Complementary Variational Principle Research Articles

An enriched model based on a complementary strain energy variational principle for stress analysis in FRP plate-strengthened beams

An enriched model based on a complementary strain energy variational principle for stress analysis in FRP plate-strengthened beams

A stress-driven computational homogenization method based on complementary potential energy variational principle for elastic composites

Based on the complementary potential energy variational principle, in this work, we proposed a stress-driven homogenization procedure to compute overall effective material properties for elastic composites with locally heterogenous micro-structures. We have developed a novel incremental variational formulation for homogenization problems of both infinitesimal and finite deformations where the macro-stress-based complementary potential energy is obtained for hyperelastic materials for a global minimization problem with respect to fine-scale displacement fluctuation field. The point of departure of our approach is a general complementary variational principle formulation that can determine material responses of elastic composites with heterogeneous micro structures. We have implemented the proposed stress-driven computational homogenization procedure with the finite element method. By comparing the numerical results with the analytical method and the strain-driven homogenization method, we find that the stress-driven homogenization offers the lower bound estimate of materials properties for elastic composites.

A reduced basis method with exact-solution certificates for steady symmetric coercive equations

Abstract We introduce a reduced basis method that computes rigorous upper and lower bounds of the energy associated with the infinite-dimensional weak solution of parametrized steady symmetric coercive partial differential equations with piecewise polynomial forcing and operators that admit decompositions that are affine in functions of parameters. The construction of the upper bound appeals to the standard primal variational argument; the construction of the lower bound appeals to the complementary variational principle. We identify algebraic conditions for the reduced basis approximation of the dual variable that results in an exact satisfaction of the dual feasibility conditions and hence a rigorous lower bound. The formulation permits an offline–online computational decomposition such that, in the online stage, the approximation and exact certificates can be evaluated in complexity independent of the underlying finite element discretization. We demonstrate the technique in two numerical examples: a one-dimensional reaction–diffusion problem with a parametrized diffusivity constant; a planar linear elasticity problem with a geometry deformation. We confirm in both cases that the method produces guaranteed upper and lower bounds of the energy at any parameter value, for any finite element discretization, and for any reduced basis approximation.

Image denoising by a direct variational minimization

In this article we introduce a novel method for the image de-noising which combines a mathematically well-posdenes of the variational modeling with the efficiency of a patch-based approach in the field of image processing. It based on a direct minimization of an energy functional containing a minimal surface regularizer that uses fractional gradient. The minimization is obtained on every predefined patch of the image, independently. By doing so, we avoid the use of an artificial time PDE model with its inherent problems of finding optimal stopping time, as well as the optimal time step. Moreover, we control the level of image smoothing on each patch (and thus on the whole image) by adapting the Lagrange multiplier using the information on the level of discontinuities on a particular patch, which we obtain by pre-processing. In order to reduce the average number of vectors in the approximation generator and still to obtain the minimal degradation, we combine a Ritz variational method for the actual minimization on a patch, and a complementary fractional variational principle. Thus, the proposed method becomes computationally feasible and applicable for practical purposes. We confirm our claims with experimental results, by comparing the proposed method with a couple of PDE-based methods, where we get significantly better denoising results specially on the oscillatory regions.

Derivation of a dual-mixed hp-finite element model for axisymmetrically loaded cylindrical shells

The two-field dual-mixed Fraeijs de Veubeke variational formulation of three-dimensional elasticity serves as the starting point of the derivation of a dimensionally reduced shell model presented in this paper. The fundamental variables of this complementary energy-based variational principle are the not a priori symmetric stress tensor and the skew-symmetric rotation tensor. The tensor of first-order stress functions is applied to satisfy translational equilibrium, while the rotation tensor plays the role of a Lagrange multiplier to ensure rotational equilibrium. The volumetric locking-free shell model uses unmodified three-dimensional constitutive equations, and no classical kinematical hypotheses are employed during the derivation. The numerical performance of the related low-order h-, and higher-order p-version finite elements developed for axisymmetrically loaded cylindrical shells is investigated by two representative model problems. It is numerically proven that no negative effect can be experienced when the thickness is small and tends to zero.

