共查询到20条相似文献,搜索用时 15 毫秒
1.
We present a new approach to the variational relaxation of functionals
of the type:where
is a continuous function with growth conditions of order p≥1 but not necessarily convex. We essentially study the case when μ is the k-dimensional Hausdorff measure restricted to a suitable piece of a k-dimensional smooth submanifold of
. 相似文献
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Nikolaos I. Katzourakis 《Calculus of Variations and Partial Differential Equations》2013,46(3-4):505-522
We establish Maximum Principles which apply to vectorial approximate minimizers of the general integral functional of Calculus of Variations. Our main result is a version of the Convex Hull Property. The primary advance compared to results already existing in the literature is that we have dropped the quasiconvexity assumption of the integrand in the gradient term. The lack of weak Lower semicontinuity is compensated by introducing a nonlinear convergence technique, based on the approximation of the projection onto a convex set by reflections and on the invariance of the integrand in the gradient term under the Orthogonal Group. Maximum Principles are implied for the relaxed solution in the case of non-existence of minimizers and for minimizing solutions of the Euler–Lagrange system of PDE. 相似文献
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J. P. Raymond 《Journal of Optimization Theory and Applications》1994,82(3):571-592
In some physical problems (mechanical problems, optimal control problems, phase transition problems, etc.), we have to minimize a functionalJ over a topological spaceU for whichJ is not sequentially lower semicontinuous. In this article, we prove new existence results for general one-dimensional vector problems of calculus of variations without any convexity condition on the integrand of the problem. In particular, we do not suppose that the integrand is split in two parts, one part depending on the gradient variable and the other part depending on the state variable, as is often supposed in recent results. In the case where the integrand is the sum of two functions, the first one depending on the gradient variable and the second one depending on the state variable, we also prove a uniqueness result without any convexity assumption with respect to the gradient variable.A preliminary version of some results given in this article was presented at the Workshop on Calculus of Variations and Nonlinear Elasticity organized at Cortona, Italy, 27–31 May 1991 by B. Dacorogna, P. Marcellini, and C. Sbordone. The author would like to thank the organizers of this workshop for their invitation. 相似文献
6.
Numerical analysis of oscillations in nonconvex problems 总被引:3,自引:0,他引:3
M. Chipot 《Numerische Mathematik》1991,59(1):747-767
Summary We study numerically the pattern of the minimizing sequences of nonconvex problems which do not admit a minimizer. 相似文献
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Faiz A. Al-Khayyal Christian Larsen Timothy Van Voorhis 《Journal of Global Optimization》1995,6(3):215-230
We present an algorithm for finding approximate global solutions to quadratically constrained quadratic programming problems. The method is based on outer approximation (linearization) and branch and bound with linear programming subproblems. When the feasible set is non-convex, the infinite process can be terminated with an approximate (possibly infeasible) optimal solution. We provide error bounds that can be used to ensure stopping within a prespecified feasibility tolerance. A numerical example illustrates the procedure. Computational experiments with an implementation of the procedure are reported on bilinearly constrained test problems with up to sixteen decision variables and eight constraints.This research was supported in part by National Science Foundation Grant DDM-91-14489. 相似文献
10.
Carsten Carstensen 《Numerische Mathematik》1999,82(4):577-597
Summary. The finite element method is a reasonable and frequently utilised tool for the spatial discretization within one time-step
in an elastoplastic evolution problem. In this paper, we analyse the finite element discretization and prove a priori and
a posteriori error estimates for variational inequalities corresponding to the primal formulation of (Hencky) plasticity.
The finite element method of lowest order consists in minimising a convex function on a subspace of continuous piecewise linear
resp. piecewise constant trial functions. An a priori error estimate is established for the fully-discrete method which shows
linear convergence as the mesh-size tends to zero, provided the exact displacement field u is smooth. Near the boundary of the plastic domain, which is unknown a priori, it is most likely that u is non-smooth. In this situation, automatic mesh-refinement strategies are believed to improve the quality of the finite
element approximation. We suggest such an adaptive algorithm on the basis of a computable a posteriori error estimate. This
estimate is reliable and efficient in the sense that the quotient of the error by the estimate and its inverse are bounded
from above. The constants depend on the hardening involved and become larger for decreasing hardening.
Received May 7, 1997 / Revised version received August 31, 1998 相似文献
11.
