arXiv subjects

Sigrid Leyendecker

Publications and source records attributed to Sigrid Leyendecker.

At least 19 recordsLinked to original sources

arXiv2026

A variational framework for bond-based peridynamics with spatially varying horizons and its asynchronous time integration

Bond-based peridynamics provides a non-local framework for modelling fracture without requiring spatial derivatives of the displacement field. However, when spatially varying horizons are used together with non-uniform discretisations, the classical single-horizon bond-based peridynamics formulation leads to asymmetric interactions between material points. These asymmetric interactions violate balance laws and can introduce non-physical artefacts such as ghost forces and spurious wave reflections. In this work, we develop a variational formulation for bond-based peridynamics with spatially varying horizons. Starting from the Lagrange-d'Alembert principle, we derive the governing equations of motion and show that the dual-horizon peridynamics formulation emerges naturally from the variation of the internal energy. Building on this variational structure, we construct asynchronous variational integrators that allow different time step sizes in different regions of the domain. This is particularly useful for dynamic fracture simulations with local refinement, where small time steps are required only near regions of high resolution or expected crack growth. Numerical examples involving wave propagation, a pre-cracked plate under tension, and the Kalthoff-Winkler impact experiment demonstrate that the proposed framework removes spurious reflections caused by non-uniform horizons, preserves physically consistent fracture patterns, and achieves results comparable to uniformly refined simulations. At the same time, the asynchronous variational integrator reduces the number of internal force evaluations compared to the standard velocity-Verlet method. The proposed approach therefore provides a consistent variational foundation and an efficient time-integration strategy for bond-based peridynamic simulations with spatially varying horizons.

cs.CE↗

arXiv2025

Modelling and simulation of electro-mechanically coupled dielectric elastomers and myocardial tissue using smoothed finite element methods

Computational modelling offers a cost-effective and time-efficient alternative to experimental studies in biomedical engineering. In cardiac electro-mechanics, finite element method (FEM)-based simulations provide valuable insights into diseased tissue behaviour and the development of assistive systems such as di-electric elastomer actuators. However, the use of automatically generated tetrahedral meshes, commonly applied due to geometric complexity, often leads to numerical issues including overly stiff responses and volume locking, particularly in incompressible materials. Smoothed finite element methods (S-FEMs) offer a promising alternative by softening the stiffness matrix through gradient smoothing over defined smoothing domains. This work extends S-FEM formulations to electro-mechanically coupled problems and compares their performance against standard linear FEM. We implement and evaluate four approaches in the Abaqus environment via custom user elements: standard linear FEM, face-based S-FEM (FS-FEM), node-based S-FEM (NS-FEM), and the hybrid face/node-based S-FEM (FSNS-FEM). Two benchmark problems are studied: the electrically induced contraction of a compressible dielectric elastomer and an incompressible, orthotropic myocardial tissue sample. Reference solutions are obtained using a mesh consisting of higher-order elements. Our results demonstrate that FSNS-FEM provides the best balance between accuracy and computational efficiency, closely matching reference data. NS-FEM produces softer results, which leads to an overestimation of the true deformation. FS-FEM and standard FEM consistently exhibit overly stiff behaviour, with pronounced volume locking in the myocardial case. These findings support the potential of S-FEMs, in particular FSNS-FEM, for accurate simulation of coupled electro-mechanical behaviour in complex biomedical applications.

cs.CE↗

arXiv2025

Second order optimality conditions in a new Lagrangian formulation for optimal control problems

It has been shown recently that optimal control problems with the dynamical constraint given by a second order system admit a regular Lagrangian formulation. This implies that the optimality conditions can be obtained in a new form based on the variational approach. In this paper we extend the first order necessary optimality conditions obtained previously to second order optimality conditions. This results in a complete characterization of the optimality conditions in a new Lagrangian form.

math.OC↗

arXiv2025

A new Lagrangian approach to optimal control of second-order systems

In this work, we propose and study a new approach to formulate the optimal control problem of second-order differential equations, with a particular interest in those derived from force-controlled Lagrangian systems. The formulation results in a new hyperregular control Langrangian and, thus, a new control Hamiltonian whose equations of motion provide necessary optimality conditions. We compare this approach to Pontryagin's maximum principle (PMP) in this setting, providing geometric insight into their relation. This leads us to define an extended Tulczyjew's triple with controls. Moreover, we study the relationship between Noether symmetries of this new formulation and those of the PMP.

math.OC↗

arXiv2025

Variational integrators for a new Lagrangian approach to control affine systems with a quadratic Lagrange term

