Multi-level substructuring combined with model order reduction methods

In many fields of engineering problems linear time-invariant dynamical systems (LTI systems) play an outstanding role. They result for instance from discretizations of the unsteady heat equation and they are also used in optimal control problems. Often the order of LTI systems is a limiting factor, since it becomes easily very large. As a consequence these systems cannot be treated efficiently without model reduction algorithms. In this paper a new approach for the combination of model order reduction methods and recent multi-level substructuring (MLS) techniques is presented. Similar multi-level substructuring methods have already been applied successfully to huge eigenvalue problems up to several millions of degrees of freedom. However, the presented approach does not make use of a modal analysis like former algorithms. Instead the original system is decomposed in smaller LTI systems which are treated with recent model reduction methods. Furthermore, the error which is induced by this substructuring approach is analysed and numerical examples based on the Oberwolfach benchmark collection are given in this paper.     

Implications of order reduction for implicit Runge-Kutta methods

Stability analysis of Runge-Kutta (RK) formulas was originally limited to linear ordinary differential equations (ODEs). More recently such analysis has been extended to include the behaviour of solutions to nonlinear problems. This extension led to additional stability requirements for RK methods. Although the class of problems has been widened, the analysis is still restricted to a fixed stepsize. In the case of differential algebraic equations (DAEs), additional order conditions must be satisfied [6] to achieve full classical ODE order and avoid possible order reduction. In this case too, a fixed stepsize analysis is employed. Such analysis may be of only limited use in quantifying the effectiveness of adaptive methods on stiff problems.In this paper we examine the phenomenon of order reduction and its implications on variable-step algorithms. We introduce a global measure of order referred to here as the observed order which is based on the average stepsize over the region of integration. This measure may be better suited to the study of stiff systems, where the stepsize selection algorithm will vary the stepsize considerably over the interval of integration. Observed order gives a better indication of the relationship between accuracy and cost. Using this measure, the observed order reduction will be seen to be less severe than that predicated by fixed stepsize order analysis.Supported by the Information Technology Research Centre of Ontario, and the Natural Science and Engineering Research Council of Canada.     

Model order reduction and error estimation with an application to the parameter-dependent eddy current equation

In product development, engineers simulate the underlying partial differential equation many times with commercial tools for different geometries. Since the available computation time is limited, we look for reduced models with an error estimator that guarantees the accuracy of the reduced model. Using commercial tools the theoretical methods proposed by G. Rozza, D.B.P. Huynh and A.T. Patera [Reduced basis approximation and a posteriori error estimation for affinely parameterized elliptic coercive partial differential equations, Arch. Comput. Methods Eng. 15 (2008), pp. 229–275] lead to technical difficulties. We present how to overcome these challenges and validate the error estimator by applying it to a simple model of a solenoid actuator that is a part of a valve.     

Model order reduction for nonlinear problems in circuit simulation

Electrical circuits usually contain nonlinear components. Hence we are interested in MOR methods that can be applied to a system of nonlinear Differential-Algebraic Equations (DAEs). In particular we consider the TPWL (Trajectory PieceWise Linear) and POD (Proper Orthogonal Decomposition) methods. While the first one fully exploits linearity, the last method needs modifications to become efficient in evaluation. We describe a particular technique based on Missing Point Estimation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Model order reduction of dynamic skeletal muscle models

Forward-dynamics simulations of three-dimensional continuum-mechanical skeletal muscle models are a complex and computationally expensive problem. Considering a fully dynamic modelling framework based on the theory of finite elasticity is challenging as the muscles' mechanical behaviour requires to consider a highly nonlinear, viscoelastic and incompressible material behaviour. The governing equations yield a nonlinear second-order differential algebraic equation (DAE), which represents a challenge to model order reduction (MOR) techniques. This contribution shows the results of the offline phase that could be obtained so far by applying a combination of the proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM). (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Model order reduction for thermomechanical problems including radiation

Thermal field problems including heat exchange by radiation lead to nonlinear system equations with a high number of inputs and outputs as radiation heat fluxes correspond to the fourth power of the temperature and thermal loads are distributed over the whole surface. In an alternative approach presented here, radiation is defined as a part of the load vector. Thus, the system matrices are constant. Furthermore, loads changing synchronously during operation are grouped into one column of the input matrix and load vector snapshots are used to consider the radiation heat fluxes. Hence, the Krylov Subspace Method can be applied to significantly reduce the system dimension and the computation times allowing transient thermal parameter studies. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Highest possible order of algebraically stable diagonally implicit runge-kutta methods

We prove that the highest possible order of an algebraically stable diagonally implicit RK-method isfour; the highest possible order of a circle contractive singly diagonally implicit RK-method isfour; the highest possible order of a circle contractive diagonally implicit RK-method issix.     

Matrix enlarging methods and their application

This paper explores several methods for matrix enlarging, where an enlarged matrixà is constructed from a given matrixA. The methods explored include matrix primitization, stretching and node splitting. Graph interpretations of these methods are provided. Solving linear problems using enlarged matrices yields the answer to the originalAx=b problem.à can exhibit several desirable properties. For example,à can be constructed so that the valence of any row and/or column is smaller than some desired number (≥4). This is beneficial for algorithms that depend on the square of the number of entries of a row or column. Most particularly, matrix enlarging can results in a reduction of the fill-in in theR matrix which occurs during orthogonal factorization as a result of dense rows. Numerical experiments support these conjectures.     

