arXiv subjects

Thomas-Peter Fries

Publications and source records attributed to Thomas-Peter Fries.

At least 19 recordsLinked to original sources

arXiv2026

Hyperelastic Membranes with Implicitly Defined, Continuously Embedded Fibers

A novel mechanical model and corresponding finite element method for anisotropic, hyperelastic, curved membranes are proposed. Hyperelastic fibers are embedded into the otherwise isotropic membrane, being relevant, for example, in reduced models for biological tissues and textiles. The geometrically nonlinear mechanics is formulated based on first principles of continuum mechanics (finite strain theory). The employed differential operators are formulated in a coordinate-free manner, through a framework known as tangential differential calculus. This enables a (semi-)implicit description of the fiber geometry through the intersection of level sets of some scalar function with the explicitly defined membrane surface. The mechanical model of the implicit fibers is then coupled to the mechanics of the membrane. For the numerical analysis, finite elements are applied such that the resulting scheme is a hybrid between classical Surface FEM and fictitious domain methods. For smooth physical fields, higher-order convergence rates are obtained and confirm the success of the numerical method.

cs.CE↗

arXiv2026

Simultaneous analysis of curved Kirchhoff beams and Kirchhoff--Love shells embedded in bulk domains

A set of curved beams and shells is geometrically implied by level sets of a scalar function over some bulk domain. The mechanical model for each structure is based on the Kirchhoff--Love theory, that is, small displacements without shear deformations are considered. These models for individual geometries are extended to bulk models, simultaneously modeling the whole set of beams/shells on all level sets. A major focus is on the numerical analysis of such models. A mixed-hybrid and higher-order accurate Bulk Trace FEM is proposed that enables the use of standard $C^0$-continuous Lagrange elements with dimensionality of the bulk domain. That is, the higher-order continuity requirements of displacement-based formulations in context of the Kirchhoff--Love theory are successfully alleviated. Several numerical tests confirm the accuracy and higher-order convergence of the proposed methodology, also qualifying as benchmark test cases in future studies.

cs.CE↗

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

Higher-order, mixed-hybrid finite elements for Kirchhoff-Love shells

A novel mixed-hybrid method for Kirchhoff-Love shells is proposed that enables the use of classical, possibly higher-order Lagrange elements in numerical analyses. In contrast to purely displacement-based formulations that require higher continuity of shape functions as in IGA, the mixed formulation features displacements and moments as primary unknowns. Thereby the continuity requirements are reduced, allowing equal-order interpolations of the displacements and moments. Hybridization enables an element-wise static condensation of the degrees of freedom related to the moments, at the price of introducing (significantly less) rotational degrees of freedom acting as Lagrange multipliers to weakly enforce the continuity of tangential moments along element edges. The mixed model is formulated coordinate-free based on the Tangential Differential Calculus, making it applicable for explicitly and implicitly defined shell geometries. All mechanically relevant boundary conditions are considered. Numerical results confirm optimal higher-order convergence rates whenever the mechanical setup allows for sufficiently smooth solutions; new benchmark test cases of this type are proposed.

cs.CE↗

arXiv2025

Simultaneous solution of incompressible Navier-Stokes flows on multiple surfaces

A mechanical model and finite element method for the simultaneous solution of Stokes and incompressible Navier-Stokes flows on multiple curved surfaces over a bulk domain are proposed. The two-dimensional surfaces are defined implicitly by all level sets of a scalar function, bounded by the three-dimensional bulk domain. This bulk domain is discretized with hexahedral finite elements which do not necessarily conform with the level sets but with the boundary. The resulting numerical method is a hybrid between conforming and non-conforming finite element methods. Taylor-Hood elements or equal-order element pairs for velocity and pressure, together with stabilization techniques, are applied to fulfil the inf-sup conditions resulting from the mixed-type formulation of the governing equations. Numerical studies confirm good agreement with independently obtained solutions on selected, individual surfaces. Furthermore, higher-order convergence rates are obtained for sufficiently smooth solutions.

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↗

arXiv2024

Geometric Uncertainty of Patient-Specific Blood Vessels and its Impact on Aortic Hemodynamics