In this work, we analyse the discretisation of a recently proposed new Lagrangian approach to optimal control problems of affine-controlled second-order differential equations with cost functions quadratic in the controls. We propose exact discrete and semi-discrete versions of the problem, providing new tools to develop numerical methods. Discrete necessary conditions for optimality are derived and their equivalence with the continuous version is proven. A family of low-order integration schemes is devised to find approximate optimality conditions, and used to solve a low-thrust orbital transfer problem. Non-trivial equivalent standard direct methods are constructed. Noether's theorem for the new Lagrangian approach is investigated in the exact and approximate cases.

math.OC↗

arXiv2024

Energy-based, geometric, and compositional formulation of fluid and plasma models

Fluid dynamics plays a crucial role in various multiphysics applications, including energy systems, electronics cooling, and biomedical engineering. Developing models for complex coupled systems can be challenging and time-consuming. In particular, ensuring the consistent integration of models from diverse physical domains requires meticulous attention. Considering the example of (electro-)magneto hydrodynamics (on a fixed spatial domain and with linear polarization and magnetization), this article demonstrates how relatively complex models can be composed from simpler parts by means of a formal language for multiphysics modeling. The Exergetic Port-Hamiltonian Systems (EPHS) modeling language features a simple graphical syntax for expressing the energy-based interconnection of subsystems. This reduces cognitive load and facilitates communication, especially in multidisciplinary environments. As the example demonstrates, existing models can be easily integrated as subsystems of new models. Specifically, an ideal fluid model is used as a subsystem of a Navier-Stokes-Fourier fluid model, which in turn is reused as a subsystem of an (electro-)magneto hydrodynamics model. The energy-based, compositional approach simplifies understanding complex models, and it makes it easy to encapsulate, reuse, and replace (parts of) models. Moreover, structural properties of EPHS models guarantee fundamental properties of thermodynamic systems, such as conservation of energy, non-negative entropy production, and Onsager reciprocal relations.

cs.CE↗

arXiv2024

Variational multirate integrators

The simulation of systems that act on multiple time scales is challenging. A stable integration of the fast dynamics requires a highly accurate approximation whereas for the simulation of the slow part, a coarser approximation is accurate enough. With regard to the general goals of any numerical method, high accuracy and low computational costs, a popular approach is to treat the slow and the fast part of a system differently. Embedding this approach in a variational framework is the keystone of this work. By paralleling continuous and discrete variational multirate dynamics, integrators are derived on a time grid consisting of macro and micro time nodes that are symplectic, momentum preserving and also exhibit good energy behaviour. The choice of the discrete approximations for the action determines the convergence order of the scheme as well as its implicit or explicit nature for the different parts of the multirate system. The convergence order is proven using the theory of variational error analysis. The performance of the multirate variational integrators is demonstrated by means of several examples.

math.NA↗

arXiv2024

Neural networks for the approximation of Euler's elastica

Euler's elastica is a classical model of flexible slender structures, relevant in many industrial applications. Static equilibrium equations can be derived via a variational principle. The accurate approximation of solutions of this problem can be challenging due to nonlinearity and constraints. We here present two neural network based approaches for the simulation of this Euler's elastica. Starting from a data set of solutions of the discretised static equilibria, we train the neural networks to produce solutions for unseen boundary conditions. We present a $\textit{discrete}$ approach learning discrete solutions from the discrete data. We then consider a $\textit{continuous}$ approach using the same training data set, but learning continuous solutions to the problem. We present numerical evidence that the proposed neural networks can effectively approximate configurations of the planar Euler's elastica for a range of different boundary conditions.

math.NA↗

arXiv2024

Exergetic Port-Hamiltonian Systems for Multibody Dynamics

Multibody dynamics simulation plays an important role in various fields, including mechanical engineering, robotics, and biomechanics. Setting up computational models however becomes increasingly challenging as systems grow in size and complexity. Especially the consistent combination of models across different physical domains usually demands a lot of attention. This motivates us to study formal languages for compositional modeling of multiphysical systems. This article shows how multibody systems, or more precisely assemblies of rigid bodies connected by lower kinematic pairs, fit into the framework of Exergetic Port-Hamiltonian Systems (EPHS). This approach is based on the hierarchical decomposition of systems into their ultimately primitive components, using a simple graphical syntax. Thereby, cognitive load can be reduced and communication is facilitated, even with non-experts. Moreover, the encapsulation and reuse of subsystems promotes efficient model development and management. In contrast to established modeling languages such as Modelica, the primitive components of EPHS are not defined by arbitrary equations. Instead, there are four kinds of components, each defined by a particular geometric structure with a clear physical interpretation. This higher-level approach could make the process of building and maintaining large-scale models simpler and also safer.

