Complementary Variational Principles Research Articles

Complementary variational principles with fractional derivatives

Complementary variational principles for a class of fractional boundary value problems are formulated. They are used for the error estimates of solutions for a general mechanical problems, first Painleve equation also given in the form with fractional derivatives and in the task of image regularization.

Complementary Principle, Algorithm, and Complete Solutions to Phase Transitions in Solids Governed by Landau-Ginzburg Equation

The complementary variational principles and associated algonrthm are presented in this paper for solving a class of nonconvex mechanics problems governed by the Landau-Ginzburg equation. The method used here is the canonical dual transfonnation developed recently in nonconvex analysis by the author. It is shown that, by this method, certain very difficult nonconvex potential variational problems can be converted into a coupled quadratic mixed variational problem. The extremality conditions of these complementary variational problems are controlled by the triality theory discovered recently. Therefore, a complementary energy principle (i.e. the canonical duality theorem with zero duality gap) is established, which leads to a potentially useful primal-dual algorithm. Applications in finite-dimensional problems are discussed. Results show that the discretized nonconvex potential variational problems in n-dimensional space can be converted into certain simple canonical (either convex or concave) dual problems. Therefore, a complete set of solutions is obtained for the Landau-Ginzburg equation in finite-dimensional space.

On the optimal shape of a Pflüger column

The optimal shape of a Pflüger column is determined by using Pontryagin's maximum principle. It is shown that the boundary value problem relevant for determining the optimal distribution of material (i.e. cross-sectional area function) along the column axis has simple eigenvalue. Necessary conditions for local extremum of column volume are reduced to a boundary-value problem for a single second order nonlinear differential equation. We examined singular points of this equation and formulated extremal complementary variational principles for it. The optimal cross-sectional area function is obtained by numerical integration and by Ritz method. The error of the analytical approximate solution obtained by Ritz method is also estimated.

Variational formulation of inverse shape design problem of heat conductors in an image plane and finite element solutions

Using an image plane introduced previously by Liu [4] the original shape design problem of heat conduction is transformed into the one with known boundary and a pair of complementary variational principles (VP) for it with complicated boundary conditions are established in terms of the heat stream function and the temperature function respectively. Based upon these VP some finite element solutions to the shape design problem of heat conduction are presented. Finally, a very interesting invariance rule of inverse problem solutions with respect to the temperature-dependent conductivity is identified and numerically demonstrated.

Orientational relaxation and flow near boundaries in nematic liquid crystals

The problem of the orientational relaxation and flow near a boundary in a nematic liquid crystal is considered. When the approximation of neglecting inertia is made, the equation for the variation in the polar angle in space and position becomes a non-linear diffusion equation. This is used for the single boundary problem to derive complementary variational principles and to give arguments which bound the solution. The introduction of the effect of inertia is included by means of a singular perturbation method. Results are given for the case of a no-slip and a perfect slip boundary with anchoring of the polar angle.

General Analytic Solutions and Complementary Variational Principles for Large Deformation Nonsmooth Mechanics

This paper presents a nonlinear dual transformation method and general complementary energy principle for solving large deformation theory of elastoplasticity governed by nonsmooth constitutive laws. It is shown that by using this method and principle, the nonconvex and nonsmooth total potential energy is dual to a smooth complementary energy functional, and fully nonlinear equilibrium equations in finite deformation problems can be converted into certain tensor equations. The algebraic relation between the first and the second Piola–Kirchhoff stresses are revealed. A closed form solution for general three-dimensional large deformation boundary value problems is obtained. The properties of this general solution are clarified by a triality extremum principle. This triality theory reveals an important phenomenon in nonconvex variational problems. Applications are illustrated by nonlinear, nonsmooth equilibrium problems in Hencky's plasticity, 3D cylindrical structures and post buckling analysis of elastoplastic bar with jumping and hardening effects. The idea and methods presented in this paper can be used and generalized to solve many nonlinear boundary value problems in finite deformation theory.

