Optimal Elastostatic Cloaks

An elastic cloak hides a hole or an inhomogeneity from elastic fields. In this paper, a formulation of the optimal design of elastic cloaks based on the adjoint state method, in which the balance of linear momentum is enforced as a constraint, is presented. The design parameters are the elastic moduli of the cloak, while the objective function is a measure of the distance between the solutions in the physical and in the virtual bodies. Both the elastic medium and the cloak are assumed to be made of isotropic linear elastic materials. In order to guarantee smooth inhomogeneous elastic moduli within the cloak a penalization term is added to the objective function. Mixed finite elements are used for discretizing the weak formulation of the optimization problem. Several numerical examples of optimal elastic cloaks designed for both single and multiple loads are presented. We consider different geometries and loading types and observe that in some cases the optimal elastic cloaks for cloaking holes (cavities) are made of auxetic materials.Comment: 28 pages, 12 figure

Soft and transferable pseudopotentials from multi-objective optimization

Ab initio pseudopotentials are a linchpin of modern molecular and condensed matter electronic structure calculations. In this work, we employ multi-objective optimization to maximize pseudopotential softness while maintaining high accuracy and transferability. To accomplish this, we develop a formulation in which softness and accuracy are simultaneously maximized, with accuracy determined by the ability to reproduce all-electron energy differences between Bravais lattice structures, whereupon the resulting Pareto frontier is scanned for the softest pseudopotential that provides the desired accuracy in established transferability tests. We employ an evolutionary algorithm to solve the multi-objective optimization problem and apply it to generate a comprehensive table of optimized norm-conserving Vanderbilt (ONCV) pseudopotentials (https://github.com/SPARC-X/SPMS-psps). We show that the resulting table is softer than existing tables of comparable accuracy, while more accurate than tables of comparable softness. The potentials thus afford the possibility to speed up calculations in a broad range of applications areas while maintaining high accuracy.Comment: 13 pages, 4 figure

Compatible-strain mixed finite element methods for nonlinear elasticity

A new family of mixed finite element methods--compatible-strain mixed finite element methods (CSFEMs)--are introduced for compressible and incompressible nonlinear elasticity problems in dimensions two and three. A Hu-Washizu-type mixed formulation is considered and the displacement, the displacement gradient, and the first Piola-Kirchhoff stress are chosen as the independent unknowns. To impose incompressibility, a pressure-like field is introduced as the fourth independent unknown. Using the Hilbert complexes of nonlinear elasticity that describe the kinematics and the kinetics of motion, we identify the solution spaces that the independent unknown fields belong to. In particular, we define the displacement in H1, the displacement gradient in H(curl), the stress in H(div), and the pressure field in L2. The test spaces of the mixed formulations are chosen to be the same as their corresponding solution spaces. In a conforming setting, we approximate the solution and the test spaces with some piecewise polynomial subspaces of them. Among these approximation spaces are the tensorial analogues of the standard Nédélec and Raviart-Thomas finite element spaces of vector fields. This approach results in mixed finite element methods that, by construction, satisfy both the Hadamard jump conditions and the continuity of traction at the discrete level regardless of the refinement level of the mesh. This, in particular, makes CSFEMs quite efficient for modeling heterogeneous solids. We assess the performance of CSFEMs by solving several numerical examples in dimensions two and three and demonstrate their good performance for bending problems, for bodies with complex geometries, for different material models, and in the nearly incompressible regime. Using CSFEMs, one can model deformations with very large strains and accurately approximate stresses and the pressure field. Moreover, in our numerical examples, we do not observe any numerical artifacts such as checkerboarding of pressure, hourglass instability, or locking.Ph.D

Size-Dependent Bending, Buckling and Free Vibration Analyses of Microscale Functionally Graded Mindlin Plates Based on the Strain Gradient Elasticity Theory

Abstract In this paper, a size-dependent microscale plate model is developed to describe the bending, buckling and free vibration behaviors of microplates made of functionally graded materials (FGMs). The size effects are captured based on the modified strain gradient theory (MSGT), and the formulation of the paper is on the basis of Mindlin plate theory. The presented model accommodates the models based upon the classical theory (CT) and the modified couple stress theory (MCST) if all or two scale parameters are set to zero, respectively. By using Hamilton's principle, the governing equations and related boundary conditions are derived. The bending, buckling and free vibration problems are considered and are solved through the generalized differential quadrature (GDQ) method. A detailed parametric and comparative study is conducted to evaluate the effects of length scale parameter, material gradient index and aspect ratio predicted by the CT, MCST and MSGT on the deflection, critical buckling load and first natural frequency of the microplate. The numerical results indicate that the model developed herein is significantly size-dependent when the thickness of the microplate is on the order of the material scale parameters