We investigate frictional contact problems for discrete linear elastic structures, in particular the quasistatic incremental problem and the rate problem. It is shown that sharp conditions on the coefficients of friction for unique solvability of these problems are the same. We also give explicit expressions of these critical bounds by using a method of optimization. For the case of two spatial dimensions the conditions are formulated as a huge set of non symmetric eigenvalue problem. A computer program for solving these problems was designed and used to compute the critical bounds for some structures of relative small size, some of which appeared in the literature. The results of a variety of numerical experiments with uniform and non uniform distributions of the frictional properties are presented. (C) 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
In a recent author's paper four classes of methods of the secant type for solving systems of nonlinear equations were proposed. They are stable with respect to linear dependence of the search directions. If the Jacobian matrix is symmetric, then two of them take into account this property. It is proved here that some four special methods from the classes converge superlinearly.
The following problem is considered. Given m + 1 points {x_{i}}_{0}^{m} in R^{n} which generate an m-dimensional linear manifold, construct for this manifold a maximally linearly independent basis that consists of vectors of the form x_{i} - x_{j}. This problem is present in, e.g., stable variants of the secant method, where it is required to approximate the Jacobian matrix f^{'} of a nonlinear mapping f by using values off computed at m + 1 points. In this case, it is also desirable to have a combination of finite differences with maximal linear independence. As a natural measure of linear independence, we consider a functional which is maximized to find an optimal combination of m pairs {x_{i}, x_{j}}. We show that the problem is not of combinatorial complexity but can be reduced to the minimum spanning tree (MST) problem, which is solved by an MST-type algorithm in O(m^{2}n) time.
We consider structural topology optimization problems including unilateral constraints arising from, for example, non-penetration conditions in contact mechanics or non-compression conditions for elastic ropes. To construct more realistic models and to hedge off possible failures or an inefficient behaviour of optimal structures, we allow parameters (for example, loads) defining the problem to be stochastic. The resulting nonsmooth stochastic optimization problem is an instance of stochastic mathematical programs with equilibrium constraints (SMPEC), or stochastic bilevel programs. The existence as well as the continuity of optimal solutions with respect to the lower bounds on the design variables are established. The question of continuity of the optimal solutions with respect to small changes in the probability measure is analysed. For a subclass of the problems considered the answer is affirmative, thus establishing the robustness of optimal solutions. © 2003 WILEY-VCH Verlag GmbH & Co. KGaA.
Phenomena such as biological growth and damage evolution can be thought of as time evolving processes, the directions of which are governed by descendent of certain goal functions. Mathematically this means using a dynamical systems approach to optimization. We extend such an approach by introducing a field quantity, representing nutrients or other non-mechanical stimuli, that modulate growth and damage evolution. The derivation of a generic model is systematic, starting from a Lyaponov-type descent condition and utilizing a Coleman-Noll strategy. A numerical algorithm for finding stationary points of the resulting dynamical system is suggested and applied to two model problems where the influence of different levels of nutrient sensitivity are observed. The paper demonstrates the use of a new modeling technique and shows its application in deriving a generic problem of growth and damage evolution. (C) 2014 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim
An alternating procedure for solving a Cauchy problem for the stationary Stokes system is presented. A convergence proof of this procedure and numerical results are included. © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
In this paper we are concerned with boundary value problems for general second order elliptic equations and systems in a polyhedral cone. We obtain point estimates of Green's matrix in different areas of the cone. The proof of these estimates is essentially based on weighted L-2 estimates for weak solutions and their derivatives. As examples, we consider the Neumann problem to the Laplace equation and the Lame system.
In this paper we are concerned with boundary value problems for general second order elliptic equations and systems in a polyhedral domain. We consider solutions in weighted Lp Sobolev spaces. A special section is dedicated to weak solutions. We prove solvability theorems and regularity assertions for the solutions. © 2003 WILEY-VCH Verlag GmbH & Co. KGaA.
We consider a general discrete structural optimization problem including unilateral constraints arising from, for example, non-penetration conditions in contact mechanics or non-compression conditions for elastic ropes. The loads applied (and, in principle, also other data such as the initial distances to the supports), are allowed to be stochastic, which we handle through a discretization of the probability space. The existence of optimal solutions to the resulting problem is established, as well as the continuity properties of the equilibrium displacements and forces with respect to the lower bounds on the design variables. The latter feature is important in topology optimization, in which one includes the possibility of vanishing structural parts by setting design variable values to zero. In design optimization computations, one usually replaces the zero lower design bound by a strictly positive number, hence rewriting the problem into a sizing form. For several such perturbations, we prove that the global optimal designs and equilibrium states converge to the correct ones as the lower bound converges to zero.
We examine the existence of solutions to incremental friction problems for a continuous elastic object in contact with obstacles. The contact is modelled by a modified Signorini impenetrability contact condition. In particular, we consider the case when the boundary conditions of the object permit the object to perform rigid body motions. This means that the friction forces have to balance the applied forces if the object is to remain stationary. The main result is friction dependent conditions on the applied force that is sufficient for the existence of solutions to the incremental friction problem. © 2007 WILEY-VCH Verlag GmbH & Co. KGaA.
We present a sufficient condition for the existence of solutions to noncoercive incremental friction problems for discrete systems in contact with obstacles. By discrete system we mean that the displacement of the object can be discribed by a finite number of displacement variables, and by noncoercive we mean that the stiffness matrix of the object is semidefinite. We have a noncoercive friction problem when the object is not fixed to a support. This means that the friction forces need to balance the applied forces if the object is to remain stationary. This is manifested in our condition for existence of solutions. This condition is a compatibility condition on the applied force field, and if it is violated there exists a nontrivial solution to a corresponding dynamical problem. © 2006 Wiley-VCH Verlag GmbH & Co. KGaA.
In this paper, an optimization method for systems described by linearizedviscous shallow water equations is proposed. It can potentially be used tooptimize bottom profiles and is based on techniques commonly found in thearea of topology optimization of structures, but with the difference that acontinuous, and not discrete, design is sought. The presentation includes aderivation and linearization of the stationary viscous shallow water equations,which are used as state equations in the optimization problem. The finalproblem definition is closely related to the Darcy–Stokes fluid flow topologyoptimization problem, from which many results are derived by analogy.
From the numerical examples it is concluded that natural–like designscan be created for proper choices of parameter values. However, presentlythe model is not physically validated and its exact domain of application isnot yet known.
A structure consisting of two elastic plates bonded by a thin and soft adhesive layer is considered. Non-linear plate models are obtained by using the method of formal asymptotic expansions. A distinguishing feature of this problem is that the only quantitative property of the adhesive that has to be taken into account is the response of the material subjected to a pure shear load.