arXiv subjects

Benjamin Marussig

Publications and source records attributed to Benjamin Marussig.

At least 19 recordsLinked to original sources

arXiv2026

Comparative study of different quadrature methods for cut elements

The quadrature of cut elements is crucial for all Finite Element Methods that do not apply boundary-fitted meshes. It should be efficient, accurate, and robust. Various approaches balancing these requirements have been published, with some available as open-source implementations. This work reviews these open-sources codes and the methods used. Furthermore, benchmarking examples are developed for 2D and 3D geometries. Implicit and explicit boundary descriptions are available for all models. The different examples test the efficiency, accuracy, versatility, and robustness of the codes. Special focus is set on the influence of the input parameter, which controls the desired quadrature order, on the actual integration error. A detailed comparison of the discussed codes is carried out. The benchmarking allows a conclusive comparison and presents a valuable tool for future code development. All tests are published in an accompanying open-source repository.

cs.CE↗

arXiv2025

A Higher-Order Time Domain Boundary Element Formulation based on Isogeometric Analysis and the Convolution Quadrature Method

An isogeometric boundary element method (BEM) is presented to solve scattering problems in an isotropic homogeneous medium. We consider wave problems governed by the scalar wave equation as in acoustics and the Lam\'e-Navier equations for elastodynamics considering the theory of linear elasticity. The underlying boundary integral equations imply time-dependent convolution integrals and allow us to determine the sought quantities in the bounded interior or the unbounded exterior after solving for the unknown Cauchy data. In the present work, the time-dependent convolution integrals are approximated by multi-stage Runge-Kutta (RK) based convolution quadratures that involve steady-state solutions in the Laplace domain. The proposed method discretizes the spatial variables in the framework of isogeometric analysis (IGA), entailing a patchwise smooth spline basis. Overall, it enables high convergence rates in space and time. The implementation scheme follows an element structure defined by the non-empty knot spans in the knot vectors and local, uniform Bernstein polynomials as basis functions. The algorithms to localize the basis functions on the elements are outlined and explained. The solutions of the mixed problems are approximated by the BEM based on a symmetric Galerkin variational formulation and a collocation method. We investigate convergence rates of the approximative solutions in a mixed space and time error norm.

cs.CE↗

arXiv2025

Isogeometric Topology Optimization Based on Topological Derivatives

Topology optimization is a valuable tool in engineering, facilitating the design of optimized structures. However, topological changes often require a remeshing step, which can become challenging. In this work, we propose an isogeometric approach to topology optimization driven by topological derivatives. The combination of a level-set method together with an immersed isogeometric framework allows seamless geometry updates without the necessity of remeshing. At the same time, topological derivatives provide topological modifications without the need to define initial holes [7]. We investigate the influence of higher-degree basis functions in both the level-set representation and the approximation of the solution. Two numerical examples demonstrate the proposed approach, showing that employing higher-degree basis functions for approximating the solution improves accuracy, while linear basis functions remain sufficient for the level-set function representation.

math.NA↗

arXiv2025

Efficient snap-to-contact computations for van der Waals interacting fibers

We consider van der Waals interactions between in-plane fibers, where the computational model employs the Lennard-Jones potential and the coarse-grained approach. The involved 6D integral over two interacting fibers is split into a 4D analytical pre-integration over cross sections and the remaining 2D numerical integration along the fibers' axes. Two section-section interaction laws are implemented, refined, and compared. Fibers are modeled using the Bernoulli-Euler beam theory and spatially discretized with isogeometric finite elements. We derive and solve the weak form of both quasi-static and dynamic boundary value problems. Four numerical examples involving highly nonlinear and dynamic snap-to-contact phenomena are scrutinized. We observe that the coarse-graining and pre-integration of interaction potentials enable the efficient modeling of complex phenomena at small length scales.

physics.comp-ph↗

arXiv2025

CAD-Integrated Electrostatic Boundary Element Simulations with Non-Conforming Higher-Order Meshes

