arXiv subjects

Hauke Gravenkamp

Publications and source records attributed to Hauke Gravenkamp.

13 recordsLinked to original sources

arXiv2026

Computing critical velocities in waveguides via multiparameter eigenvalue problems

In waveguide dynamics and moving-load problems (e.g., high-speed trains), critical velocities indicate the onset of strong vibration amplification. In systems that are invariant in the direction of motion, these velocities can be identified from dispersion relations as points where the phase and group velocities of a propagating mode coincide. Finding such points indirectly by tracing dispersion curves can be cumbersome and potentially unreliable for multimodal systems with complex branch interactions. We present a direct method for computing critical velocities in such scenarios, specifically in the context of semi-analytical methods. Starting from a polynomial parameter-dependent eigenvalue problem for the wavenumber-frequency relation, incorporating the additional condition of equal phase and group velocities yields a singular polynomial multiparameter eigenvalue problem that can be linearized and solved using established algorithms. The proposed approach enables the simultaneous computation of all critical points without requiring the tracing of dispersion curves. Its performance is demonstrated by several benchmark problems, confirming the accurate and robust identification of critical velocities.

math.NA↗

arXiv2025

A modified Levenberg-Marquardt method for estimating the elastic material parameters of polymer waveguides using residuals between autocorrelated frequency responses

In this contribution, we address the estimation of the frequency-dependent elastic parameters of polymers in the ultrasound range, which is formulated as an inverse problem. This inverse problem is implemented as a nonlinear regression-type optimization problem, in which the simulation signals are fitted to the measurement signals. These signals consist of displacement responses in waveguides, focusing on hollow cylindrical geometries to enhance the simulation efficiency. To accelerate the optimization and reduce the number of model evaluations and wait times, we propose two novel methods. First, we introduce an adaptation of the Levenberg-Marquardt method derived from a geometrical interpretation of the least-squares optimization problem. Second, we introduce an improved objective function based on the autocorrelated envelopes of the measurement and simulation signals. Given that this study primarily relies on simulation data to quantify optimization convergence, we aggregate the expected ranges of realistic material parameters and derive their distributions to ensure the reproducibility of optimizations with proper measurements. We demonstrate the effectiveness of our objective function modification and step adaptation for various materials with isotropic material symmetry by comparing them with a state-of-the-art optimization method. In all cases, our method reduces the total number of model evaluations, thereby shortening the time to identify the material parameters.

cs.CE↗

arXiv2025

Computing leaky waves in semi-analytical waveguide models by exponential residual relaxation

Semi-analytical methods for the modeling of guided waves in structures of constant cross-section lead to frequency-dependent polynomial eigenvalue problems for the wavenumbers and mode shapes. Solving these eigenvalue problems for a range of frequencies results in continuous eigencurves that are of relevance in practical applications of ultrasonic measurement systems. Recent research has shown that eigencurves of parameter-dependent eigenvalue problems can alternatively be computed as solutions of a system of ordinary differential equations, which are obtained by postulating an exponentially decaying residual of a modal solution. This general concept for solving parameter-dependent matrix equations is, in this context, known as Zeroing Neural Networks or Zhang Neural Networks (ZNN). We exploit this idea to develop an efficient method for computing the dispersion curves of plate structures coupled to unbounded solid or fluid media. In these scenarios, the alternative formulation is particularly useful since the boundary conditions give rise to nonlinear terms that severely hinder the application of traditional solvers.

physics.class-ph↗

arXiv2025

A posteriori error estimates for the finite element approximation of the convection-diffusion-reaction equation based on the variational multiscale concept

In this study, we employ the variational multiscale (VMS) concept to develop a posteriori error estimates for the stationary convection-diffusion-reaction equation. The variational multiscale method is based on splitting the continuous part of the problem into a resolved scale (coarse scale) and an unresolved scale (fine scale). The unresolved scale (also known as the sub-grid scale) is modeled by choosing it proportional to the component of the residual orthogonal to the finite element space, leading to the orthogonal sub-grid scale (OSGS) method. The idea is then to use the modeled sub-grid scale as an error estimator, considering its contribution in the element interiors and on the edges. We present the results of the a priori analysis and two different strategies for the a posteriori error analysis for the OSGS method. Our proposal is to use a scaled norm of the sub-grid scales as an a posteriori error estimate in the so-called stabilized norm of the problem. This norm has control over the convective term, which is necessary for convection-dominated problems. Numerical examples show the reliable performance of the proposed error estimator compared to other error estimators belonging to the variational multiscale family.

