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  • 1.
    Andersson, Lars-Erik
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Pinto da Costa, A.
    University of Lisbon, Portugal.
    Agwa, M. A.
    Zagazig University, Egypt.
    Existence and uniqueness for frictional incremental and rate problems - sharp critical bounds2016In: Zeitschrift für angewandte Mathematik und Mechanik, ISSN 0044-2267, E-ISSN 1521-4001, Vol. 96, no 1, p. 78-105Article in journal (Refereed)
    Abstract [en]

    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

  • 2.
    Burdakov, Oleg
    Computing Center of the USSR Academy of Sciences, Moscow.
    On Superlinear Convergence of Some Stable Variants of the Secant Method1986In: Zeitschrift für angewandte Mathematik und Mechanik, ISSN 0044-2267, E-ISSN 1521-4001, Vol. 66, no 2, p. 615-622Article in journal (Refereed)
    Abstract [en]

    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.

  • 3.
    Burdakov, Oleg
    CERFACS, Parallel Algorithms Team, Toulouse, France.
    On using the minimum spanning tree algorithm for optimal secant approximation of derivatives1996In: Zeitschrift für angewandte Mathematik und Mechanik, ISSN 0044-2267, E-ISSN 1521-4001, Vol. 76, no S3, p. 389-390Article in journal (Refereed)
    Abstract [en]

    The following problem is considered. Given m + 1 points {xi}0m in Rn which generate an m-dimensional linear manifold, construct for this manifold a maximally linearly independent basis that consists of vectors of the form xi - xj. 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 {xi, xj}. 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(m2n) time.

  • 4.
    Evgrafov, A.
    et al.
    Chalmers University of Technology, Göteborg University, Department of Mathematics, 412 96 Goteborg, Sweden.
    Patriksson, M.
    Chalmers University of Technology, Göteborg University, Department of Mathematics, 412 96 Goteborg, Sweden.
    Petersson, Joakim
    Linköping University, The Institute of Technology. Linköping University, Department of Management and Engineering, Mechanics .
    Stochastic structural topology optimization: Existence of solutions and sensitivity analyses2003In: Zeitschrift für angewandte Mathematik und Mechanik, ISSN 0044-2267, E-ISSN 1521-4001, Vol. 83, no 7, p. 479-492Article in journal (Refereed)
    Abstract [en]

    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.

  • 5.
    Klarbring, Anders
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mechanical Engineering, Engineering Mechanics.
    Pang, J-S
    The Discrete Steady Sliding Problem1999In: Zeitschrift für angewandte Mathematik und Mechanik, ISSN 0044-2267, E-ISSN 1521-4001, Vol. 79, no 2, p. 75-90Article in journal (Other (popular science, discussion, etc.))
  • 6.
    Klarbring, Anders
    et al.
    Linköping University, Department of Management and Engineering, Solid Mechanics. Linköping University, Faculty of Science & Engineering.
    Torstenfelt, Bo
    Linköping University, Department of Management and Engineering, Solid Mechanics. Linköping University, Faculty of Science & Engineering.
    Satha, Ganarupan
    Linköping University, Department of Management and Engineering, Mechanics. Linköping University, The Institute of Technology.
    Nutrient modulated structural design with application to growth and degradation2015In: Zeitschrift für angewandte Mathematik und Mechanik, ISSN 0044-2267, E-ISSN 1521-4001, Vol. 95, no 11, p. 1323-1334Article in journal (Refereed)
    Abstract [en]

    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

  • 7.
    Kozlov, Vladimir
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Baravdish, George
    Linköping University, The Institute of Technology. Linköping University, Department of Science and Technology.
    Johansson, Tomas
    Lesnic, Daniel
    An alternating method for the stationary Stokes system2006In: Zeitschrift für angewandte Mathematik und Mechanik, ISSN 0044-2267, E-ISSN 1521-4001, Vol. 86, no 4, p. 268-280Article in journal (Refereed)
    Abstract [en]

    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.

  • 8.
    Maz´ya, Vladimir G.
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Rossmann, J
    Linkoping Univ, Dept Math, S-58183 Linkoping, Sweden Univ Rostock, Dept Math, D-18051 Rostock, Germany.
    Point estimates for Green's matrix to boundary value problems for second order elliptic systems in a polyhedral cone2002In: Zeitschrift für angewandte Mathematik und Mechanik, ISSN 0044-2267, E-ISSN 1521-4001, Vol. 82, no 5, p. 291-316Article in journal (Refereed)
    Abstract [en]

    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.

  • 9.
    Maz´ya, Vladimir G.
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Rossmann, J.
    Roßmann, J., University of Rostock, Dept. of Mathematics, 18051 Rostock, Germany.
    Weighted Lp estimates of solutions to boundary value problems for second order elliptic systems in polyhedral domains2003In: Zeitschrift für angewandte Mathematik und Mechanik, ISSN 0044-2267, E-ISSN 1521-4001, Vol. 83, no 7, p. 435-467Article in journal (Refereed)
    Abstract [en]

    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.

  • 10.
    Patriksson, M
    et al.
    Chalmers Univ Technol, Dept Math, SE-41296 Gothenburg, Sweden Linkoping Univ, SE-58183 Linkoping, Sweden.
    Petersson, J
    Linköping University, Department of Mechanical Engineering, Fluid and Mechanical Engineering Systems. Linköping University, The Institute of Technology.
    Existence and continuity of optimal solutions to some structural topology optimization problems including unilateral constraints and stochastic loads2002In: Zeitschrift für angewandte Mathematik und Mechanik, ISSN 0044-2267, E-ISSN 1521-4001, Vol. 82, no 7, p. 435-459Article in journal (Refereed)
    Abstract [en]

    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.

  • 11.
    Rietz, Andreas
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Incremental friction problems for continuous elastic bodies with discretized contact2007In: Zeitschrift für angewandte Mathematik und Mechanik, ISSN 0044-2267, E-ISSN 1521-4001, Vol. 87, no 4, p. 290-302Article in journal (Refereed)
    Abstract [en]

    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.

  • 12.
    Rietz, Andreas
    et al.
    Siemens.
    Andersson, Lars-Erik
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Noncoercive incremental friction problems for discrete systems2006In: Zeitschrift für angewandte Mathematik und Mechanik, ISSN 0044-2267, E-ISSN 1521-4001, Vol. 86, no 12, p. 932-950Article in journal (Refereed)
    Abstract [en]

    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.

  • 13.
    Wiker, Niclas
    et al.
    Linköping University, Department of Management and Engineering, Mechanics. Linköping University, The Institute of Technology.
    Klarbring, Anders
    Linköping University, Department of Management and Engineering, Mechanics. Linköping University, The Institute of Technology.
    Borrvall, Thomas
    Linköping University, Department of Management and Engineering. Linköping University, The Institute of Technology.
    Simplified shallow water flow equations and their potential use in bottom profile optimization2007In: Zeitschrift für angewandte Mathematik und Mechanik, ISSN 0044-2267, E-ISSN 1521-4001, Vol. 87, no 11-12, p. 792-808Article in journal (Refereed)
    Abstract [en]

    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.

  • 14.
    Åslund, Jan
    Linköping University, The Institute of Technology. Linköping University, Department of Electrical Engineering, Vehicular Systems.
    Asymptotic analysis of adhesively bonded nonlinearly elastic plates2005In: Zeitschrift für angewandte Mathematik und Mechanik, ISSN 0044-2267, E-ISSN 1521-4001, Vol. 85, no 1, p. 22-35Article in journal (Refereed)
    Abstract [en]

    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.

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