Universitätsbibliothek
Weimar Open Access

Optimale Positionierung von Messeinrichtungen an Staumauern zur Bauwerksüberwachung

Optimal positioning of sensors for the monitoring of water dams

In the paper presented, reinforced concrete shells of revolution are analyzed in both meridional and circumferential directions. Taking into account the physical non-linearity of the material, the internal forces and the deflections of the shell as well as the strain distribution at the cross-sections are calculated. The behavior of concrete under compression is described by linear and non-linear stress-strain relations. The description of the behavior of concrete under tension must account for tension stiffening effects. A tri-linear function is used to formulate the material law of reinforcement. The problem cannot be solved analytically due to the physical non-linearity. Thus a numerical solution is formulated by means of the LAGRANGE Principle of the minimum of the total potential energy. The kinematically admissible field of deformation is defined by the displacements u in the meridional and w in the radial direction. These displacements must satisfy the equations of compatibility and the kinematical boundary conditions of the shell. The strains are linearly distributed across the wall thickness. The strain energy depends on the specific of the material behavior. Using integral formulations of the material law [1], the strain energy of each part of the cross-section is defined as a function of the strains at the boundaries of the cross-sections. The shell is discretised in the meridional direction. Various methods of numerical differentiation and numerical integration are applied in order to determine the deformations and the strain energy. The unknown displacements u and w are calculated by a non-restricted extremum problem based on the minimum of the total potential energy. From mathematical point of view, the objective function is a convex function, thus the minimum can be determined without difficulty. The advantage of this formulation is that unlike non-linear methods with path-following algorithms the calculation does not have to account for changing stiffness and load increments. All iterations necessary to find the solution are integrated into the “Solver”. The model presented provides many ways of investigating the influence of various material parameters on the stresses and deformations of the entire shell structure.

Using a quaternionic reformulation of the electrical impedance equation, we consider a two-dimensional separable-variables conductivity function and, posing two different techniques, we obtain a special class of Vekua equation, whose general solution can be approach by virtue of Taylor series in formal powers, for which is possible to introduce an explicit Bers generating sequence.

In this note, we describe quite explicitly the Howe duality for Hodge systems and connect it with the well-known facts of harmonic analysis and Clifford analysis. In Section 2, we recall briefly the Fisher decomposition and the Howe duality for harmonic analysis. In Section 3, the well-known fact that Clifford analysis is a real refinement of harmonic analysis is illustrated by the Fisher decomposition and the Howe duality for the space of spinor-valued polynomials in the Euclidean space under the so-called L-action. On the other hand, for Clifford algebra valued polynomials, we can consider another action, called in Clifford analysis the H-action. In the last section, we recall the Fisher decomposition for the H-action obtained recently. As in Clifford analysis the prominent role plays the Dirac equation in this case the basic set of equations is formed by the Hodge system. Moreover, analysis of Hodge systems can be viewed even as a refinement of Clifford analysis. In this note, we describe the Howe duality for the H-action. In particular, in Proposition 1, we recognize the Howe dual partner of the orthogonal group O(m) in this case as the Lie superalgebra sl(2 1). Furthermore, Theorem 2 gives the corresponding multiplicity free decomposition with an explicit description of irreducible pieces.

In this paper we consider the time independent Klein-Gordon equation on some conformally flat 3-tori with given boundary data. We set up an explicit formula for the fundamental solution. We show that we can represent any solution to the homogeneous Klein-Gordon equation on the torus as finite sum over generalized 3-fold periodic elliptic functions that are in the kernel of the Klein-Gordon operator. Furthermore we prove Cauchy and Green type integral formulas and set up a Teodorescu and Cauchy transform for the toroidal Klein-Gordon operator. These in turn are used to set up explicit formulas for the solution to the inhomogeneous version of the Klein-Gordon equation on the 3-torus.

In this paper we present rudiments of a higher dimensional analogue of the Szegö kernel method to compute 3D mappings from elementary domains onto the unit sphere. This is a formal construction which provides us with a good substitution of the classical conformal Riemann mapping. We give explicit numerical examples and discuss a comparison of the results with those obtained alternatively by the Bergman kernel method.

The present article proposes an alternative way to compute the torsional stiffness based on three-dimensional continuum mechanics instead of applying a specific theory of torsion. A thin, representative beam slice is discretized by solid finite elements. Adequate boundary conditions and coupling conditions are integrated into the numerical model to obtain a proper answer on the torsion behaviour, thus on shear center, shear stress and torsional stiffness. This finite element approach only includes general assumptions of beam torsion which are independent of cross-section geometry. These assumptions essentially are: no in-plane deformation, constant torsion and free warping. Thus it is possible to achieve numerical solutions of high accuracy for arbitrary cross-sections. Due to the direct link to three-dimensional continuum mechanics, it is possible to extend the range of torsion analysis to sections which are composed of different materials or even to heterogeneous beams on a high scale of resolution. A brief study follows to validate the implementation and results are compared to analytical solutions.

The numerical simulation of microstructure models in 3D requires, due to enormous d.o.f., significant resources of memory as well as parallel computational power. Compared to homogeneous materials, the material hetrogeneity on microscale induced by different material phases demand for adequate computational methods for discretization and solution process of the resulting highly nonlinear problem. To enable an efficient/scalable solution process of the linearized equation systems the heterogeneous FE problem will be described by a FETI-DP (Finite Element Tearing and Interconnecting - Dual Primal) discretization. The fundamental FETI-DP equation can be solved by a number of different approaches. In our approach the FETI-DP problem will be reformulated as Saddle Point system, by eliminating the primal and Lagrangian variables. For the reduced Saddle Point system, only defined by interior and dual variables, special Uzawa algorithms can be adapted for iteratively solving the FETI-DP saddle-point equation system (FETI-DP SPE). A conjugate gradient version of the Uzawa algorithm will be shown as well as some numerical tests regarding to FETI-DP discretization of small examples using the presented solution technique. Furthermore the inversion of the interior-dual Schur complement operator can be approximated using different techniques building an adequate preconditioning matrix and therewith leading to substantial gains in computing time efficiency.

In nonlinear simulations the loading is, in general, applied in an incremental way. Path-following algorithms are used to trace the equilibrium path during the failure process. Standard displacement controlled solution strategies fail if snap-back phenomena occur. In this contribution, a path-following algorithm based on the dissipation of the inelastic energy is presented which allows for the simulation of snap-backs. Since the constraint is defined in terms of the internal energy, the algorithm is not restricted to continuum damage models. Furthermore, no a priori knowledge about the final damage distribution is required. The performance of the proposed algorithm is illustrated using nonlinear mesoscale simulations.