EXACT AUGMENTED LAGRANGIAN FUNCTION FOR NONLINEAR PROGRAMMING PROBLEMS WITH INEQUALITY CONSTRAINTS

An exact augmented Lagrangian function for the nonlinear nonconvex programming problems with inequality constraints was discussed. Under suitable hypotheses, the relationship was established between the local unconstrained minimizers of the augmented Lagrangian function on the space of problem variables and the local minimizers of the original constrained problem. Furthermore, under some assumptions, the relationship was also established between the global solutions of the augmented Lagrangian function on some compact subset of the space of problem variables and the global solutions of the constrained problem. Therefore, from the theoretical point of view, a solution of the inequality constrained problem and the corresponding values of the Lagrange multipliers can be found by the well-known method of multipliers which resort to the unconstrained minimization of the augmented Lagrangian function presented.     

Local and Global Injective Solution of the Rotationally Symmetric Sphere Problem

There are problems in the classical linear theory of elasticity whose closed form solutions, while satisfying the governing equations of equilibrium together with well-posed boundary conditions, predict the existence of regions, often quite small, inside the body where material overlaps. Of course, material overlapping is not physically realistic, and one possible way to prevent it combines linear theory with the requirement that the deformation field be injective. A formulation of minimization problems in classical linear elasticity proposed by Fosdick and Royer [3] imposes this requirement through a Lagrange multiplier technique. An existence theorem for minimizers of plane problems is also presented. In general, however, it is not certain that such minimizers exist. Here, the Euler–Lagrange equations corresponding to a family of three-dimensional problems is investigated. In classical linear elasticity, these problems do not have bounded solutions inside a body of anisotropic material for a range of material parameters. For another range of parameters, bounded solutions do exist but yield stresses that are infinite at a point inside the body. In addition, these solutions are not injective in a region surrounding this point, yielding unrealistic behavior such as overlapping of material. Applying the formulation of Fosdick and Royer on this family of problems, it is shown that both the displacements and the constitutive part of the stresses are bounded for all values of the material parameters and that the injectivity constraint is preserved. In addition, a penalty functional formulation of the constrained elastic problems is proposed, which allows to devise a numerical approach to compute the solutions of these problems. The approach consists of finding the displacement field that minimizes an augmented potential energy functional. This augmented functional is composed of the potential energy of linear elasticity theory and of a penalty functional divided by a penalty parameter. A sequence of solutions is then constructed, parameterized by the penalty parameter, that converges to a function that satisfies the first variation conditions for a minimizer of the constrained minimization problem when this parameter tends to infinity. This approach has the advantages of being mathematically appealling and computationally simple to implement.     

Dimensionality of Local Minimizers of the Interaction Energy

In this work we consider local minimizers (in the topology of transport distances) of the interaction energy associated with a repulsive–attractive potential. We show how the dimensionality of the support of local minimizers is related to the repulsive strength of the potential at the origin.     

Equilibrium Configurations of Epitaxially Strained Elastic Films: Second Order Minimality Conditions and Qualitative Properties of Solutions

We consider a variational model introduced in the physical literature to describe the epitaxial growth of an elastic film over a thick flat substrate when a lattice mismatch between the two materials is present. We study quantitative and qualitative properties of equilibrium configurations, that is, of local and global minimizers of the free-energy functional. More precisely, we determine analytically the critical threshold for the local minimality of the flat configuration and we also prove several results concerning its global minimality. The non-occurrence of singularities in non-flat global minimizers is also addressed. One of the main results of the paper is a new sufficient condition for local minimality, which provides the first extension of the classical criteria based on the positivity of the second variation to the context of functionals with bulk and surface energies.     

Symmetry Groups of the Planar Three-Body Problem and Action-Minimizing Trajectories

We consider periodic and quasi-periodic solutions of the three-body problem with homogeneous potential from the point of view of equivariant calculus of variations. First, we show that symmetry groups of the Lagrangian action functional can be reduced to groups in a finite explicitly given list, after a suitable change of coordinates. Then, we show that local symmetric minimizers are always collisionless, without any assumption on the group other than the fact that collisions are not forced by the group itself. Moreover, we describe some properties of the resulting symmetric collisionless minimizers (Lagrange, Euler, Hill-type orbits and Chenciner–Montgomery figure-eights).     

New exact penalty function for solving constrained finite min-max problems

This paper introduces a new exact and smooth penalty function to tackleconstrained min-max problems.By using this new penalty function and adding justone extra variable,a constrained min-max problem is transformed into an unconstrainedoptimization one.It is proved that,under certain reasonable assumptions and when thepenalty parameter is sufficiently large,the minimizer of this unconstrained optimizationproblem is equivalent to the minimizer of the original constrained one.Numerical resultsdemonstrate that this penalty function method is an effective and promising approach forsolving constrained finite min-max problems.     