Kinematic modeling of flexure hinges and compliant mechanisms

In this paper, the kinematic performance of flexure hinges and compliant mechanisms calculated by conventional modeling techniques are compared. As these exhibit certain drawbacks with regard to control strategies, mainly large number of degrees of freedom or unacceptable errors, a novel modeling approach for flexure hinges is presented. Instead of the entire flexure hinge only its significant regions are modeled by 3-D structural solids. These master patterns are positioned appropriately and connected by rigid constraint conditions to build a compliant mechanism. The resulting model is characterized by considerably fewer degrees of freedom than a full solid model as well as a marginal deviation of the deflection compared to that of pseudo-rigid-body models, 3-D tapered finite beam elements and analytical Timoshenko beam theory. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stage reduction on P-stable Numerov type methods of eighth order

We present an implicit hybrid two step method for the solution of second order initial value problem. It costs only six function evaluations per step and attains eighth algebraic order. The method satisfy the P-stability property requiring one stage less. We conclude dealing with implementation issues for the methods of this type and give some first pleasant results from numerical tests.     

Solving the order reduction phenomenon in variable step size quasi-consistent Nordsieck methods

The phenomenon is studied of reducing the order of convergence by one in some classes of variable step size Nordsieck formulas as applied to the solution of the initial value problem for a first-order ordinary differential equation. This phenomenon is caused by the fact that the convergence of fixed step size Nordsieck methods requires weaker quasi-consistency than classical Runge-Kutta formulas, which require consistency up to a certain order. In other words, quasi-consistent Nordsieck methods on fixed step size meshes have a higher order of convergence than on variable step size ones. This fact creates certain difficulties in the automatic error control of these methods. It is shown how quasi-consistent methods can be modified so that the high order of convergence is preserved on variable step size meshes. The regular techniques proposed can be applied to any quasi-consistent Nordsieck methods. Specifically, it is shown how this technique performs for Nordsieck methods based on the multistep Adams-Moulton formulas, which are the most popular quasi-consistent methods. The theoretical conclusions of this paper are confirmed by the numerical results obtained for a test problem with a known solution.     

Runge-Kutta methods without order reduction for linear initial boundary value problems

Summary. It is well-known the loss of accuracy when a Runge–Kutta method is used together with the method of lines for the full discretization of an initial boundary value problem. We show that this phenomenon, called order reduction, is caused by wrong boundary values in intermediate stages. With a right choice, the order reduction can be avoided and the optimal order of convergence in time is achieved. We prove this fact for time discretizations of abstract initial boundary value problems based on implicit Runge–Kutta methods. Moreover, we apply these results to the full discretization of parabolic problems by means of Galerkin finite element techniques. We present some numerical examples in order to confirm that the optimal order is actually achieved. Received July 10, 2000 / Revised version received March 13, 2001 / Published online October 17, 2001     

Faxén relations of arbitrary order and their application

Summary For a particle in Stokes flow the relation between the expansion tensors appearing in the perturbation velocity to gradients of the undisturbed (though arbitrary) velocity are called Faxén relations. The expansion tensors themselves turn out to be defined in terms of moments of the stress and velocity distribution, evaluated for any surface completely enclosing the particles. Since these moments vanish if the stress and velocity fields are regular inside the particle the Faxén relations can be used advantageously, if the classical method of reflections for the hydrodynamic interaction between two particles applies, i.e. if the particles are sufficiently far apart. Assuming this to be so for two interacting spheres, a general recursion formula for the expansion tensors of the (j+1)st reflection in terms of expansion tensors of thejth reflection is derived via the Faxén relations. In this way the interaction problem is reduced to a bookkeeping one. This is demonstrated by calculating the friction tensors to the fourth power of the inverse interparticle distance. Furthermore by evaluating the stresslet for a suspension of pair-interacting spheres we obtain an estimate of the suspension viscosity up to quadratic terms in the volume fraction.

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Zusammenfassung Für ein Teilchen in schleichender Strömung werden die Beziehungen zwischen den Entwicklungstensoren der Störgeschwindigkeit zu den Gradienten des ungestörten (aber beliebigen) Geschwindigkeitsfeldes Faxén Relationen genannt. Die Entwicklungstensoren selbst ergeben sich als Momente der Spannungs- und Geschwindigkeitsverteilung, berechnet über irgendeine beliebige Oberfläche, die das Teilchen umschließt. Da diese Momente für jedes Geschwindigkeits- und Spannungsfeld verschwinden, das im Innern des Teilchens regulär ist, können die Faxén Relationen vorteilhaft angewendet werden, wann immer die Reflexions-methode für 2 Teilchen gültig ist, d.h. die Teilchen müssen genügend weit voneinander entfernt sein. Mit Hilfe der Faxén Relationen wird für wechselwirkende Kugeln im letztgenannten Fall eine Rekursionsformel für die Entwicklungstensoren der (j+1)ten Reflexion abgeleitet, die nur von den Entwicklungstensoren derjten Reflexion abhängt. Dadurch braucht man bei dem hydrodynamischen Wechselwirkungsproblem nur noch genau Buch zu führen. Dies wird dadurch demonstriert, daß die Reibungstensoren bis zur vierten Potenz des inversen Kugelabstandes berechnet werden. Indem wir auch das Stresslet in einer Suspension paarweise wechselwirkender Kugeln berechnen, erhalten wir eine Abschätzung der Suspensionsviskosität einschließlich Gliedern quadratisch in der Konzentration.

     

Arc graphs and their possible application to sparse matrix problems

In continuation of earlier work on the graph algorithmic language GRAAL, a new type of graph representation is introduced involving solely the arcs and their incidence relations. In line with the set theoretical foundation of GRAAL, the are graph structure is defined in terms of four Boolean mappings over the power set of the ares. A simple data structure is available for are graphs requiring only storage of the order of the cardinality of the are set. As an application, the LU decomposition of large sparse matrices and the solution of the corresponding linear systems are formulated in terms of are graphs and their operators, and experimental results involving these algorithms are presented.