Based on the authors’ previous work which established theoretical foundations of two, conceptual, successive convex relaxation
methods, i.e., the SSDP (Successive Semidefinite Programming) Relaxation Method and the SSILP (Successive Semi-Infinite Linear Programming)
Relaxation Method, this paper proposes their implementable variants for general quadratic optimization problems. These problems
have a linear objective function c
T
x to be maximized over a nonconvex compact feasible region F described by a finite number of quadratic inequalities. We introduce two new techniques, “discretization” and “localization,”
into the SSDP and SSILP Relaxation Methods. The discretization technique makes it possible to approximate an infinite number
of semi-infinite SDPs (or semi-infinite LPs) which appeared at each iteration of the original methods by a finite number of
standard SDPs (or standard LPs) with a finite number of linear inequality constraints. We establish:?•Given any open convex set U containing F, there is an implementable discretization of the SSDP (or SSILP) Relaxation Method
which generates a compact convex set C such that F⊆C⊆U in a finite number of iterations.?The localization technique is for the cases where we are only interested in upper bounds on the optimal objective value (for
a fixed objective function vector c) but not in a global approximation of the convex hull of F. This technique allows us to generate a convex relaxation of F that is accurate only in certain directions in a neighborhood of the objective direction c. This cuts off redundant work to make the convex relaxation accurate in unnecessary directions. We establish:?•Given any positive number ε, there is an implementable localization-discretization of the SSDP (or SSILP) Relaxation Method
which generates an upper bound of the objective value within ε of its maximum in a finite number of iterations.
Received: June 30, 1998 / Accepted: May 18, 2000?Published online September 20, 2000 相似文献
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Summary In this paper, methods for numerical verifications of solutions for elliptic equations in nonconvex polygonal domains are studied. In order to verify solutions using computer, it is necessary to determine some constants which appear in a priori error estimations. We propose some methods for determination of these constants. In numerical examples, calculating these constants for anL-shaped domain, we verify the solution of a nonlinear elliptic equation. 相似文献
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Antonio Corbo Esposito Riccardo De Arcangelis 《Annali di Matematica Pura ed Applicata》1993,164(1):155-193
A comparison between some relaxation methods of an integral functional is carried out. The following relaxed functionals of the variational integral I(, u)=
:
相似文献
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I. Chryssoverghi 《Journal of Optimization Theory and Applications》1985,45(1):73-88
In this paper, we consider an optimal control problem for distributed systems governed by parabolic equations. The state equations are nonlinear in the control variable; the constraints and the cost functional are generally nonconvex. Relaxed controls are used to prove existence and derive necessary conditions for optimality. To compute optimal controls, a descent method is applied to the resulting relaxed problem. A numerical method is also given for approximating a special class of relaxed controls, notably those obtained by the descent method. Convergence proofs are given for both methods, and a numerical example is provided. 相似文献
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Carsten Carstensen 《Mathematical Methods in the Applied Sciences》1993,16(11):819-835
The three-dimensional interface problem with the homogeneous Lamé system in an unbounded exterior domain and holonomic material behaviour in a bounded interior Lipschitz domain is considered. Existence and uniqueness of solutions of the interface problem are obtained rewriting the exterior problem in terms of boundary integral operators following the symmetric coupling procedure. The numerical approximation of the solutions consists in coupling of the boundary element method (BEM) and the finite element method (FEM). A Céa-like error estimate is presented for the discrete solutions of the numerical procedure proving its convergence. 相似文献
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We prove some relaxation results in the spirit of Anza Hafsa and Mandallena for integral functionals arising in the study of coherent thermochemical equilibria for multiphase solids. The energy density exhibits an explicit dependence on the deformation gradient and on a vector field representing the chemical composition. The deformation gradient satisfies a determinant type constraint and the chemical composition a constraint on the modulus. To cite this article: E. Zappale, H. Zorgati, C. R. Acad. Sci. Paris, Ser. I 347 (2009). 相似文献
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In this article,the authors obtain an integral representation for the relaxation of the functional
F(x,u,Ω):={∫^f(x,u(x),εu(x))dx Ω if u∈W^1,1(Ω,R^N), +∞ otherwise, in the space of functions of bounded deformation,with respect to L^1-convergence.Here Eu represents the absolutely continuous part of the symmetrized distributional derivative Eu.f(x,p,ξ)satisfying weak convexity assumption. 相似文献 20.
Hilbeth P. Azikri de Deus Claudio R. Ávila S. Jr. Ivan Moura Belo André T. Beck 《Applied Mathematical Modelling》2012
The numerical simulation of the mechanical behavior of industrial materials is widely used for viability verification, improvement and optimization of designs. Elastoplastic models have been used to forecast the mechanical behavior of different materials. The numerical solution of most elastoplastic models comes across problems of ill-condition matrices. A complete representation of the nonlinear behavior of such structures involves the nonlinear equilibrium path of the body and handling of singular (limit) points and/or bifurcation points. Several techniques to solve numerical problems associated to these points have been disposed in the specialized literature. Two examples are the load-controlled Newton–Raphson method and displacement controlled techniques. However, most of these methods fail due to convergence problems (ill-conditioning) in the neighborhood of limit points, specially when the structure presents snap-through or snap-back equilibrium paths. This study presents the main ideas and formalities of the Tikhonov regularization method and shows how this method can be used in the analysis of dynamic elastoplasticity problems. The study presents a rigorous mathematical demonstration of existence and uniqueness of the solution of well-posed dynamic elastoplasticity problems. The numerical solution of dynamic elastoplasticity problems using Tikhonov regularization is presented in this paper. The Galerkin method is used in this formulation. Effectiveness of Tikhonov’s approach in the regularization of the solution of elastoplasticity problems is demonstrated by means of some simple numerical examples. 相似文献
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