连续体结构拓扑优化的一种改进变密度法及其应用

针对连续体结构拓扑优化设计变密度方法SIMP和RAMP,因惩罚函数选取的不合理而导致拓扑结构形式不甚合理的问题,本文提出了一种新的惩罚函数,并基于此函数导出了相应的迭代设计公式,几个典型考题的数值结果,说明了方法的可行性和有效性.     

Another Theorem of Classical Solvability ‘In Small’ for One-Dimensional Variational Problems

In this paper we suggest a direct method for studying local minimizers of one-dimensional variational problems which naturally complements the classical local theory. This method allows us both to recover facts of the classical local theory and to resolve a number of problems which were previously unreachable. The basis of these results is a regularity theory (a priori estimates and compactness in C ¹) for solutions of obstacle problems with sufficiently close obstacles. In these problems we establish that solutions exist and inherit regularity of the obstacles even under assumptions on integrands that are much weaker than those required in the classical local theory.     

Symmetry Groups and Non-Planar Collisionless Action-Minimizing Solutions of the Three-Body Problem in Three-Dimensional Space

Periodic and quasi-periodic solutions of the n-body problem can be found as minimizers of the Lagrangian action functional restricted to suitable spaces of symmetric paths. The main purpose of this paper is to develop a systematic approach to the equivariant minimization for the three-body problem in three-dimensional space. First we give a finite complete list of symmetry groups fitting to the minimization of the action, with the property that any other symmetry group can be reduced to be isomorphic to one of these representatives. A second step is to prove that the resulting (local and global) symmetric action-minimizers are always collisionless (when they are not already bound to collisions). Furthermore, we prove some results which address the question of whether minimizers are planar or non-planar; as a consequence of our theory we will give general criteria for a symmetry group to yield planar or homographic minimizers (either homographic or not, as in the Chenciner-Montgomery eight solution). On the other hand we will provide a rigorous proof of the existence of some interesting one-parameter families of periodic and quasi-periodic non-planar orbits. These include the choreographic Marchal's P₁₂ family with equal masses – together with a less-symmetric choreographic family (which anyway probably coincides with the P₁₂ family).     

无网格局部Petrov-Galerkin方法中本质边界条件的处理

鉴于无网格局部Petrov-Galerkin方法(MLPG)形函数的非插值性质,将一种新的本质边界处理方案——完全变换法与MLPG结合,通过变换矩阵修正形函数,使其满足Kronecker-δ条件,实现了本质边界的精确实施。进一步与MLPG中通常处理边界的罚方法作了比较研究,数值结果表明新方法的可靠性与精确度。     

Higher Integrability for Minimizers of the Mumford–Shah Functional

We prove higher integrability for the gradient of local minimizers of the Mumford–Shah energy functional, providing a positive answer to a conjecture of De Giorgi (Free discontinuity problems in calculus of variations. Frontiers in pure and applied mathematics, North-Holland, Amsterdam, pp 55–62, 1991).     

Vortices for a Rotating Toroidal Bose–Einstein Condensate

We construct local minimizers of the Gross–Pitaevskii energy, introduced to model Bose–Einstein condensates (BEC) in the Thomas–Fermi regime which are subject to a uniform rotation. Our sample domain is taken to be a solid torus of revolution in with starshaped cross-section. We show that for angular speeds ω_ε = O(|ln ε|) there exist local minimizers of the energy which exhibit vortices, for small enough values of the parameter ε. These vortices concentrate at one or several planar arcs (represented by integer multiplicity rectifiable currents) which minimize a line energy, obtained as a Γ-limit of the Gross–Pitaevskii functional. The location of these limiting vortex lines can be described under certain geometrical hypotheses on the cross-sections of the torus.     

APPLICATION OF PENALTY FUNCTION METHOD IN ISOPARANIETRIC HYBRID FINITE ELEMENT ANALYSIS

IntroductionSince T.H.H.Pain firstly puts forward hybrid element method[1]in1964, the researchand application of hybrid element have got great development. T.H.H.Painet al.havemade important pioneer works in the domain of the research on incompatible displacementstructure hybrid elementstress pattern[2,3]in recentdecades. Reference [4] had provided theoptimizing design concept of hybrid element further and established standardization methodof incompatible displacement structure hybrid eleme…     

Variational theory for spatial rods

The simplest theory of spatial rods is presented in a variational setting and certain necessary conditions for minimizers of the potential energy are derived. These include the Weierstrass and Legendre inequalities, which require that the vector describing curvature and twist belong to a domain of convexity of the strain energy function.     

