96 research outputs found

    Minimization variational principles for acoustics, elastodynamics, and electromagnetism in lossy inhomogeneous bodies at fixed frequency

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    The classical energy minimization principles of Dirichlet and Thompson are extended as minimization principles to acoustics, elastodynamics and electromagnetism in lossy inhomogeneous bodies at fixed frequency. This is done by building upon ideas of Cherkaev and Gibiansky, who derived minimization variational principles for quasistatics. In the absence of free current the primary electromagnetic minimization variational principles have a minimum which is the time-averaged electrical power dissipated in the body. The variational principles provide constraints on the boundary values of the fields when the moduli are known. Conversely, when the boundary values of the fields have been measured, then they provide information about the values of the moduli within the body. This should have application to electromagnetic tomography. We also derive saddle point variational principles which correspond to variational principles of Gurtin, Willis, and Borcea.Comment: 32 pages 0 figures (Previous version omitted references

    On the forces that cable webs under tension can support and how to design cable webs to channel stresses

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    In many applications of Structural Engineering the following question arises: given a set of forces f1,f2,…,fN\mathbf{f}_1,\mathbf{f}_2,\dots,\mathbf{f}_N applied at prescribed points x1,x2,…,xN\mathbf{x}_1,\mathbf{x}_2,\dots,\mathbf{x}_N, under what constraints on the forces does there exist a truss structure (or wire web) with all elements under tension that supports these forces? Here we provide answer to such a question for any configuration of the terminal points x1,x2,…,xN\mathbf{x}_1,\mathbf{x}_2,\dots,\mathbf{x}_N in the two- and three-dimensional case. Specifically, the existence of a web is guaranteed by a necessary and sufficient condition on the loading which corresponds to a finite dimensional linear programming problem. In two-dimensions we show that any such web can be replaced by one in which there are at most PP elementary loops, where elementary means the loop cannot be subdivided into subloops, and where PP is the number of forces f1,f2,…,fN\mathbf{f}_1,\mathbf{f}_2,\dots,\mathbf{f}_N applied at points strictly within the convex hull of x1,x2,…,xN\mathbf{x}_1,\mathbf{x}_2,\dots,\mathbf{x}_N. In three-dimensions we show that, by slightly perturbing f1,f2,…,fN\mathbf{f}_1,\mathbf{f}_2,\dots,\mathbf{f}_N, there exists a uniloadable web supporting this loading. Uniloadable means it supports this loading and all positive multiples of it, but not any other loading. Uniloadable webs provide a mechanism for distributing stress in desired ways.Comment: 18 pages, 8 figure

    On periodic homogenization of highly contrasted elastic structures

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    While homogenization of periodic linear elastic structures is now a well-known procedure when the stiffness of the material varies inside fixed bounds, no homogenization formula is known which enables us to compute the effective properties of highly contrasted structures. Examples have been given in which the effective energy involves the strain-gradient but no general formula provides this strain-gradient dependence. Some formulas have been proposed which involve such terms and provide a small correction to the classical effective energy still when the stiffness of the material varies inside fixed bounds. The goal of this paper is to check the applicability of these formulas for highly contrasted structures. To that aim we focus on structures whose limit energy is already known and we compare the energies given by (i) the convergence results, (ii) the corrective formulas and (iii) by a direct numerical simulation of the complete structure

    A 1D continuum model for beams with pantographic microstructure: asymptotic micro-macro identification and numerical results

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    In the standard asymptotic micro-macro identification theory, starting from a De Saint-Venant cylinder, it is possible to prove that, in the asymptotic limit, only flexible, inextensible, beams can be obtained at the macro-level. In the present paper we address the following problem: is it possible to find a microstructure producing in the limit, after an asymptotic micro-macro identification procedure, a continuum macro-model of a beam which can be both extensible and flexible? We prove that under certain hypotheses, exploiting the peculiar features of a pantographic microstructure, this is possible. Among the most remarkable features of the resulting model we find that the deformation energy is not of second gradient type only because it depends, like in the Euler beam model, upon the Lagrangian curvature, i.e. the projection of the second gradient of the placement function upon the normal vector to the deformed line, but also because it depends upon the projection of the second gradient of the placement on the tangent vector to the deformed line, which is the elongation gradient. Thus, a richer set of boundary conditions can be prescribed for the pantographic beam model. Phase transition and elastic softening are exhibited as well. Using the resulting planar 1D continuum limit homogenized macro-model, by means of FEM analyses, we show some equilibrium shapes exhibiting highly non-standard features. Finally, we conceive that pantographic beams may be used as basic elements in double scale metamaterials to be designed in future

    A mathematical and numerical framework for ultrasonically-induced Lorentz force electrical impedance tomography

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    We provide a mathematical analysis and a numerical framework for Lorentz force electrical conductivity imaging. Ultrasonic vibration of a tissue in the presence of a static magnetic field induces an electrical current by the Lorentz force. This current can be detected by electrodes placed around the tissue; it is proportional to the velocity of the ultrasonic pulse, but depends nonlinearly on the conductivity distribution. The imaging problem is to reconstruct the conductivity distribution from measurements of the induced current. To solve this nonlinear inverse problem, we first make use of a virtual potential to relate explicitly the current measurements to the conductivity distribution and the velocity of the ultrasonic pulse. Then, by applying a Wiener filter to the measured data, we reduce the problem to imaging the conductivity from an internal electric current density. We first introduce an optimal control method for solving such a problem. A new direct reconstruction scheme involving a partial differential equation is then proposed based on viscosity-type regularization to a transport equation satisfied by the current density field. We prove that solving such an equation yields the true conductivity distribution as the regularization parameter approaches zero. We also test both schemes numerically in the presence of measurement noise, quantify their stability and resolution, and compare their performance

    Boundary Conditions in Porous Media: a Variational Approach

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    A general set of boundary conditions at fluid-permeable interfaces between dissimilar fluid-filled porous matrices is established starting from an extended Hamilton-Rayleigh principle. These conditions do include friction and inertial effects. Once linearized, they encompass boundary conditions relative to volume Darcy-Brinkman and to surface Saffman-Beavers-Joseph dissipation effects

    The obstacle problem in masonry structures and cable nets

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    We consider the problem of finding a net that supports prescribed forces applied at prescribed points, yet avoids certain obstacles, with all the elements of the net under compression (strut net) or under tension (cable web). In the case of masonry structures, for instance, this consists in finding a strut net that supports the forces, is contained within the physical structure, and avoids regions that may be not accessible due, for instance, to the presence of holes. We solve such a problem in the two-dimensional case, where the prescribed forces are applied at the vertices of a convex polygon, and we treat the cases of both single and multiple obstacles. By approximating the obstacles by polygonal regions, the task reduces to identifying the feasible domain in a linear programming problem. For a single obstacle we show how the region őď\Gamma available to the obstacle can be enlarged as much as possible in the sense that there is no other strut net, having a region őď1\Gamma_1 available to the obstacle with őď1‚äāőď\Gamma_1 \subset\Gamma. The case where some of the forces are reactive, unprescribed but reacting to the other prescribed forces, is also treated. It again reduces to identifying the feasible domain in a linear programming problem. Finally, one may allow a subset of the reactive forces to each act not at a prescribed point, but rather at any point on a prescribed line segment. Then the task reduces to identifying the feasible domain in a quadratic programming problem.Comment: 22 pages, 12 figures, 3 supplemental video
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