GEOMS: A software package for the numerical integration of general model equations of multibody systems

In this paper we present the new numerical algorithm GEOMS for the numerical integration of the most general form of the equations of motion of multibody systems, including nonholonomic constraints and possible redundancies in the constraints, as they may appear in industrial applications. Besides the numerical integration it offers some additional features like stabilization of the model equations, use of different decomposition strategies, or checking and correction of the initial values with respect to their consistency. Furthermore, GEOMS preserves hidden constraints and (possibly) existing solution invariants if they are provided as equations. We will also demonstrate the performance and the applicability of GEOMS for two mechanical examples of different degrees of complexity

GEOMS: a software package for the numerical integration of general model equations of multibody systems

In this paper we present the new numerical algorithm GEOMS for the numerical integration of the most general form of the equations of motion of multibody systems, including nonholonomic constraints and possible redundancies in the constraints, as they may appear in industrial applications. Besides the numerical integration it offers some additional features like stabilization of the model equations, use of different decomposition strategies, or checking and correction of the initial values with respect to their consistency. Furthermore, GEOMS preserves hidden constraints and (possibly) existing solution invariants if they are provided as equations. We will also demonstrate the performance and the applicability of GEOMS for two mechanical examples of different degrees of complexity

Optimal control of robot guided laser material treatment

In this article we will consider the optimal control of robot guided laser material treatments, where the discrete multibody system model of a robot is coupled with a PDE model of the laser treatment. We will present and discuss several optimization approaches of such optimal control problems and its properties in view of a robust and suitable numerical solution. We will illustrate the approaches in an application to the surface hardening of steel

M001 - The Simple Pendulum (v1.0)

The considerations in this report The Simple Pendulum are part of the example collection which can be found in http://www3.math.tu-berlin.de/multiphysics/Examples/. The aim is to investigate different formulations, i.e., regularized formulations or also index reduced formulations, of the model equations in combination with different numerical solvers with respect to its applicability, efficiency, accuracy, and robustness

Optimal control of robot-guided laser material treatment

In this article we will consider the optimal control of robot guided laser material treatments, where the discrete multibody system model of a robot is coupled with a PDE model of the laser treatment. We will present and discuss several optimization approaches of such optimal control problems and its properties in view of a robust and suitable numerical solution. We will illustrate the approaches in an application to the surface hardening of steel

Regularization of nonlinear equations of motion of multibody systems by index reduction with preserving the solution manifold

Different types of solution behavior for equations of motion of multibody systems with respect to deviate from the solution manifold and numerical instabilities are considered. An algorithm is presented that reduces the index of linear and nonlinear equations of motion of multibody systems in the usually used form by preserving all information about the solution manifold. The reduction is obtained by analyzing only the constraint matrix, the mass matrix and the transformation matrix. This technique allows the construction of a strangeness-free form which is suitable for numerical integration using stiff ODE solvers. The here presented algorithm is the generalization of the already developed algorithm for linear equations of motion. The obtained results are illustrated by a numerical example

Signature Method Based Regularization and Numerical Integration of DAEs

Modeling and simulation of dynamical systems often leads to differential-algebraic equations (DAEs) which can be seen as differential equations, where every solution has to satisfy constraints which are contained in the DAE. In general not all these constraints are stated explicitly as equations or can be obtained by algebraic manipulations but are hidden in the DAE and can be obtained from certain derivatives of (parts of) the DAE. Due to those hidden constraints a direct numerical integration of DAEs in general leads to instabilities and possibly non-convergence of numerical methods. Therefore, a regularization or remodeling of the model equations is required. In this article we present three approaches for the regularization of DAEs that are based on the Signature method, which is a structural analysis for DAEs. Furthermore, we present a software package suited for the proposed regularizations and illustrate their efficiency on two examples

M007 - A Skateboard (v1.0)

The considerations in this report A Skateboard are part of the example collection which can be found in http://www3.math.tu-berlin.de/multiphysics/Examples/. The aim is to investigate different formulations, i.e., regularized formulations or also index reduced formulations, of the model equations in combination with different numerical solvers with respect to its applicability, efficiency, accuracy, and robustness

M002 - Slider Crank (v1.0)

The considerations in this report Slider Crank are part of the example collection which can be found in http://www3.math.tu-berlin.de/multiphysics/Examples/. The aim is to investigate different formulations, i.e., regularized formulations or also index reduced formulations, of the model equations in combination with different numerical solvers with respect to its applicability, efficiency, accuracy, and robustness

A Combined Structural-Algebraic Approach for the Regularization of Coupled Systems of DAEs

The automated modeling of multi-physical dynamical systems is usually realized by coupling different subsystems together via certain interface or coupling conditions. This approach results in large-scale high-index differential-algebraic equations (DAEs). Since the direct numerical simulation of these kind of systems leads to instabilities and possibly non-convergence of the numerical methods a regularization or remodeling of the system is required. In many simulation environments a kind of structural analysis based on the sparsity pattern of the system is used to determine the index and a reduced system model. However, this approach is not reliable for certain problem classes, in particular we show that it is not suited for coupled systems of DAEs. We will present a new approach for the regularization of coupled dynamical systems that combines the structural analysis, in particular the Signature Method of Pryce, with classical algebraic regularization techniques and thus allows to handle so-called structurally singular systems and also enables a proper treatment of redundancies or inconsistencies in the system