math.NA↗

arXiv2025

A variational multiscale approach to goal-oriented error estimation in finite element analysis of convection-diffusion-reaction equation problems

This paper presents a goal-oriented a posteriori error estimation framework for linear functionals in the stabilized finite element discretization of the stationary convection-diffusion-reaction (CDR) equation. The theoretical framework for error estimation is based on the variational multiscale (VMS) concept, where the solution is decomposed into resolved (finite element) and unresolved (sub-grid) scales. In this work, we propose an orthogonal sub-grid scale (OSGS) method for a goal-oriented error estimation in VMS discretizations. In the OSGS approach, the space of the sub-grid scales (SGSs) is orthogonal to the finite element space. The error is estimated in the quantity of interest, given by the linear functional $Q(u)$ of the unknown $u$. If the SGS $u'$ is estimated, the error in the quantity of interest can be approximated by $Q(u')$. Our approach is compared with a duality-based a posteriori error estimation method, which requires the solution of an additional auxiliary problem. The results indicate that both methods yield similar error estimates, whereas the VMS-based explicit approach is computationally less expensive than the duality-based implicit approach. Numerical tests demonstrated the effectiveness of our proposed error estimation techniques in terms of the quantity of interest functionals.

math.NA↗

arXiv2024

A Sylvester equation approach for the computation of zero-group-velocity points in waveguides

Eigenvalues of parameter-dependent quadratic eigenvalue problems form eigencurves. The critical points on these curves, where the derivative vanishes, are of practical interest. A particular example is found in the dispersion curves of elastic waveguides, where such points are called zero-group-velocity (ZGV) points. Recently, it was revealed that the problem of computing ZGV points can be modeled as a multiparameter eigenvalue problem (MEP), and several numerical methods were devised. Due to their complexity, these methods are feasible only for problems involving small matrices. In this paper, we improve the efficiency of these methods by exploiting the link to the Sylvester equation. This approach enables the computation of ZGV points for problems with much larger matrices, such as multi-layered plates and three-dimensional structures of complex cross-sections.

math.NA↗

arXiv2024

Computation of leaky waves in layered structures coupled to unbounded media by exploiting multiparameter eigenvalue problems

We present a semi-analytical approach to compute quasi-guided elastic wave modes in horizontally layered structures radiating into unbounded fluid or solid media. This problem is of relevance, e.g., for the simulation of guided ultrasound in embedded plate structures or seismic waves in soil layers over an elastic half-space. We employ a semi-analytical formulation to describe the layers, thus discretizing the thickness direction by means of finite elements. For a free layer, this technique leads to a well-known quadratic eigenvalue problem for the mode shapes and corresponding horizontal wavenumbers. Rigorously incorporating the coupling conditions to account for the adjacent half-spaces gives rise to additional terms that are nonlinear in the wavenumber. We show that the resulting nonlinear eigenvalue problem can be cast in the form of a multiparameter eigenvalue problem whose solutions represent the wave numbers in the plate and in the half-spaces. The multiparameter eigenvalue problem is solved numerically using recently developed algorithms.

math.NA↗

arXiv2023

Automatic 3D modeling by combining SBFEM and transfinite element shape functions

The scaled boundary finite element method (SBFEM) has recently been employed as an efficient means to model three-dimensional structures, in particular when the geometry is provided as a voxel-based image. To this end, an octree decomposition of the computational domain is deployed and each cubic cell is treated as an SBFEM subdomain. The surfaces of each subdomain are discretized in the finite element sense. We improve on this idea by combining the semi-analytical concept of the SBFEM with certain transition elements on the subdomains' surfaces. Thus, we avoid the triangulation of surfaces employed in previous works and consequently reduce the number of surface elements and degrees of freedom. In addition, these discretizations allow coupling elements of arbitrary order such that local p-refinement can be achieved straightforwardly.

math.NA↗

arXiv2023

Stabilized finite elements for the solution of the Reynolds equation considering cavitation

The Reynolds equation, combined with the Elrod algorithm for including the effect of cavitation, resembles a nonlinear convection-diffusion-reaction (CDR) equation. Its solution by finite elements is prone to oscillations in convection-dominated regions, which are present whenever cavitation occurs. We propose a stabilized finite-element method that is based on the variational multiscale method and exploits the concept of orthogonal subgrid scales. We demonstrate that this approach only requires one additional term in the weak form to obtain a stable method that converges optimally when performing mesh refinement.

