynamic analysis of shear- and torsion-free rods using isogeometric discretization and outlier removal

In this paper, we present a discrete formulation of nonlinear shear- and torsion-free rods introduced by Gebhardt and Romero in [20] that uses isogeometric discretization and robust time integration. Omitting the director as an independent variable field, we reduce the number of degrees of freedom and obtain discrete solutions in multiple copies of the Euclidean space (R^3), which is larger than the corresponding multiple copies of the manifold (R^3 x S^2) obtained with standard Hermite finite elements. For implicit time integration, we choose the same integration scheme as Gebhardt and Romero in [20] that is a hybrid form of the midpoint and the trapezoidal rules. In addition, we apply a recently introduced approach for outlier removal by Hiemstra et al. [26] that reduces high-frequency content in the response without affecting the accuracy, ensuring robustness of our nonlinear discrete formulation. We illustrate the efficiency of our nonlinear discrete formulation for static and transient rods under different loading conditions, demonstrating good accuracy in space, time and the frequency domain. Our numerical example coincides with a relevant application case, the simulation of mooring lines.

cs.CE↗

arXiv2023

Towards higher-order accurate mass lumping in explicit isogeometric analysis for structural dynamics

We present a mass lumping approach based on an isogeometric Petrov-Galerkin method that preserves higher-order spatial accuracy in explicit dynamics calculations irrespective of the polynomial degree of the spline approximation. To discretize the test function space, our method uses an approximate dual basis, whose functions are smooth, have local support and satisfy approximate bi-orthogonality with respect to a trial space of B-splines. The resulting mass matrix is ``close'' to the identity matrix. Specifically, a lumped version of this mass matrix preserves all relevant polynomials when utilized in a Galerkin projection. Consequently, the mass matrix can be lumped (via row-sum lumping) without compromising spatial accuracy in explicit dynamics calculations. We address the imposition of Dirichlet boundary conditions and the preservation of approximate bi-orthogonality under geometric mappings. In addition, we establish a link between the exact dual and approximate dual basis functions via an iterative algorithm that improves the approximate dual basis towards exact bi-orthogonality. We demonstrate the performance of our higher-order accurate mass lumping approach via convergence studies and spectral analyses of discretized beam, plate and shell models.

cs.CE↗

arXiv2022

A variational approach based on perturbed eigenvalue analysis for improving spectral properties of isogeometric multipatch discretizations

A key advantage of isogeometric discretizations is their accurate and well-behaved eigenfrequencies and eigenmodes. For degree two and higher, however, optical branches of spurious outlier frequencies and modes may appear due to boundaries or reduced continuity at patch interfaces. In this paper, we introduce a variational approach based on perturbed eigenvalue analysis that eliminates outlier frequencies without negatively affecting the accuracy in the remainder of the spectrum and modes. We then propose a pragmatic iterative procedure that estimates the perturbation parameters in such a way that the outlier frequencies are effectively reduced. We demonstrate that our approach allows for a much larger critical time-step size in explicit dynamics calculations. In addition, we show that the critical time-step size obtained with the proposed approach does not depend on the polynomial degree of spline basis functions.

math.NA↗

arXiv2022

Variationally consistent mass scaling for explicit time-integration schemes of lower- and higher-order finite element methods

In this paper, we propose a variationally consistent technique for decreasing the maximum eigenfrequencies of structural dynamics related finite element formulations. Our approach is based on adding a symmetric positive-definite term to the mass matrix that follows from the integral of the traction jump across element boundaries. The added term is weighted by a small factor, for which we derive a suitable, and simple, element-local parameter choice. For linear problems, we show that our mass-scaling method produces no adverse effects in terms of spatial accuracy and orders of convergence. We illustrate these properties in one, two and three spatial dimension, for quadrilateral elements and triangular elements, and for up to fourth order polynomials basis functions. To extend the method to non-linear problems, we introduce a linear approximation and show that a sizeable increase in critical time-step size can be achieved while only causing minor (even beneficial) influences on the dynamic response.

math.NA↗

arXiv2021

Leveraging spectral analysis to elucidate membrane locking and unlocking in isogeometric finite element formulations of the curved Euler-Bernoulli beam

In this paper, we take a fresh look at using spectral analysis for assessing locking phenomena in finite element formulations. We propose to "measure" locking by comparing the difference between eigenvalue and mode error curves computed on coarse meshes with "asymptotic" error curves computed on "overkill" meshes, both plotted with respect to the normalized mode number. To demonstrate the intimate relation between membrane locking and spectral accuracy, we focus on the example of a circular ring discretized with isogeometric curved Euler-Bernoulli beam elements. We show that the transverse-displacement-dominating modes are locking-prone, while the circumferential-displacement-dominating modes are naturally locking-free. We use eigenvalue and mode errors to assess five isogeometric finite element formulations in terms of their locking-related efficiency: the displacement-based formulation with full and reduced integration and three locking-free formulations based on the B-bar, discrete strain gap and Hellinger-Reissner methods. Our study shows that spectral analysis uncovers locking-related effects across the spectrum of eigenvalues and eigenmodes, rigorously characterizing membrane locking in the displacement-based formulation and unlocking in the locking-free formulations.

cs.CE↗

arXiv2020

The Quad Layout Immersion: A Mathematically Equivalent Representation of a Surface Quadrilateral Layout

Quadrilateral layouts on surfaces are valuable in texture mapping, and essential in generation of quadrilateral meshes and in fitting splines. Previous work has characterized such layouts as a special metric on a surface or as a meromorphic quartic differential with finite trajectories. In this work, a surface quadrilateral layout is alternatively characterized as a special immersion of a cut representation of the surface into the Euclidean plane. We call this a quad layout immersion. This characterization, while posed in smooth topology, naturally generalizes to piecewise-linear representations. As such, it mathematically describes and generalizes integer grid maps, which are common in computer graphics settings. Finally, the utility of the representation is demonstrated by computationally extracting quadrilateral layouts on surfaces of interest.

cs.CG↗