On penalty function methods in the finite-element analysis of flow problems

In this paper the penalty function method is reviewed in the general context of solving constrained minimization problems. Mathematical properties, such as the existence of a solution to the penalty problem and convergence of the solution of a penalty problem to the solution of the original problem, are studied for the general case. Then the results are extended to a penalty function formulation of the Stokes and Navier-Stokes equations. Conditions for the equivalence of two penalty-finite element models of fluid flow are established, and the theoretical error estimates are verified in the case of Stokes's problem.     

Weak Local Minimizers in Finite Elasticity

This paper deals with necessary conditions and sufficient conditions for a weak local minimum of the energy of a hyperelastic body. We consider anisotropic bodies of arbitrary shape, subject to prescribed displacements on a given portion of the boundary. As an example, we consider the uniaxial stretching of a cylinder, in the two cases of compressible and incompressible material. In both cases we find that there is a continuous path across the natural state, made of local energy minimizers. For the Blatz-Ko compressible material and for the Mooney-Rivlin incompressible material, explicit estimates of the minimizing path are given and compared with those available in the literature. Dedicated to the memory of Victor J. Mizel.     

A Double Bubble in a Ternary System with Inhibitory Long Range Interaction

The free energy of a ternary system with a self-organization property includes an interface energy and a longer ranging, inhibitory interaction energy. In a planar domain, if the two energies are properly balanced and two of the three constituents make up an equal but small fraction, the free energy admits a local minimizer that is shaped like a perturbed double bubble. Most difficulties in the proof of this result are related to the triple junction phenomenon that the three constituents of the ternary system meet at a point. Two techniques are developed to deal with the triple junction. First, one defines restricted classes of perturbed double bubbles. Each perturbed double bubble in a restricted class is obtained from a standard double bubble by a special perturbation. The two triple junction points of the standard double bubble can only move along the line connecting them, in opposite directions, and by the same distance. The second technique is the use of the so called internal variables. These variables derive from the more geometric quantities that describe perturbed double bubbles in restricted classes. The advantage of the internal variables is that they are only subject to linear constraints, and perturbed double bubbles in a restricted class represented by internal variables are elements of a Hilbert space. A local minimizer of the free energy in each restricted class is found as a fixed point of a nonlinear equation by a contraction mapping argument. The second variation at the fixed point within its restricted class is positive. This perturbed double bubble satisfies three of the four equations for critical points of the free energy. The unsolved equation is the 120 degree angle condition at triple junction points. Then perform another minimization among the local minimizers from all restricted classes. A minimum of minimizers emerges and solves all the equations for critical points.     

Threshold-based Quasi-static Brittle Damage Evolution

We introduce models for static and quasi-static damage in elastic materials, based on a strain threshold, and then investigate the relationship between these threshold models and the energy-based models introduced in Francfort and Marigo (Eur J Mech A Solids 12:149–189, 1993) and Francfort and Garroni (Ration Mech Anal 182(1):125–152, 2006). A somewhat surprising result is that, while classical solutions for the energy models are also threshold solutions, this is shown not to be the case for nonclassical solutions, that is, solutions with microstructure. A new and arguably more physical definition of solutions with microstructure for the energy-based model is then given, in which the energy minimality property is satisfied by sequences of sets that generate the effective elastic tensors, rather than by the tensors themselves. We prove existence for this energy-based problem, and show that these solutions are also threshold solutions. A by-product of this analysis is that all local minimizers, in both the classical setting and for the new microstructure definition, are also global minimizers.     

Steady States in Galactic Dynamics

By minimizing the energy for the Vlasov-Poisson system under a constraint, Guo and Rein have constructed a large class of isotropic, spherically symmetric steady states. They have shown that an isolated minimizer is automatically dynamically stable under general (i.e., not necessarily symmetric) perturbations. The main result of this work is to remove the assumption that the minimizer must be isolated, so minimizers are stable even if they are not isolated. It is also shown that the Lagrange multipliers associated with all minimizers have the same value. Finally, an example where two distinct minimizers exist is studied numerically.     

Polyconvexity and Existence Theorem for Nonlinearly Elastic Shells

We present an existence theorem for a large class of nonlinearly elastic shells with low regularity in the framework of a two-dimensional theory involving the mean and Gaussian curvatures. We restrict our discussion to hyperelastic materials, that is to elastic materials possessing a stored energy function. Under some specific conditions of polyconvexity, coerciveness and growth of the stored energy function, we prove the existence of global minimizers. In addition, we define a general class of polyconvex stored energy functions which satisfies a coerciveness inequality.