arXiv subjects

Tobias Thoma

Publications and source records attributed to Tobias Thoma.

7 recordsLinked to original sources

arXiv2025

Discrete nonlinear elastodynamics in a port-Hamiltonian framework

We provide a fully nonlinear port-Hamiltonian formulation for discrete elastodynamical systems as well as a structure-preserving time discretization. The governing equations are obtained in a variational manner and represent index-1 differential algebraic equations. Performing an index reduction one obtains the port-Hamiltonian state space model, which features the nonlinear strains as an independent state next to position and velocity. Moreover, hyperelastic material behavior is captured in terms of a nonlinear stored energy function. The model exhibits passivity and losslessness and has an underlying symmetry yielding the conservation of angular momentum. We perform temporal discretization using the midpoint discrete gradient, such that the beneficial properties are inherited by the developed time stepping scheme in a discrete sense. The numerical results obtained in a representative example are demonstrated to validate the findings.

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arXiv2025

Optimal Trajectory Control of Geometrically Exact Strings with Space-Time Finite Elements

In this contribution, we present a variational space-time formulation which generates an optimal feed-forward controller for geometrically exact strings. More concretely, the optimization problem is solved with an indirect approach, and the space-time finite element method translates the problem to a set of algebraic equations. Thereby, only the positional field and the corresponding adjoint variable field are approximated by continuous shape functions, which makes the discretization of a velocity field unnecessary. In addition, the variational formulation can be solved using commercial or open source finite element packages. The entire approach can also be interpreted as a multiple-shooting method for solving the optimality conditions based on the semi-discrete problem. The performance of our approach is demonstrated by a numerical test.

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arXiv2024

On the velocity-stress formulation for geometrically nonlinear elastodynamics and its structure-preserving discretization

We consider the dynamics of an elastic continuum under large deformation but small strain. Such systems can be described by the equations of geometrically nonlinear elastodynamics in combination with the St. Venant-Kirchhoff material law. The velocity-stress formulation of the problem turns out to have a formal port-Hamiltonian structure. In contrast to the linear case, the operators of the problem are modulated by the displacement field which can be handled as a passive variable and integrated along with the velocities. A weak formulation of the problem is derived and essential boundary conditions are incorporated via Lagrange multipliers. This variational formulation explicitly encodes the transfer between kinetic and potential energy in the interior as well as across the boundary, thus leading to a global power balance and ensuring passivity of the system. The particular geometric structure of the weak formulation can be preserved under Galerkin approximation via appropriate mixed finite elements. In addition, a fully discrete power balance can be obtained by appropriate time discretization. The main properties of the system and its discretization are shown theoretically and demonstrated by numerical tests.

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arXiv2023

Port-Hamiltonian formulation and structure-preserving discretization of hyperelastic strings

Port-Hamiltonian (PH) systems provide a framework for modeling, analysis and control of complex dynamical systems, where the complexity might result from multi-physical couplings, non-trivial domains and diverse nonlinearities. A major benefit of the PH representation is the explicit formulation of power interfaces, so-called ports, which allow for a power-preserving interconnection of subsystems to compose flexible multibody systems in a modular way. In this work, we present a PH representation of geometrically exact strings with nonlinear material behaviour. Furthermore, using structure-preserving discretization techniques a corresponding finite-dimensional PH state space model is developed. Applying mixed finite elements, the semi-discrete model retains the PH structure and the ports (pairs of velocities and forces) on the discrete level. Moreover, discrete derivatives are used in order to obtain an energy-consistent time-stepping method. The numerical properties of the newly devised model are investigated in a representative example. The developed PH state space model can be used for structure-preserving simulation and model order reduction as well as feedforward and feedback control design.

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arXiv2022

Structure preserving discontinuous Galerkin approximation of one-dimensional port-Hamiltonian systems

In this article, we present the structure-preserving discretization of linear one-dimensional port-Hamiltonian (PH) systems of two conservation laws using discontinuous Galerkin (DG) methods. We recall the DG discretization procedure which is based on a subdivision of the computational domain, an elementwise weak formulation with up to two integration by parts, and the interconnection of the elements using several numerical fluxes. We present the interconnection of the element models, which is power preserving in the case of conservative (unstabilized) numerical fluxes, and we set up the resulting global PH state space model. We discuss the properties of the obtained models, including the effect of the flux stabilization parameter on the spectrum. Finally, we show simulations with different parameters for a boundary controlled linear hyperbolic system.

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arXiv2021

Explicit Port-Hamiltonian FEM-Models for Linear Mechanical Systems with Non-Uniform Boundary Conditions

In this contribution we present how to obtain explicit state space models in port-Hamiltonian form when a mixed finite element method is applied to a linear mechanical system with non-uniform boundary conditions. The key is to express the variational problem based on the principle of virtual power, with both the Dirichlet (velocity) and Neumann (stress) boundary conditions imposed in a weak sense. As a consequence, the formal skew-adjointness of the system operator becomes directly visible after integration by parts, and, after compatible FE discretization, the boundary degrees of freedom of both causalities appear as explicit inputs in the resulting state space model. The rationale behind our formulation is illustrated using a lumped parameter example, and numerical experiments on a one-dimensional rod show the properties of the approach in practice.

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arXiv2021

Symplectic discrete-time energy-based control for nonlinear mechanical systems

In this article we present a novel discrete-time design approach which reduces the deteriorating effects of sampling on stability and performance in digitally controlled nonlinear mechanical systems. The method is motivated by recent results for linear systems, where feedback imposes closed-loop behavior that exactly represents the symplectic discretization of a desired target system. In the nonlinear case, both the second order accurate representation of the sampling process and the definition of the target dynamics stem from the application of the implicit midpoint rule. The implicit nature of the resulting state feedback requires the numerical solution of an in general nonlinear system of algebraic equations in every sampling interval. For an implementation with pure position feedback, the velocities/momenta have to be approximated in the sampling instants, which gives a clear interpretation of our approach in terms of the Störmer-Verlet integration scheme on a staggered grid. We present discrete-time versions of impedance or energy shaping plus damping injection control as well as computed torque tracking control. Both the Hamiltonian and the Lagrangian perspective are adopted. Besides a linear example to introduce the concept, the simulations with a planar two link robot model illustrate the performance and stability gain compared to the discrete implementations of continuous-time control laws. A short analysis of computation times shows the real-time capability of our method.

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