We present a design through analysis workflow that enables virtual prototyping of electric devices. A CAD plugin establishes the interaction between design and analysis, allowing the preparation of analysis models and the visualization of its results within the design environment. The simulations utilize a fast boundary element method (BEM) that allows for non-conforming and higher-order meshes. Our numerical experiments investigate the accuracy of the approach and its sensitivity to the initial CAD representation. Overall, the workflow enables a close link between design and analysis, where the non-conforming higher-order BEM approach provides accurate results and significantly simplifies the interaction.

cs.CE↗

arXiv2025

Employing Continuous Integration inspired workflows for benchmarking of scientific software -- a use case on numerical cut cell quadrature

In the field of scientific computing, one often finds several alternative software packages (with open or closed source code) for solving a specific problem. These packages sometimes even use alternative methodological approaches, e.g., different numerical discretizations. If one decides to use one of these packages, it is often not clear which one is the best choice. To make an informed decision, it is necessary to measure the performance of the alternative software packages for a suitable set of test problems, i.e. to set up a benchmark. However, setting up benchmarks ad-hoc can become overwhelming as the parameter space expands rapidly. Very often, the design of the benchmark is also not fully set at the start of some project. For instance, adding new libraries, adapting metrics, or introducing new benchmark cases during the project can significantly increase complexity and necessitate laborious re-evaluation of previous results. This paper presents a proven approach that utilizes established Continuous Integration tools and practices to achieve high automation of benchmark execution and reporting. Our use case is the numerical integration (quadrature) on arbitrary domains, which are bounded by implicitly or parametrically defined curves or surfaces in 2D or 3D.

cs.SE↗

arXiv2025

Isogeometric Analysis for 2D Magnetostatic Computations with Multi-level B\'{e}zier Extraction for Local Refinement

Local refinement is vital for efficient numerical simulations. In the context of Isogeometric Analysis (IGA), hierarchical B-splines have gained prominence. The work applies the methodology of truncated hierarchical B-splines (THB-splines) as they keep additional properties. The framework is further enriched with B\'{e}zier extraction, resulting in the multi-level B\'{e}zier extraction method. We apply this discretization method to 2D magnetostatic problems. The implementation is based on an open-source Octave/MATLAB IGA code called GeoPDEs, which allows us to compare our routines with globally refined spline models as well as locally refined ones where the solver does not rely on B\'{e}zier extraction.

cs.CE↗

arXiv2025

Comparison of Integration Methods for Cut Elements

Using an interface inserted in a background mesh is an alternative way of constructing a complex geometrical shape with a relative low meshing efforts. However, this process may require special treatment of elements cut by the interface. Our study focuses on comparing the integration of cut elements defined by implicit and parametric curves. We investigate the efficiency and robustness of open-source tools such as Algoim [5](a library for quadrature on implicitly defined geometries) and Ginkgo [2](a library for isogeometric analysis on Boolean operations with a parametric description) with numerical examples computing the area defined by the interface and benchmarks for 2D elasticity problem using the open-source code GeoPDEs [7]. It is concluded that none of the two interface descriptions is preferable with respect to the quality of the integration. Thus, the choice of the interface type depends only on the studied problem and the available curve description, but not on the numerical aspects of the integration.

cs.CE↗

arXiv2024

Reducing Meshing Requirements for Electrostatic Problems using a Galerkin Boundary Element Method

This work focuses on model preparation for electrostatic simulations of CAD designs to realize a rapid virtual prototyping concept. We present a boundary element method (BEM) allowing discontinuous fields between surfaces. The corresponding edges of the CAD model are enhanced with the data required to integrate over non-conforming elements. Finally, we generate a mesh for each CAD surface. The approach is verified via numerical experiments and shows excellent agreement with conforming BEM results.

cs.CE↗

arXiv2023

Fast immersed boundary method based on weighted quadrature

Combining sum factorization, weighted quadrature, and row-based assembly enables efficient higher-order computations for tensor product splines. We aim to transfer these concepts to immersed boundary methods, which perform simulations on a regular background mesh cut by a boundary representation that defines the domain of interest. Therefore, we present a novel concept to divide the support of cut basis functions to obtain regular parts suited for sum factorization. These regions require special discontinuous weighted quadrature rules, while Gauss-like quadrature rules integrate the remaining support. Two linear elasticity benchmark problems confirm the derived estimate for the computational costs of the different integration routines and their combination. Although the presence of cut elements reduces the speed-up, its contribution to the overall computation time declines with h-refinement.