In the context of numerical simulations of the vascular system, local geometric uncertainties have not yet been examined in sufficient detail due to model complexity and the associated large numerical effort. Such uncertainties are related to geometric modeling errors resulting from computed tomography imaging, segmentation and meshing. This work presents a methodology to systematically induce local modifications and perform a sufficient number of blood flow simulations to draw statistically relevant conclusions on the most commonly employed quantities of interest, such as flow rates or wall shear stress. The surface of a structured hexahedral mesh of a patient-specific aorta is perturbed by displacement maps defined via Gaussian random fields to stochastically model the local uncertainty of the boundary. Three different cases are studied, with the mean perturbation magnitude of $0.25$, $0.5$ and $1.0~$mm. Valid, locally perturbed meshes are constructed via an elasticity operator that extends surface perturbations into the interior. Otherwise, identical incompressible flow problems are solved on these meshes, taking physiological boundary conditions and Carreau fluid parameters into account. Roughly $300\,000$ three-dimensional non-stationary blood flow simulations are performed for the three different perturbation cases to estimate the probability distributions of the quantities of interest. Convergence studies justify the spatial resolution of the employed meshes. Overall, the results suggest that moderate geometric perturbations result in reasonable engineering accuracy (relative errors in single-digit percentage range) of the quantities of interest, with higher sensitivity for gradient-related measures, noting that the observed errors are not negligible.

physics.med-ph↗

arXiv2024

SynthAorta: A 3D Mesh Dataset of Parametrized Physiological Healthy Aortas

The effects of the aortic geometry on its mechanics and blood flow, and subsequently on aortic pathologies, remain largely unexplored. The main obstacle lies in obtaining patient-specific aorta models, an extremely difficult procedure in terms of ethics and availability, segmentation, mesh generation, and all of the accompanying processes. Contrastingly, idealized models are easy to build but do not faithfully represent patient-specific variability. Additionally, a unified aortic parametrization in clinic and engineering has not yet been achieved. To bridge this gap, we introduce a new set of statistical parameters to generate synthetic models of the aorta. The parameters possess geometric significance and fall within physiological ranges, effectively bridging the disciplines of clinical medicine and engineering. Smoothly blended realistic representations are recovered with convolution surfaces. These enable high-quality visualization and biological appearance, whereas the structured mesh generation paves the way for numerical simulations. The only requirement of the approach is one patient-specific aorta model and the statistical data for parameter values obtained from the literature. The output of this work is SynthAorta, a dataset of ready-to-use synthetic, physiological aorta models, each containing a centerline, surface representation, and a structured hexahedral finite element mesh. The meshes are structured and fully consistent between different cases, making them imminently suitable for reduced order modeling and machine learning approaches.

physics.med-ph↗

arXiv2023

Simultaneous Analysis of Continuously Embedded Reissner-Mindlin Shells in 3D Bulk Domains

A mechanical model and numerical method for the simultaneous analysis of Reissner-Mindlin shells with geometries implied by a continuous set of level sets (isosurfaces) over some three-dimensional bulk domain is presented. A three-dimensional mesh in the bulk domain is used in a tailored FEM formulation where the elements are by no means conforming to the level sets representing the shape of the individual shells. However, the shell geometries are bounded by the intersection curves of the level sets with the boundary of the bulk domain so that the boundaries are meshed conformingly. This results in a method which was coined Bulk Trace FEM before. The simultaneously considered, continuously embedded shells may be useful in the structural design process or for the continuous reinforcement of bulk domains. Numerical results confirm higher-order convergence rates.

cs.CE↗

arXiv2023

On the Simultaneous Solution of Structural Membranes on all Level Sets within a Bulk Domain

A mechanical model and numerical method for structural membranes implied by all isosurfaces of a level-set function in a three-dimensional bulk domain are proposed. The mechanical model covers large displacements in the context of the finite strain theory and is formulated based on the tangential differential calculus. Alongside curved two-dimensional membranes embedded in three dimensions, also the simpler case of curved ropes (cables) in two-dimensional bulk domains is covered. The implicit geometries (shapes) are implied by the level sets and the boundaries of the structures are given by the intersection of the level sets with the boundary of the bulk domain. For the numerical analysis, the bulk domain is discretized using a background mesh composed by (higher-order) elements with the dimensionality of the embedding space. The elements are by no means aligned to the level sets, i.e., the geometries of the structures, which resembles a fictitious domain method, most importantly the Trace FEM. The proposed numerical method is a hybrid of the classical FEM and fictitious domain methods which may be labeled as "Bulk Trace FEM". Numerical studies confirm higher-order convergence rates and the potential for new material models with continuously embedded sub-structures in bulk domains.

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

A Unified Finite Strain Theory for Membranes and Ropes

The finite strain theory is reformulated in the frame of the Tangential Differential Calculus (TDC) resulting in a unification in a threefold sense. Firstly, ropes, membranes and three-dimensional continua are treated with one set of governing equations. Secondly, the reformulated boundary value problem applies to parametrized and implicit geometries. Therefore, the formulation is more general than classical ones as it does not rely on parametrizations implying curvilinear coordinate systems and the concept of co- and contravariant base vectors. This leads to the third unification: TDC-based models are applicable to two fundamentally different numerical approaches. On the one hand, one may use the classical Surface FEM where the geometry is discretized by curved one-dimensional elements for ropes and two-dimensional surface elements for membranes. On the other hand, it also applies to recent Trace FEM approaches where the geometry is immersed in a higher-dimensional background mesh. Then, the shape functions of the background mesh are evaluated on the trace of the immersed geometry and used for the approximation. As such, the Trace FEM is a fictitious domain method for partial differential equations on manifolds. The numerical results show that the proposed finite strain theory yields higher-order convergence rates independent of the numerical methodology, the dimension of the manifold, and the geometric representation type.