Complementary variational principles for generalized dynamic continuum models

Complementary variational principles are presented for a range of dynamic continuum models based on fully coupled inhomogeneous, anisotropic poroelasticity, extended to include coupled thermal effects and viscoelastic behaviour. The dual principles are all developed in Laplace transform space. Previous results using variational principles to derive bounds on effective permeability for a rigid porous solid for both steady and unsteady loading are generalized to the anisotropic case. As an illustration of the power of the dual principles, bounds are placed on the Stress Intensity Factor for a sample fracture problem in dynamic thermoelasticity, in transform space. Very good bounds are obtained by using an optimization method with the dual principles. Results for a specific inhomogeneous material are shown, and compared with published results in the literature concerning so called Functionally Gradient Materials.

Complementary variational principles for radiation heat transfer in an absorbing medium

The radiation heat flux in a plane slab of an absorbing medium is evaluated by determining upper and lower bounds via complementary variational principles. The results obtained are in excellent agreement with those corresponding to other more cumbersome solution procedures (for small values of the spacing parameterd).

The variational principles of coupled systems in photoelasticity

This paper presents an energy principle, zero different principle of coupled systems in photoelasticity, from which the potential energy, the complementary energy, generalized potential energy and generalized complementary energy variational principles of the coupled systems in photoelasticity are derived.

Complementary solutions of direct current flow problems

An error-based approach is used for the systematic derivation of complementary variational principles for the problem of steady current flow in conductors. Using two-dimensional finite-element solution results, the significance of combined presentation of complementary computational results is discussed, and the distribution of the numerical error is displayed graphically. It is shown that the limits of the theory of complementary variational principles are decreed by boundary specifications, and that, under certain circumstances, a treatment is available for transcending them. Finally, the error-based approach is applied to the circuit version of the direct-current flow problem.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Numerical methods for inverse problem of heat conduction with unknown boundary based on variational principles with variable domain

In this article a variable-domain variational approach to the entitled problem is presented. A pair of complementary variational principles with a variable domain in terms of temperature and heat-streamfunction are first established. Based on them, two methods of solution—generalized Ritz method and variable-domain FEM—both capable of handling problems with unknown boundaries, are suggested. Then, three sample numerical examples have been tested. The computational process is quite stable, and the results are encouraging. This variational approach can be extended straightforwardly to 3-D inverse problems as well as to other problems in mathematical physics.

Unified symmetric finite element and boundary integral variational coupling methods for linear fluid–structure interaction

AbstractThis article concerns the development of energy‐based variational formulations and their corresponding finite element–boundary element Rayleigh–Ritz approximations for solving the time‐harmonic vibration and scattering problem of an inhomogeneous penetrable fluid or solid object immersed in a compressible, inviscid, homogeneous fluid. The resulting coupled finite element and boundary integral methods (FEM‐BEM) have the following attractive features: (1) Separate direct and complementary variational principles lead naturally to several alternative structure variable and fluid variable methodologies. (2) The solution in the exterior region is represented by a combined single‐ and double‐layer potential which ensures the validity of the methods for all wave numbers; even though this representation introduces hypersingular integrals, for actual computations the hypersingular operator may be rewritten in terms of single‐layer potentials, which can be integrated by standard techniques. (3) Since the discretized equations for the interior region and for the boundary are derived from the first variation of bilinear functionals the resulting algebraic systems of equations are always symmetric. In addition, the transition conditions across the interface are natural. This allows one to approximate the solutions within the interior and exterior regions independently, without imposing any boundary constraints.