cs.CE↗

arXiv2023

Teaching mechanics with individual exercise assignments and automated correction

Solving exercise problems by yourself is a vital part of developing a mechanical understanding. Yet, most mechanics lectures have more than 200 participants, so the workload for manually creating and correcting assignments limits the number of exercises. The resulting example pool is usually much smaller than the number of participants, making verifying whether students can solve problems themselves considerably harder. At the same time, unreflected copying of tasks already solved does not foster the understanding of the subject and leads to a false self-assessment. We address these issues by providing a scalable approach for creating, distributing, and correcting exercise assignments for problems related to statics, strength of materials, dynamics, and hydrostatics. The overall concept allows us to provide individual exercise assignments for each student. A quantitative survey among students of our recent statics lecture assesses the acceptance of our teaching tool. The feedback indicates a clear added value for the lecture, which fosters self-directed and reflective learning.

physics.ed-ph↗

arXiv2023

Multi-level B\'{e}zier extraction of truncated hierarchical B-splines for isogeometric analysis

Multivariate B-splines and Non-uniform rational B-splines (NURBS) lack adaptivity due to their tensor product structure. Truncated hierarchical B-splines (THB-splines) provide a solution for this. THB-splines organize the parameter space into a hierarchical structure, which enables efficient approximation and representation of functions with different levels of detail. The truncation mechanism ensures the partition of unity property of B-splines and defines a more scattered set of basis functions without overlapping on the multi-level spline space. Transferring these multi-level splines into B\'{e}zier elements representation facilitates straightforward incorporation into existing finite element (FE) codes. By separating the multi-level extraction of the THB-splines from the standard B\'{e}zier extraction, a more general independent framework applicable to any sequence of nested spaces is created. The operators for the multi-level structure of THB-splines and the operators of B\'{e}zier extraction are constructed in a local approach. Adjusting the operators for the multi-level structure from an element point of view and multiplying with the B\'{e}zier extraction operators of those elements, a direct map between B\'{e}zier elements and a hierarchical structure is obtained. The presented implementation involves the use of an open-source Octave/MATLAB isogeometric analysis (IGA) code called GeoPDEs. A basic Poisson problem is presented to investigate the performance of multi-level B\'{e}zier extraction compared to a standard THB-spline approach.

cs.CE↗

arXiv2022

Fast formation and assembly for spline-based 3D fictitious domain methods

Standard finite element methods employ an element-wise assembly strategy. The element's contribution to the system matrix is formed by a loop over quadrature points. This concept is also used in fictitious domain methods, which perform simulations on a simple tensor-product background mesh cut by a boundary representation that defines the domain of interest. Considering such $d$-dimensional background meshes based on splines of degree $p$ with maximal smoothness, $C^{p-1}$, the cost of setting up the system matrix is $\mathcal{O}\left(p^{3d}\right)$ per degree of freedom. Alternative assembly and formation techniques can significantly reduce this cost. In particular, the combination of (1) sum factorization, (2) weighted quadrature, and (3) row-based assembly yields a cost of $\mathcal{O}\left(p^{d+1}\right)$ for non-cut background meshes. However, applying this fast approach to cut background meshes is an open challenge since they do not have a tensor-product structure. This work presents techniques that allow the treatment of cut background meshes and thus the application of fast formation and assembly to fictitious domain methods. First, a discontinuous version of weighted quadrature is presented, which introduces a discontinuity into a cut test function's support. The cut region can be treated separately from the non-cut counterpart; the latter can be assembled by the fast concepts. A three-dimensional example investigates the accuracy and efficiency of the proposed concept and demonstrates its speed-up compared to conventional formation and assembly.

