Ingo Schäfer

Fractional Calculus via Functional Calculus: Theory and Applications

AbstractThis paper demonstrates the power of the functional-calculus definition oflinear fractional (pseudo-)differential operators via generalised Fouriertransforms.Firstly, we describe in detail how to get global causal solutions of linearfractional differential equations via this calculus. The solutions arerepresented as convolutions of the input functions with the related impulseresponses. The suggested method via residue calculus separates an impulseresponse automatically into an exponentially damped (possibly oscillatory)part and a `slow relaxation. If an impulse response is stable it becomesautomatically causal, otherwise one has to add a homogeneous solution to getcausality.Secondly, we present examples and, moreover, verify the approach alongexperiments on viscolelastic rods. The quality of the method as an effectivefew-parameter model is impressively demonstrated: the chosen referenceexample PTFE (Teflon) shows that in contrast to standard classical modelsour model describes the behaviour in a wide frequency range within theaccuracy of the measurement. Even dispersion effects are included.Thirdly, we conclude the paper with a survey of the required theory. Therethe attention is directed to the extension from the L2-approachon the space of distributions n

NUMERICAL METHODS FOR WAVE SCATTERING PHENOMENA BY MEANS OF DIFFERENT BOUNDARY INTEGRAL FORMULATIONS

{mathcal{D}}

Optimization and limitations of a preconditioned multi-level fast multipole algorithm for acoustical calculations

n′.

Modelling of coils using fractional derivatives

Coils exposed to eddy current and hysteresis losses are conventionally described by an inductance with equivalent core-loss resistance connected in parallel. The value of the equivalent core-loss resistance depends on the working frequency and the external wiring. Thus the model is less than satisfactory. The authors propose to describe loss inductance using fractional derivatives containing both a loss term and a storage term.After introducing the theory of fractional derivatives, the operating mode of the fractional coil model is explained by the example of an RLC oscillating circuit. Subsequent measurements of a series resonant circuit with a lossy coil impressively confirm the theoretical model with regard to both the frequency and time domains.

Impulse Responses of Fractional Damped Systems

Different numerical approaches for the physical phenomena of scattering waves from an obstacle are presented. They are based on different integral formulations. Fluid structure interaction effects are numerically treatable as well. We use the Boundary Element Method (BEM) in different approaches because the inherently satisfied Sommerfeld radiation condition makes sure that no reflecting waves from boundaries at infinity occur. One of the biggest disadvantages of numerical methods like BEM is the fact that they have difficulties with handling the high frequency range. For the high frequency range approximations like the Kirchhoff–Helmholtz integral equation have to be used. With varying assumptions of the reflecting behavior of the structure different approaches for the higher frequency range are obtained, where the explicit solving of a system of equations is not necessary. Another high frequency approach is the plane wave approximation which is compared with the Kirchhoff approach of the first kind. Additionally a modified Kirchhoff approach is introduced. Because the incident pressure on the scatterers surface is known the integral is evaluated analytically on triangular patches. The discretization is no longer frequency dependent and the size of the patches only depends on the curvature of the structure. Large planar parts can be discretized with one element only. This leads to a substantial advantage in terms of calculation time over the traditional Kirchhoff approach. Like the traditional approach this procedure is valid under the assumption of high frequency or far field conditions.

Fractional differential equations and viscoelastic damping

The problem of non-uniqueness (NU) of the solution of exterior acoustic problems when using the boundary element method (BEM) is well known. Methods like the Burton-Miller technique or the CHIEF method are used to solve this challenge at the expense of more complex procedures for handling hypersingular integrals and/or higher computing times due to higher complexity of the algorithm or additional equations. The dual surface method, commonly used for electromagnetic problems, was adapted for acoustic radiation and scattering problems. The basic principles of methods to solve the NU problem are outlined and results for different models and solution procedures are presented, taking into account quality, solution time, and the numerical advantages when using iterative solvers.

IMPLEMENT ATION AND RESULTS OF A MASS INERTIA COUPLING AS AN EXTENSION OF THE BEM FOR THIN SHELLS

The Multi-Level Fast Multipole Method (MLFMM) allows the computation of acoustical problems based on the Boundary Element Method (BEM) where the discretized models of the corresponding structures may consist of a huge number of elements. The required calculation time and the memory requirements are much less when compared with conventional methods because the algorithm uses a level-based composition of the potentials from different point sources to acoustic multipoles, which highly accelerates the computation of the matrix-vector-products required for iterative solvers. A multi-level single-order variation of the algorithm was extended to a multi-level adaptive-order version, which was analyzed and optimized with respect to quality, performance and parallelization issues. The iterative solvers used with the MLFMM will be combined with appropriate preconditioners for reducing the number of iterations and improving the performance. The insights gained will be presented using different test cases and the res...