cs.CE↗

arXiv2017

Higher-order surface FEM for incompressible Navier-Stokes flows on manifolds

Stationary and instationary Stokes and Navier-Stokes flows are considered on two-dimensional manifolds, i.e., on curved surfaces in three dimensions. The higher-order surface FEM is used for the approximation of the geometry, velocities, pressure, and Lagrange multiplier to enforce tangential velocities. Individual element orders are employed for these various fields. Stream-line upwind stabilization is employed for flows at high Reynolds numbers. Applications are presented which extend classical benchmark test cases from flat domains to general manifolds. Highly accurate solutions are obtained and higher-order convergence rates are confirmed.

cs.CE↗

arXiv2016

Conformal higher-order remeshing schemes for implicitly defined interface problems

A new higher-order accurate method is proposed that combines the advantages of the classical $p$-version of the FEM on body-fitted meshes with embedded domain methods. A background mesh composed by higher-order Lagrange elements is used. Boundaries and interfaces are described implicitly by the level set method and are within elements. In the elements cut by the boundaries or interfaces, an automatic decomposition into higher-order accurate sub-elements is realized. Therefore, the zero level sets are detected and meshed in a first step which is called reconstruction. Then, based on the topological situation in the cut element, higher-order sub-elements are mapped to the two sides of the boundary or interface. The quality of the reconstruction and the mapping largely determines the properties of the resulting, automatically generated conforming mesh. It is found that optimal convergence rates are possible although the resulting sub-elements are not always well-shaped.

math.NA↗

arXiv2016

Stable Isogeometric Analysis of Trimmed Geometries

We explore extended B-splines as a stable basis for isogeometric analysis with trimmed parameter spaces. The stabilization is accomplished by an appropriate substitution of B-splines that may lead to ill-conditioned system matrices. The construction for non-uniform knot vectors is presented. The properties of extended B-splines are examined in the context of interpolation, potential, and linear elasticity problems and excellent results are attained. The analysis is performed by an isogeometric boundary element formulation using collocation. It is argued that extended B-splines provide a flexible and simple stabilization scheme which ideally suits the isogeometric paradigm.

math.NA↗

arXiv2015

Isogeometric Boundary Element Analysis with elasto-plastic inclusions. Part 1: Plane problems

In this work a novel approach is presented for the isogeometric Boundary Element analysis of domains that contain inclusions with different elastic properties than the ones used for computing the fundamental solutions. In addition the inclusion may exhibit inelastic material behavior. In this paper only plane stress/strain problems are considered. In our approach the geometry of the inclusion is described using NURBS basis functions. The advantage over currently used methods is that no discretization into cells is required in order to evaluate the arising volume integrals. The other difference to current approaches is that Kernels of lower singularity are used in the domain term. The implementation is verified on simple finite and infinite domain examples with various boundary conditions. Finally a practical application in geomechanics is presented.

math.NA↗

arXiv2015

The Isogeometric Nystr\"om Method

In this paper the isogeometric Nystr\"om method is presented. It's outstanding features are: it allows the analysis of domains described by many different geometrical mapping methods in computer aided geometric design and it requires only pointwise function evaluations just like isogeometric collocation methods. The analysis of the computational domain is carried out by means of boundary integral equations, therefor only the boundary representation is required. The method is thoroughly integrated into the isogeometric framework. For example, the regularization of the arising singular integrals performed with local correction as well as the interpolation of the pointwise existing results are carried out by means of Bezier elements. The presented isogeometric Nystr\"om method is applied to practical problems solved by the Laplace and the Lame-Navier equation. Numerical tests show higher order convergence in two and three dimensions. It is concluded that the presented approach provides a simple and flexible alternative to currently used methods for solving boundary integral equations, but has some limitations.

math.NA↗

arXiv2014

Boundary Element Analysis with trimmed NURBS and a generalized IGA approach

A novel approach to the simulation with the boundary element method using trimmed NURBS patches is presented. The advantage of this approach is its efficiency and easy implementation. The analysis with trimmed NURBS is achieved by double mapping. The variation of the unknowns on the boundary is specified in a local coordinate system and is completely independent of the description of the geometry. The method is tested on a branched tunnel and the results compared with those obtained from a conventional analysis. The conclusion is that the proposed approach is superior in terms of number of unknowns and effort required.

math.NA↗