eess.SY↗

arXiv2024

Exergetic Port-Hamiltonian Systems Modeling Language

Mathematical modeling of real-world physical systems requires the consistent combination of a multitude of physical laws and phenomenological models. This challenging task can be greatly simplified by hierarchically decomposing systems into ultimately simple components. Moreover, the use of diagrams for expressing the decomposition helps make the process more intuitive and facilitates communication, even with non-experts. As an important requirement, models have to respect fundamental physical laws such as the first and the second law of thermodynamics. While some existing modeling frameworks make such guarantees based on structural properties of their models, they lack a formal graphical syntax. We present a compositional and thermodynamically consistent modeling language with a graphical syntax. In terms of its semantics, we essentially endow port-Hamiltonian systems with additional structural properties and a fixed physical interpretation, ensuring thermodynamic consistency in a manner closely related to the metriplectic or GENERIC formalism. While port-Hamiltonian systems are inspired by graphical modeling with bond graphs, neither the link between the two, nor bond graphs themselves, can be easily formalized. In contrast, our syntax is based on a refinement of the well-studied operad of undirected wiring diagrams. By combining a compositional, graphical syntax with an energy-based, thermodynamic approach, the presented modeling language simplifies the understanding, reuse, and modification of complex physical models.

eess.SY↗

arXiv2024

A new Lagrangian approach to control affine systems with a quadratic Lagrange term

In this work, we consider optimal control problems for mechanical systems on vector spaces with fixed initial and free final state and a quadratic Lagrange term. Specifically, the dynamics is described by a second order ODE containing an affine control term and we allow linear coordinate changes in the configuration space. Classically, Pontryagin's maximum principle gives necessary optimality conditions for the optimal control problem. For smooth problems, alternatively, a variational approach based on an augmented objective can be followed. Here, we propose a new Lagrangian approach leading to equivalent necessary optimality conditions in the form of Euler-Lagrange equations. Thus, the differential geometric structure (similar to classical Lagrangian dynamics) can be exploited in the framework of optimal control problems. In particular, the formulation enables the symplectic discretisation of the optimal control problem via variational integrators in a straightforward way.

math.OC↗

arXiv2023

Discrete Adjoint Method for Variational Integration of Constrained ODEs and its application to Optimal Control of Geometrically Exact Beam Dynamics

Direct methods for the simulation of optimal control problems apply a specific discretization to the dynamics of the problem, and the discrete adjoint method is suitable to calculate corresponding conditions to approximate an optimal solution. While the benefits of structure preserving or geometric methods have been known for decades, their exploration in the context of optimal control problems is a relatively recent field of research. In this work, the discrete adjoint method is derived for variational integrators yielding structure preserving approximations of the dynamics firstly in the ODE case and secondly for the case in which the dynamics is subject to holonomic constraints. The convergence rates are illustrated by numerical examples. Thirdly, the discrete adjoint method is applied to geometrically exact beam dynamics, represented by a holonomically constrained PDE.

math.OC↗

arXiv2023

ShaRPy: Shape Reconstruction and Hand Pose Estimation from RGB-D with Uncertainty

Despite their potential, markerless hand tracking technologies are not yet applied in practice to the diagnosis or monitoring of the activity in inflammatory musculoskeletal diseases. One reason is that the focus of most methods lies in the reconstruction of coarse, plausible poses, whereas in the clinical context, accurate, interpretable, and reliable results are required. Therefore, we propose ShaRPy, the first RGB-D Shape Reconstruction and hand Pose tracking system, which provides uncertainty estimates of the computed pose, e.g., when a finger is hidden or its estimate is inconsistent with the observations in the input, to guide clinical decision-making. Besides pose, ShaRPy approximates a personalized hand shape, promoting a more realistic and intuitive understanding of its digital twin. Our method requires only a light-weight setup with a single consumer-level RGB-D camera yet it is able to distinguish similar poses with only small joint angle deviations in a metrically accurate space. This is achieved by combining a data-driven dense correspondence predictor with traditional energy minimization. To bridge the gap between interactive visualization and biomedical simulation we leverage a parametric hand model in which we incorporate biomedical constraints and optimize for both, its pose and hand shape. We evaluate ShaRPy on a keypoint detection benchmark and show qualitative results of hand function assessments for activity monitoring of musculoskeletal diseases.

