## Berichte der Arbeitsgruppe Technomathematik (AGTM Report)

### Refine

#### Year of publication

#### Keywords

- Boltzmann Equation (7)
- Numerical Simulation (5)
- Elastoplastizität (4)
- Elastoplasticity (3)
- Hysterese (3)
- lattice Boltzmann method (3)
- low Mach number limit (3)
- Elastic BVP (2)
- Elastisches RWP (2)
- Elastoplastisches RWP (2)

- 265
- Model Reduction Techniques for Frequency Averaging in Radiative Heat Transfer (2006)
- We study model reduction techniques for frequency averaging in radiative heat transfer. Especially, we employ proper orthogonal decomposition in combination with the method of snapshots to devise an automated a posteriori algorithm, which helps to reduce significantly the dimensionality for further simulations. The reliability of the surrogate models is tested and we compare the results with two other reduced models, which are given by the approximation using the weighted sum of gray gases and by an frequency averaged version of the so-called \(\mathrm{SP}_n\) model. We present several numerical results underlining the feasibility of our approach.

- 264
- Parameter optimization for a stress-strain correction scheme (2005)
- A gradient based algorithm for parameter identification (least-squares) is applied to a multiaxial correction method for elastic stresses and strains at notches. The correction scheme, which is numerically cheap, is based on Jiang's model of elastoplasticity. Both mathematical stress-strain computations (nonlinear PDE with Jiang's constitutive material law) and physical strain measurements have been approximized. The gradient evaluation with respect to the parameters, which is large-scale, is realized by the automatic forward differentiation technique.

- 263
- A multiaxial stress-strain correction scheme (2005)
- A method to correct the elastic stress tensor at a fixed point of an elastoplastic body, which is subject to exterior loads, is presented and analysed. In contrast to uniaxial corrections (Neuber or ESED), our method takes multiaxial phenomena like ratchetting or cyclic hardening/softening into account by use of Jiang's model. Our numerical algorithm is designed for the case that the scalar load functions are piecewise linear and can be used in connection with critical plane/multiaxial rainflow methods in high cycle fatigue analysis. In addition, a local existence and uniqueness result of Jiang's equations is given.

- 262
- Regularized Fixed-Point Iterations for Nonlinear Inverse Problems (2005)
- In this paper we introduce a derivative-free, iterative method for solving nonlinear ill-posed problems \(Fx=y\), where instead of \(y\) noisy data \(y_\delta\) with \(|| y-y_\delta ||\leq \delta\) are given and \(F:D(F)\subseteq X \rightarrow Y\) is a nonlinear operator between Hilbert spaces \(X\) and \(Y\). This method is defined by splitting the operator \(F\) into a linear part \(A\) and a nonlinear part \(G\), such that \(F=A+G\). Then iterations are organized as \(A u_{k+1}=y_\delta-Gu_k\). In the context of ill-posed problems we consider the situation when \(A\) does not have a bounded inverse, thus each iteration needs to be regularized. Under some conditions on the operators \(A\) and \(G\) we study the behavior of the iteration error. We obtain its stability with respect to the iteration number \(k\) as well as the optimal convergence rate with respect to the noise level \(\delta\), provided that the solution satisfies a generalized source condition. As an example, we consider an inverse problem of initial temperature reconstruction for a nonlinear heat equation, where the nonlinearity appears due to radiation effects. The obtained iteration error in the numerical results has the theoretically expected behavior. The theoretical assumptions are illustrated by a computational experiment.

- 261
- Initial Temperature Reconstruction for a Nonlinear Heat Equation: Application to Radiative Heat Transfer (2005)
- Consider a cooling process described by a nonlinear heat equation. We are interested to recover the initial temperature from temperature measurements which are available on a part of the boundary for some time. Up to now even for the linear heat equation such a problem has been usually studied as a nonlinear ill-posed operator equation, and regularization methods involving Frechet derivatives have been applied. We propose a fast derivative-free iterative method. Numerical results are presented for the glass cooling process, where nonlinearity appears due to radiation.

- 260
- On an asymptotic expansion for porous media flow of Carreau fluids (2004)
- Porous media flow of polymers with Carreau law viscosities and their application to enhanced oil recovery (EOR) is considered. Applying the homogenization method leads to a nonlinear two-scale problem. In case of a small difference between the Carreau and the Newtonian case an asymptotic expansion based on the small deviation of the viscosity from the Newtonian case is introduced. For uni-directional pressure gradients, which is a reasonable assumption in applications like EOR, auxiliary problems to decouple the micro- from the macrovariables are derived. The microscopic flow field obtained by the proposed approach is compared to the solution of the two-scale problem. Finite element calculations for an isotropic and an anisotropic pore cell geometries are used to validate the accuracy and speed-up of the proposed approach. The order of accuracy has been studied by performing the simulations up to the third order expansion for the isotropic geometry.

- 259
- Algebraic Systems Theory (2004)
- Control systems are usually described by differential equations, but their properties of interest are most naturally expressed in terms of the system trajectories, i.e., the set of all solutions to the equations. This is the central idea behind the so-called "behavioral approach" to systems and control theory. On the other hand, the manipulation of linear systems of differential equations can be formalized using algebra, more precisely, module theory and homological methods ("algebraic analysis"). The relationship between modules and systems is very rich, in fact, it is a categorical duality in many cases of practical interest. This leads to algebraic characterizations of structural systems properties such as autonomy, controllability, and observability. The aim of these lecture notes is to investigate this module-system correspondence. Particular emphasis is put on the application areas of one-dimensional rational systems (linear ODE with rational coefficients), and multi-dimensional constant systems (linear PDE with constant coefficients).

- 258
- A Model for Spherical SH-Wave Propagation in Self-reinforced Linearly Elastic Media (2003)
- The original publication is available at www.springerlink.com. This original publication also contains further results. We study a spherical wave propagating in radius- and latitude-direction and oscillating in latitude-direction in case of fibre-reinforced linearly elastic material. A function system solving Euler's equation of motion in this case and depending on certain Bessel and associated Legendre functions is derived.

- 257
- Regularized Multiresolution Recovery of the Mass Density Distribution from Satellite Data of the Earth's Gravitational Field (2003)
- The inverse problem of recovering the Earth's density distribution from satellite data of the first or second derivative of the gravitational potential at orbit height is discussed. This problem is exponentially ill-posed. In this paper a multiscale regularization technique using scaling functions and wavelets constructed for the corresponding integro-differential equations is introduced and its numerical applications are discussed. In the numerical part the second radial derivative of the gravitational potential at 200 km orbit height is calculated on a point grid out of the NASA/GSFC/NIMA Earth Geopotential Model (EGM96). Those simulated derived data out of SGG satellite measurements are taken for convolutions with the introduced scaling functions yielding a multiresolution analysis of harmonic density variations in the Earth's crust.

- 255
- Smoothing Splines in Multiscale Geopotential Determination from Satellite Data (2003)
- SST (satellite-to-satellite tracking) and SGG (satellite gravity gradiometry) provide data that allows the determination of the first and second order radial derivative of the earth's gravitational potential on the satellite orbit, respectively. The modeling of the gravitational potential from such data is an exponentially ill-posed problem that demands regularization. In this paper, we present the numerical studies of an approach, investigated in [24] and [25], that reconstructs the potential with spline smoothing. In this case, spline smoothing is not just an approximation procedure but it solves the underlying compact operator equation of the SST-problem and the SGG-problem. The numerical studies in this paper are performed for a simplified geometrical scenario with simulated data, but the approach is designed to handle first or second order radial derivative data on a real satellite orbit.