A discontinuous Galerkin formulation for nonlinear analysis of multilayered shells refined theories

A novel pure penalty discontinuous Galerkin method is proposed for the geometrically nonlinear analysis of multilayered composite plates and shells, modelled via high-order refined theories. The approach allows to build different two-dimensional equivalent single layer structural models, which are obtained by expressing the covariant components of the displacement field through-the-thickness via Taylor’s polynomial expansion of different order. The problem governing equations are deduced starting from the geometrically nonlinear principle of virtual displacements in a total Lagrangian formulation. They are addressed with a pure penalty discontinuous Galerkin method using Legendre polynomials trial functions. The resulting nonlinear algebraic system is solved by a Newton–Raphson arc-length linearization scheme. Numerical tests involving plates and shells are proposed to validate the method, by comparison with literature benchmark problems and finite element solutions, and to assess its features. The obtained results demonstrate the accuracy of the method as well as the effectiveness of high-order elements

Optimization of shell structure acoustics

This thesis analyzes a mathematical model for shell structure acoustics, and develops and implements the adjoint equations for this model. The adjoint equations allow the computation of derivatives with respect to large parameter sets in shape optimization problems where the thickness and mid-surface of the shell are computed so as to generate a radiated sound field subject to broad-band design requirements. The structure and acoustics are modeled, respectively, via the Naghdi shell equations, and thin boundary integral equations, with full coupling at the shell mid-surface. In this way, the three-dimensional structural-acoustic equations can be posed as a problem on the two-dimensional mid-surface of the shell. A wide variety of shapes can thus be explored without re-meshing, and the acoustic field can be computed anywhere in the exterior domain with little additional effort. The problem is discretized using triangular MITC shell elements and piecewise-linear Galerkin boundary elements, coupled with a simple one-to-one scheme. Prior optimization work on coupled shell-acoustics problems has been focused on applications with design requirements over a small range of frequencies. These problems are amenable to a number of simplifying assumptions. In particular, it is often assumed that the structure is dense enough that the air pressure loading can be neglected, or that the structural motions can be expanded in a basis of low-frequency eigenmodes. Optimization of this kind can be done with reasonable success using a small number of shape parameters because simple modal analysis permits a reasonable knowledge of the parts of the design that will require modification. None of these assumptions are made in this thesis. By using adjoint equations, derivatives of the radiated field can be efficiently computed with respect to large numbers of shape parameters, allowing a much richer space of shapes, and thus, a broader range of design problems to be considered. The adjoint equation approach developed in this thesis is applied to the computation of optimal mid-surfaces and shell thicknesses, using a large shape parameter set, proportional in size to the number of degrees of freedom in the underlying finite element discretization

Hp-spectral Methods for Structural Mechanics and Fluid Dynamics Problems

We consider the usage of higher order spectral element methods for the solution of problems in structures and fluid mechanics areas. In structures applications we study different beam theories, with mixed and displacement based formulations, consider the analysis of plates subject to external loadings, and large deformation analysis of beams with continuum based formulations. Higher order methods alleviate the problems of locking that have plagued finite element method applications to structures, and also provide for spectral accuracy of the solutions. For applications in computational fluid dynamics areas we consider the driven cavity problem with least squares based finite element methods. In the context of higher order methods, efficient techniques need to be devised for the solution of the resulting algebraic systems of equations and we explore the usage of element by element bi-orthogonal conjugate gradient solvers for solving problems effectively along with domain decomposition algorithms for fluid problems. In the context of least squares finite element methods we also explore the usage of Multigrid techniques to obtain faster convergence of the the solutions for the problems of interest. Applications of the traditional Lagrange based finite element methods with the Penalty finite element method are presented for modelling porous media flow problems. Finally, we explore applications to some CFD problems namely, the flow past a cylinder and forward facing step

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells

Hybridizable discontinuous Galerkin method for nonlinear elasticity

University of Minnesota Ph.D. dissertation. November 2017. Major: Mathematics. Advisor: Bernardo Cockburn. 1 computer file (PDF); 1viii, 128 pages.Hybridizable discontinuous Galerkin method for nonlinear elasticit