The calculation of short-circuit reactance of transformers by complementary variational principles

A T- Omega method for steady magnetic fields is developed. It is able to overcome the difficulties in the implementation of the complementary functional in the complementary variational principles to obtain the upper bound of the reactance and the energy of a magnetic system. The short-circuit reactances of a variety of transformer productions with different structure and different power have been calculated accurately by the method presented. The maximum relative errors of the calculation results are 2% compared with those obtained by experiments.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Self adaptive mesh generation for 3-D finite element calculation

A three-dimensional, fully self-adaptive finite-element procedure is presented, based on the use of complementary variational principles. The use of the dual formulation ensures overall accuracy and provides a way of implementing localized mesh refinement. The initial problem meshing is achieved using a mesh generation technique based on the method of Delaunay tessellation. This work is an extension of techniques previously used successfully for two-dimensional representations. >

ERROR BOUNDS IN DUAL &amp; COMPLEMENTARY EDDY‐CURRENT SYSTEMS

The formulation of electromagnetic field problems using dual and complementary energy methods is now well established, and the advantages of the error bounds to system energy for static systems that it provides are widely realized. When considering eddy‐current problems the picture is less clear, even for the restricted case of time harmonic fields. Several interesting papers have been published that address the particular question of whether or not error bounds can exist for such dissipative systems, but as yet no consensus view has emerged. This contribution is an attempt to help clarify the position. It will be shown that through the use of complementary variational principles it is possible to establish pairs of functionals that do indeed rigourously provide error bounds. Ilustrative examples show that it is usually possible to provide error bounds to eddy‐current system parameters so long as due care is taken to pre‐determine whether or not a particular problem may be regarded as either current driven, or voltage driven, from the circuit point of view.

Quantitative error estimates for material idealization of torsion problems

We establish a posteriori quantitative error estimates for effects of material idealization on solutions of torsion problems. The estimates are derived by employing complementary variational principles.

Geometric nonlinearity: potential energy, complementary energy, and the gap function

Dual minimum principles for displacements and stresses are well established for linear variational problems and also for nonlinear (and monotone) constitutive laws. This paper studies the problem of geometric nonlinearity. By introducing a gap function, we recover complementary variational principles in the equilibrium problems of mathematical physics. When the gap function is nonnegative those become minimum principles. The theory is based on convex analysis, and the applications made here are to nonlinear mechanics.

Variational methods for cellular models.

Two complementary variational principles are derived for the one-electron Schr\"odinger or local-density-functional equation in a closed cell of arbitrary shape, for external Neumann and Dirichlet boundary conditions, respectively. The surface operators scrR and ${\mathrm{scrR}}^{\mathrm{\ensuremath{-}}1}$, respectively, are stationary in these two variational principles. Subject to a condition of compatibility of the boundary conditions, these results are combined to give variational equations that are valid for an arbitrary cluster of atomic cells, assumed to fill space within an outer boundary. Cell interface terms agree with prior variational derivations for discontinuous functions. It is shown that structure constants of multiple scattering theory can be used within the variational formalism, giving contracted Hermitian matrix equations linearized in energy. New variational prescriptions are given for two broad classes of applications: (i) electron scattering by a cluster, or bound states using scattering theory; (ii) embedding a cluster in a substrate.

Error-based derivation of complementary formulations for the eddy-current problem

An error-based approach is developed for complementary formulations of eddy-current problems corresponding to a mismatch in constitutive relationships. The derivation accounts systematically for the various problem specifications, including boundary conditions. It is shown that under steady harmonic conditions energy bounds cannot be established; the problem gives rise to complementary variational principles that are not extremal, as well as a single extremal principle that involves both complementary fields simultaneously. The treatment results in a general framework that places previously proposed formulations in proper perspective, and that accounts for differences among them.

Determination of electromagnetic energy and parameters by complementary and complementary-dual energy method

The complementary and complementary-dual variational principles and their expressions in static electric and magnetic field are discussed. This paper gives a general complementary-dual variational principle. By expanding the various methods of constructing approximate function, including the R-function introduced by the authors ,some experience and approaches of the energy method are summarized. The complementary and complementary - dual finite element methods are also investigated.