163 research outputs found
Mixed Finite Elements of Higher-Order in Elastoplasticity
In this paper a higher-order mixed finite element method for elastoplasticity
with linear kinematic hardening is analyzed. Thereby, the non-differentiability
of the involved plasticity functional is resolved by a Lagrange multiplier
leading to a three field formulation. The finite element discretization is
conforming in the displacement field and the plastic strain but potentially
non-conforming in the Lagrange multiplier as its Frobenius norm is only
constrained in a certain set of Gauss quadrature points. A discrete inf-sup
condition with constant 1 and the well posedness of the discrete mixed problem
are shown. Moreover, convergence and guaranteed convergence rates are proved
with respect to the mesh size and the polynomial degree, which are optimal for
the lowest order case. Numerical experiments underline the theoretical results.Comment: 20 page
Optimal control of the sweeping process over polyhedral controlled sets
The paper addresses a new class of optimal control problems governed by the
dissipative and discontinuous differential inclusion of the sweeping/Moreau
process while using controls to determine the best shape of moving convex
polyhedra in order to optimize the given Bolza-type functional, which depends
on control and state variables as well as their velocities. Besides the highly
non-Lipschitzian nature of the unbounded differential inclusion of the
controlled sweeping process, the optimal control problems under consideration
contain intrinsic state constraints of the inequality and equality types. All
of this creates serious challenges for deriving necessary optimality
conditions. We develop here the method of discrete approximations and combine
it with advanced tools of first-order and second-order variational analysis and
generalized differentiation. This approach allows us to establish constructive
necessary optimality conditions for local minimizers of the controlled sweeping
process expressed entirely in terms of the problem data under fairly
unrestrictive assumptions. As a by-product of the developed approach, we prove
the strong -convergence of optimal solutions of discrete
approximations to a given local minimizer of the continuous-time system and
derive necessary optimality conditions for the discrete counterparts. The
established necessary optimality conditions for the sweeping process are
illustrated by several examples
Stable numerical methodology for variational inequalities with application in quantitative finance and computational mechanics
Coercivity is a characteristic property of the bilinear term in a weak form of a partial differential equation in both infinite space and the corresponding finite space utilized by a numerical scheme. This concept implies \textit{stability} and \textit{well-posedness} of the weak form in both the exact solution and the numerical solution. In fact, the loss of this property especially in finite dimension cases leads to instability of the numerical scheme. This phenomenon occurs in three major families of problems consisting of advection-diffusion equation with dominant advection term, elastic analysis of very thin beams, and associated plasticity and non-associated plasticity problems. There are two main paths to overcome the loss of coercivity, first manipulating and stabilizing a weak form to ensure that the discrete weak form is coercive, second using an automatically stable method to estimate the solution space such as the Discontinuous Petrov Galerkin (DPG) method in which the optimal test space is attained during the design of the method in such a way that the scheme keeps the coercivity inherently. In this dissertation, A stable numerical method for the aforementioned problems is proposed. A stabilized finite element method for the problem of migration risk problem which belongs to the family of the advection-diffusion problems is designed and thoroughly analyzed. Moreover, DPG method is exploited for a wide range of valuing option problems under the black-Scholes model including vanilla options, American options, Asian options, double knock barrier options where they all belong to family of advection-diffusion problem, and elastic analysis of Timoshenko beam theory. Besides, The problem of American option pricing, migration risk, and plasticity problems can be categorized as a free boundary value problem which has their extra complexity, and optimization theory and variational inequality are the main tools to study these families of the problems. Thus, an overview of the classic definition of variational inequalities and different tools and methods to study analytically and numerically this family of problems is provided and a novel adjoint sensitivity analysis of variational inequalities is proposed
Topology optimization for incremental elastoplasticity: a phase-field approach
We discuss a topology optimization problem for an elastoplastic medium. The
distribution of material in a region is optimized with respect to a given
target functional taking into account compliance. The incremental elastoplastic
problem serves as state constraint. We prove that the topology optimization
problem admits a solution. First-order optimality conditions are obtained by
considering a regularized problem and passing to the limit
Efficient fast Fourier transform-based solvers for computing the thermomechanical behavior of applied materials
The mechanical behavior of many applied materials arises from their microstructure. Thus, to aid the design, development and industrialization of new materials, robust computational homogenization methods are indispensable. The present thesis is devoted to investigating and developing FFT-based micromechanics solvers for efficiently computing the (thermo)mechanical response of nonlinear composite materials with complex microstructures
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