cs.CE↗

arXiv2022

Surface Patches with Rounded Corners

We analyze surface patches with a corner that is rounded in the sense that the partial derivatives at that point are antiparallel. Sufficient conditions for $G^1$ smoothness are given, which, up to a certain degenerate case, are also necessary. Further, we investigate curvature integrability and present examples

cs.CG↗

arXiv2021

Immersed Boundary-Conformal Isogeometric Method for Linear Elliptic Problems

We present a novel isogeometric method, namely the Immersed Boundary-Conformal Method (IBCM), that features a layer of discretization conformal to the boundary while employing a simple background mesh for the remaining domain. In this manner, we leverage the geometric flexibility of the immersed boundary method with the advantages of a conformal discretization, such as intuitive control of mesh resolution around the boundary, higher accuracy per degree of freedom, automatic satisfaction of interface kinematic conditions, and the ability to strongly impose Dirichlet boundary conditions. In the proposed method, starting with a boundary representation of a geometric model, we extrude it to obtain a corresponding conformal layer. Next, a given background B-spline mesh is cut with the conformal layer, leading to two disconnected regions: an exterior region and an interior region. Depending on the problem of interest, one of the two regions is selected to be coupled with the conformal layer through Nitsche's method. Such a construction involves Boolean operations such as difference and union, which therefore require proper stabilization to deal with arbitrarily cut elements. In this regard, we follow our precedent work called the minimal stabilization method [1]. In the end, we solve several 2D benchmark problems to demonstrate improved accuracy and expected convergence with IBCM. Two applications that involve complex geometries are also studied to show the potential of IBCM, including a spanner model and a fiber-reinforced composite model. Moreover, we demonstrate the effectiveness of IBCM in an application that exhibits boundary-layer phenomena.

math.NA↗

arXiv2021

Fast formation and assembly of isogeometric Galerkin matrices for trimmed patches

This work explores the application of the fast assembly and formation strategy from [8, 17] to trimmed bi-variate parameter spaces. Two concepts for the treatment of basis functions cut by the trimming curve are investigated: one employs a hybrid Gauss-point-based approach, and the other computes discontinuous weighted quadrature rules. The concepts' accuracy and efficiency are examined for the formation of mass matrices and their application to L2-projection. Significant speed-ups compared to standard element by element finite element formation are observed. There is no clear preference between the concepts proposed. While the discontinuous weighted scheme scales favorably with the degree of the basis, it also requires additional effort for computing the quadrature weights. The hybrid Gauss approach does not have this overhead, which is determined by the complexity of the trimming curve. Hence, it is well-suited for moderate degrees, whereas discontinuous-weightedquadrature has potential for high degrees, in particular, if the related weights are computed in parallel.

cs.CE↗

arXiv2020

A Consistent Higher-Order Isogeometric Shell Formulation

Shell analysis is a well-established field, but achieving optimal higher-order convergence rates for such simulations is a difficult challenge. We present an isogeometric Kirchhoff-Love shell framework that treats every numerical aspect in a consistent higher-order accurate way. In particular, a single trimmed B-spline surface provides a sufficiently smooth geometry, and the non-symmetric Nitsche method enforces the boundary conditions. A higher-order accurate reparametrization of cut knot spans in the parameter space provides a robust, higher-order accurate quadrature for (multiple) trimming curves, and the extended B-spline concept controls the conditioning of the resulting system of equations. Besides these components ensuring all requirements for higher-order accuracy, the presented shell formulation is based on tangential differential calculus, and level-set functions define the trimming curves. Numerical experiments confirm that the approach yields higher-order convergence rates, given that the solution is sufficiently smooth.

cs.CE↗

arXiv2019

Advances in the Treatment of Trimmed CAD Models due to Isogeometric Analysis

Trimming is a core technique in geometric modeling. Unfortunately, the resulting objects do not take the requirements of numerical simulations into account and yield various problems. This paper outlines principal issues of trimmed models and highlights different analysis-suitable strategies to address them. It is discussed that these concepts not only provide important computational tools for isogeometric analysis, but can also improve the treatment of trimmed models in a design context.

cs.GR↗