Material Optimization for Nonlinearly Elastic Planar Beams

We consider the problem of an optimal distribution of soft and hard material for nonlinearly elastic planar beams. We prove that under gravitational force the optimal distribution involves no microstructure and is ordered, and we provide numerical simulations confirming and extending this observation.


Similar Publications

Sensitivity analysis (SA) is a procedure for studying how sensitive are the output results of large-scale mathematical models to some uncertainties of the input data. The models are described as a system of partial differential equations. Often such systems contain a large number of input parameters. Read More


The purpose of this work is to construct a simple, efficient and accurate well-balanced numerical scheme for one-dimensional (1D) blood flow in large arteries with varying geometrical and mechanical properties. As the steady states at rest are not relevant for blood flow, we construct two well-balanced hydrostatic reconstruction techniques designed to preserve low-Shapiro number steady states that may occur in large network simulations. The Shapiro number S h = u/c is the equivalent of the Froude number for shallow water equations and the Mach number for compressible Euler equations. Read More


This article describes the implementation of an all-in-one numerical procedure within the runtime StarPU. In order to limit the complexity of the method, for the sake of clarity of the presentation of the non-classical task-driven programming environnement, we have limited the numerics to first order in space and time. Results show that the task distribution is efficient if the tasks are numerous and individually large enough so that the task heap can be saturated by tasks which computational time covers the task management overhead. Read More


In this work, we derive particle schemes, based on micro-macro decomposition, for linear kinetic equations in the diffusion limit. Due to the particle approximation of the micro part, a splitting between the transport and the collision part has to be performed, and the stiffness of both these two parts prevent from uniform stability. To overcome this difficulty, the micro-macro system is reformulated into a continuous PDE whose coefficients are no longer stiff, and depend on the time step $\Delta t$ in a consistent way. Read More


The main focus of the present work is the inclusion of spatial adaptivity for snapshot computation in the offline phase of model order reduction utilizing Proper Orthogonal Decomposition (POD-MOR). For each time level, the snapshots live in different finite element spaces, which means in a fully discrete setting that the snapshots are vectors of different length. In order to overcome this obstacle, we present a discretization independent POD reduced order model, which is motivated from a continuous perspective and is set up explicitely without interpolation of the snapshots. Read More


This paper discusses the properties of certain risk estimators recently proposed to choose regularization parameters in ill-posed problems. A simple approach is Stein's unbiased risk estimator (SURE), which estimates the risk in the data space, while a recent modification (GSURE) estimates the risk in the space of the unknown variable. It seems intuitive that the latter is more appropriate for ill-posed problems, since the properties in the data space do not tell much about the quality of the reconstruction. Read More


We consider implementations of high-order finite difference Weighted Essentially Non-Oscillatory (WENO) schemes for the Euler equations in cylindrical and spherical coordinate systems with radial dependence only. The main concern of this work lies in ensuring both high-order accuracy and conservation. Three different spatial discretizations are assessed: one that is shown to be high-order accurate but not conservative, one conservative but not high-order accurate, and a new approach that is both high-order accurate and conservative. Read More


This paper investigates oscillation-free stability conditions of numerical methods for linear parabolic partial differential equations with some example extrapolations to nonlinear equations. Not clearly understood, numerical oscillations can create infeasible results. Since oscillation-free behavior is not ensured by stability conditions, a more precise condition would be useful for accurate solutions. Read More


Problems of stabilization of the unstable cycle of one-dimensional complex dynamical system are briefly discussed. These questions reduced to the problem of description of the ranges of polynomials $q(z) = q_1z + q_2z^2 +\dots + q_nz^n$ defined in the unit disk and normalized by the conditions $q(1) = 1 $ and this is the main subject of the present paper. Read More


The main goal of this paper is to investigate the order reduction phenomenon that appears in the integral deferred correction (InDC) methods based on implicit-explicit (IMEX) Runge-Kutta (R-K) schemes when applied to a class of stiff problems characterized by a small positive parameter $\varepsilon$, called singular perturbation problems (SPPs). In particular, an error analysis is presented for these implicit-explicit InDC (InDC-IMEX) methods when applied to SPPs. In our error estimate, we expand the global error in powers of $\varepsilon$ and show that its coefficients are global errors of the corresponding method applied to a sequence of differential algebraic systems. Read More