math.NA↗

arXiv2023

Notes on osculations and mode tracing in semi-analytical waveguide modeling

The dispersion curves of (elastic) waveguides frequently exhibit crossings and osculations (also known as veering, repulsion, or avoided crossing). Osculations are regions in the dispersion diagram where curves approach each other arbitrarily closely without ever crossing before veering apart. In semi-analytical (undamped) waveguide models, dispersion curves are obtained as solutions to discretized parameterized Hermitian eigenvalue problems. In the mathematical literature, it is known that such eigencurves can exhibit crossing points only if the corresponding matrix flow (parameter-dependent matrix) is uniformly decomposable. We discuss the implications for the solution of the waveguide problem. In particular, we make use of a simple algorithm recently suggested in the literature for decomposing matrix flows. We also employ a method for mode tracing based on approximating the eigenvalue problem for individual modes by an ordinary differential equation that can be solved by standard procedures.

physics.class-ph↗

arXiv2023

Computing zero-group-velocity points in anisotropic elastic waveguides: Globally and locally convergent methods

Dispersion curves of elastic waveguides exhibit points where the group velocity vanishes while the wavenumber remains finite. These are the so-called zero-group-velocity (ZGV) points. As the elastodynamic energy at these points remains confined close to the source, they are of practical interest for nondestructive testing and quantitative characterization of structures. These applications rely on the correct prediction of the ZGV points. In this contribution, we first model the ZGV resonances in anisotropic plates based on the appearance of an additional modal solution. The resulting governing equation is interpreted as a two-parameter eigenvalue problem. Subsequently, we present three complementary numerical procedures capable of computing ZGV points in arbitrary nondissipative elastic waveguides in the conventional sense that their axial power flux vanishes. The first method is globally convergent and guarantees to find all ZGV points but can only be used for small problems. The second procedure is a very fast, generally-applicable, Newton-type iteration that is locally convergent and requires initial guesses. The third method combines both kinds of approaches and yields a procedure that is applicable to large problems, does not require initial guesses and is likely to find all ZGV points. The algorithms are implemented in "GEW ZGV computation" (doi: 10.5281/zenodo.7537442).

physics.class-ph↗

arXiv2021

A massively parallel explicit solver for elasto-dynamic problems exploiting octree meshes

Typical areas of application of explicit dynamics are impact, crash test, and most importantly, wave propagation simulations. Due to the numerically highly demanding nature of these problems, efficient automatic mesh generators and transient solvers are required. To this end, a parallel explicit solver exploiting the advantages of balanced octree meshes is introduced. To avoid the hanging nodes problem encountered in standard finite element analysis (FEA), the scaled boundary finite element method (SBFEM) is deployed as a spatial discretization scheme. Consequently, arbitrarily shaped star-convex polyhedral elements are straightforwardly generated. Considering the scaling and transformation of octree cells, the stiffness and mass matrices of a limited number of unique cell patterns are pre-computed. A recently proposed mass lumping technique is extended to 3D yielding a well-conditioned diagonal mass matrix. This enables us to leverage the advantages of explicit time integrator, i.e., it is possible to efficiently compute the nodal displacements without the need for solving a system of linear equations. We implement the proposed scheme together with a central difference method (CDM) in a distributed computing environment. The performance of our parallel explicit solver is evaluated by means of several numerical benchmark examples, including complex geometries and various practical applications. A significant speedup is observed for these examples with up to one billion of degrees of freedom and running on up to 16,384 computing cores.

cs.CE↗

arXiv2019

High order transition elements: The xNy-element concept -- Part I: Statics

Advanced transition elements are of utmost importance in many applications of the finite element method (FEM) where a local mesh refinement is required. Considering problems that exhibit singularities in the solution, an adaptive hp-refinement procedure must be applied. Even today, this is a very demanding task especially if only quadrilateral/hexahedral elements are deployed and consequently the hanging nodes problem is encountered. These element types, are, however, favored in computational mechanics due to the improved accuracy compared to triangular/tetrahedral elements. Therefore, we propose a compatible transition element - xNy-element - which provides the capability of coupling different element types. The adjacent elements can exhibit different element sizes, shape function types, and polynomial orders. Thus, it is possible to combine independently refined h- and p-meshes. The approach is based on the transfinite mapping concept and constitutes an extension/generalization of the pNh-element concept. By means of several numerical examples, the convergence behavior is investigated in detail, and the asymptotic rates of convergence are determined numerically. Overall, it is found that the proposed approach provides very promising results for local mesh refinement procedures.

math.NA↗