The Sixth Copper Mountain Conference on Multigrid Methods, part 1

The Sixth Copper Mountain Conference on Multigrid Methods was held on 4-9 Apr. 1993, at Copper Mountain, CO. This book is a collection of many of the papers presented at the conference and as such represents the conference proceedings. NASA LaRC graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection clearly shows its rapid trend to further diversity and depth

Numerical Methods for Optimal Transport and Elastic Shape Optimization

In this thesis, we consider a novel unbalanced optimal transport model incorporating singular sources, we develop a numerical computation scheme for an optimal transport distance on graphs, we propose a simultaneous elastic shape optimization problem for bone tissue engineering, and we investigate optimal material distributions on thin elastic objects. The by now classical theory of optimal transport admits a metric between measures of the same total mass. Various generalizations of this so-called Wasserstein distance have been recently studied in the literature. In particular, these have been motivated by imaging applications, where the mass-preserving condition is too restrictive. Based on the Benamou Brenier formulation we present a novel unbalanced optimal transport model by introducing a source term in the continuity equation, which is incorporated in the path energy by a squared L2-norm in time of a functional with linear growth in space. As a key advantage of our model, this source term functional allows singular sources in space. We demonstrate the existence of constant speed geodesics in the space of Radon measures. Furthermore, for a numerical computation scheme, we apply a proximal splitting algorithm for a finite element discretization. On discrete spaces, Maas introduced a Benamou Brenier formulation, where a kinetic energy is defined via an appropriate (e.g., logarithmic) averaging of mass on nodes and momentum on edges. Concerning a numerical optimization scheme, this, unfortunately, couples all these variables on the graph. We propose a conforming finite element discretization in time and prove convergence of corresponding path energy minimizing curves. To apply a proximal splitting algorithm, we introduce suitable auxiliary variables. Besides similar projections as for the classical optimal transport distance and additional simple operations, this allows us to separate the nonlinearity given by the averaging operator to projections onto three-dimensional convex sets, the associated (e.g., logarithmic) cones. In elastic shape optimization, we are usually concerned with finding a subdomain maximizing the mechanical stability w.r.t. given forces acting onto a larger domain of interest. Motivated by a biomechanical application in bone tissue engineering, where recently biologically degradable polymers have been explored as bone substitutes, we propose a simultaneous elastic shape optimization problem to guarantee stiffness of the polymer implant and of the complementary set where new bone tissue will grow first. Under the assumption that the microstructure of the scaffold is periodic, we optimize a single microcell. We define a novel cost functional depending on specific entries of the homogenized elasticity tensors of polymer and regrown bone. Additionally, the perimeter is penalized for regularizing the interface of the scaffold. For a numerical optimization scheme, we choose a phase-field model, which allows a diffuse approximation of the elastic objects and the perimeter by the Modica Mortola functional. We also incorporate further biomechanically relevant constraints like the diffusivity of the regrown bone. Finally, we investigate shape optimization problems for thin elastic objects. For a numerical discretization, we take into account the discrete Kirchhoff triangle (DKT) element for parametric surfaces and approximate the material distribution by a phase-field. To describe equilibrium deformations for a given force, we study different corresponding state equations. In particular, we consider nonlinear elasticity combining membrane and bending models. Furthermore, a special focus is on pure bending isometries, which can be efficiently approximated by the DKT element. We also analyze a one-dimensional model of nonlinear elastic planar beams, where our numerical simulations confirm and extend a theoretical classification result of the optimal design

Stability Analysis of Plates and Shells

This special publication contains the papers presented at the special sessions honoring Dr. Manuel Stein during the 38th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference held in Kissimmee, Florida, Apdl 7-10, 1997. This volume, and the SDM special sessions, are dedicated to the memory of Dr. Manuel Stein, a major pioneer in structural mechanics, plate and shell buckling, and composite structures. Many of the papers presented are the work of Manny's colleagues and co-workers and are a result, directly or indirectly, of his influence. Dr. Stein earned his Ph.D. in Engineering Mechanics from Virginia Polytechnic Institute and State University in 1958. He worked in the Structural Mechanics Branch at the NASA Langley Research Center from 1943 until 1989. Following his retirement, Dr. Stein continued his involvement with NASA as a Distinguished Research Associate