cs.CV↗

arXiv2022

Optimal control of dielectric elastomer actuated multibody dynamical systems

In this work, a simulation model for the optimal control of dielectric elastomer actuated flexible multibody dynamics systems is presented. The Dielectric Elastomer Actuator (DEA) behaves like a flexible artificial muscles in soft robotics. It is modeled as an electromechanically coupled geometrically exact beam, where the electric charges serve as control variables. The DEA-beam is integrated as an actuator into multibody systems consisting of rigid and flexible components. The model also represents contact interaction via unilateral constraints between the beam actuator and e.g. a rigid body during the grasping process of a soft robot. Specifically for the DEA, a work conjugated electric displacement and strain-like electric variables are derived for the Cosserat beam. With a mathematically concise and physically representative formulation, a reduced free energy function is developed for the beam-DEA. In the optimal control problem, an objective function is minimized while the dynamic balance equations for the multibody system have to be fulfilled together with the complementarity conditions for the contact and boundary conditions. The optimal control problem is solved via a direct transcription method, transforming it into a constrained nonlinear optimization problem. The beam is firstly semidiscretized with 1D finite elements and then the multibody dynamics is temporally discretized with a variational integrator leading to the discrete Euler-Lagrange equations, which are further reduced with the null space projection. The discrete Euler-Lagrange equations and the boundary conditions serve as equality constraints, whereas the contact constraints are treated as inequality constraints in the optimization of the discretized objective. The effectiveness of the developed model is demonstrated by three numerical examples, including a cantilever beam, a soft robotic worm and a soft grasper.

cs.CE↗

arXiv2022

EPHS: A Port-Hamiltonian Modelling Language

A prevalent theme throughout science and engineering is the ongoing paradigm shift away from isolated systems towards open and interconnected systems. Port-Hamiltonian theory developed as a synthesis of geometric mechanics and network theory. The possibility to model complex multiphysical systems via interconnection of simpler components is often advertised as one of its most attractive features. The development of a port-Hamiltonian modelling language however remains a topic which has not been sufficiently addressed. We report on recent progress towards the formalization and implementation of a modelling language for exergetic port-Hamiltonian systems. Its diagrammatic syntax inspired by bond graphs and its functorial semantics together enable a modular and hierarchical approach to model specification.

eess.SY↗

arXiv2022

Exergetic Port-Hamiltonian Systems: Navier-Stokes-Fourier Fluid

The Exergetic Port-Hamiltonian Systems modeling language combines a graphical syntax inspired by bond graphs with a port-Hamiltonian semantics akin to the GENERIC formalism. The syntax enables the modular and hierarchical specification of the composition pattern of lumped and distributed-parameter models. The semantics reflects the first and second law of thermodynamics as structural properties. Interconnected and hierarchically defined models of multiphysical thermodynamic systems can thus be expressed in a formal language accessible to humans and computers alike. We discuss a composed model of the Navier-Stokes-Fourier fluid on a fixed spatial domain as an example of an open distributed-parameter system. At the top level, the system comprises five subsystems which model kinetic energy storage, internal energy storage, thermal conduction, bulk viscosity, and shear viscosity.

eess.SY↗

arXiv2021

A spatially adaptive phase-field model of fracture

Phase-field models of fracture introduce smeared cracks of width commensurate with a regularisation length parameter $ε$ and obeying a minimum energy principle. Mesh adaptivity naturally suggests itself as a means of supplying spatial resolution were needed while simultaneously keeping the computational size of the model as small as possible. Here, a variational-based spatial adaptivity is proposed for a phase-field model of fracture. The conventional phase-field model is generalised by allowing a spatial variation of the regularisation length $ε$ in the energy functional. The optimal spatial variation of the regularisation length then follows by energy minimisation in the same manner as the displacement and phase fields. The extended phase-field model is utilised as a basis for an adaptive mesh refinement strategy, whereby the mesh size is required to resolve the optimal length parameter locally. The resulting solution procedure is implemented in the framework of the finite element library FEniCS. Selected numerical tests suggest that the spatially adaptive phase-field model exhibits the same convergence rate as the conventional phase-field model, albeit with a vastly superior constant, which results in considerable computational savings.

cs.CE↗

arXiv2021

Exergetic Port-Hamiltonian Systems: Modelling Basics

Port-Hamiltonian systems theory provides a structured approach to modelling, optimization and control of multiphysical systems. Yet, its relationship to thermodynamics seems to be unclear. The Hamiltonian is traditionally thought of as energy, although its meaning is exergy. This insight yields benefits: 1. Links to the GENERIC structure are identified, making it relatively easy to borrow ideas from a popular framework for nonequilibrium thermodynamics. 2. The port-Hamiltonian structure combined with a suitable bond-graph syntax is expected to become a main ingredient in thermodynamic optimization methods akin to exergy analysis and beyond. The intuitive nature of exergy and diagrammatic language facilitates interdisciplinary communication that is necessary for implementing sustainable energy systems and processes. Port-Hamiltonian systems are cyclo-passive, meaning that a power-balance equation immediately follows from their definition. For exergetic port-Hamiltonian systems, cyclo-passivity is synonymous with degradation of energy and follows from the first and the second law of thermodynamics being encoded as structural properties.

physics.class-ph↗