Complementarity, polarity and triality in non‐smooth, non–convex and non–conservative Hamilton systems

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Yang Gao David 2001Complementarity, polarity and triality in non‐smooth, non–convex and non–conservative Hamilton systemsPhil. Trans. R. Soc. A.3592347–2367http://doi.org/10.1098/rsta.2001.0855SectionRestricted accessComplementarity, polarity and triality in non‐smooth, non–convex and non–conservative Hamilton systems David Yang Gao David Yang Gao Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA () Google Scholar Find this author on PubMed Search for more papers by this author David Yang Gao David Yang Gao Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA () Google Scholar Find this author on PubMed Search for more papers by this author Published:15 December 2001https://doi.org/10.1098/rsta.2001.0855AbstractThis paper presents a unified critical–point theory in non–smooth, non–convex and dissipative Hamilton systems. The canonical dual/polar transformation methods and the associated bi–duality and triality theories proposed recently in non–convex variational problems are generalized into fully nonlinear dissipative dynamical systems governed by non–smooth constitutive laws and boundary conditions. It is shown that, by this method, non–smooth and non–convex Hamilton systems can be reformulated into certain smooth dual, complementary and polar variational problems. Based on a newly proposed polar Hamiltonian, a nice bipolarity variational principle is established for three–dimensional non–smooth elastodynamical systems, and a potentially powerful complementary variational principle can be used for solving unilateral variational inequality problems governed by non–smooth boundary conditions. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Qiu Z and Xia H (2021) Symplectic perturbation series methodology for non-conservative linear Hamiltonian system with damping, Acta Mechanica Sinica, 10.1007/s10409-021-01076-0, 37:6, (983-996), Online publication date: 1-Jun-2021. Gao D and Ali E (2019) A Novel Canonical Duality Theory for Solving 3-D Topology Optimization Problems Advances in Mathematical Methods and High Performance Computing, 10.1007/978-3-030-02487-1_13, (209-246), . Gao D (2019) Canonical Duality-Triality Theory: Unified Understanding for Modeling, Problems, and NP-Hardness in Global Optimization of Multi-Scale Systems Advances in Mathematical Methods and High Performance Computing, 10.1007/978-3-030-02487-1_1, (3-50), . Gao D (2018) On topology optimization and canonical duality method, Computer Methods in Applied Mechanics and Engineering, 10.1016/j.cma.2018.06.027, 341, (249-277), Online publication date: 1-Nov-2018. Gao D, Ruan N and Latorre V (2017) Canonical Duality-Triality Theory: Bridge Between Nonconvex Analysis/Mechanics and Global Optimization in Complex System Canonical Duality Theory, 10.1007/978-3-319-58017-3_1, (1-47), . Gao D (2015) Analytical solutions to general anti-plane shear problems in finite elasticity, Continuum Mechanics and Thermodynamics, 10.1007/s00161-015-0412-y, 28:1-2, (175-194), Online publication date: 1-Mar-2016. Gao D (2009) Canonical duality theory: Unified understanding and generalized solution for global optimization problems, Computers & Chemical Engineering, 10.1016/j.compchemeng.2009.06.009, 33:12, (1964-1972), Online publication date: 1-Dec-2009. Gao D and Sherali H (2009) Canonical Duality Theory: Connections between Nonconvex Mechanics and Global Optimization Advances in Applied Mathematics and Global Optimization, 10.1007/978-0-387-75714-8_8, (257-326), . Penot J (2009) Ekeland Duality as a Paradigm Advances in Applied Mathematics and Global Optimization, 10.1007/978-0-387-75714-8_10, (349-376), . Yuan Y (2008) Optimal solutions to a class of nonconvex minimization problems with linear inequality constraints, Applied Mathematics and Computation, 10.1016/j.amc.2008.04.016, 203:1, (142-152), Online publication date: 1-Sep-2008. Gao D and Yu H (2008) Multi-scale modelling and canonical dual finite element method in phase transitions of solids, International Journal of Solids and Structures, 10.1016/j.ijsolstr.2007.08.027, 45:13, (3660-3673), Online publication date: 1-Jun-2008. Gao D (2007) Duality-Mathematics Wiley Encyclopedia of Electrical and Electronics Engineering, 10.1002/047134608X.W2412.pub2 Gao D (2016) Complementary Principle, Algorithm, and Complete Solutions to Phase Transitions in Solids Governed by Landau-Ginzburg Equation, Mathematics and Mechanics of Solids, 10.1177/1081286504038455, 9:3, (285-305), Online publication date: 1-Jun-2004. Gao D (2003) Perfect duality theory and complete solutions to a class of global optimization problems*, Optimization, 10.1080/02331930310001611501, 52:4-5, (467-493), Online publication date: 1-Aug-2003. Gao D (2003) Nonconvex Semi-Linear Problems and Canonical Duality Solutions Advances in Mechanics and Mathematics, 10.1007/978-1-4613-0247-6_5, (261-312), . This Issue15 December 2001Volume 359Issue 1789Theme Issue ‘Non-smooth mechanics’ compiled by F. G. Pfeiffer Article InformationDOI:https://doi.org/10.1098/rsta.2001.0855Published by:Royal SocietyPrint ISSN:1364-503XOnline ISSN:1471-2962History: Published online15/12/2001Published in print15/12/2001 License: Citations and impact Keywordsnon–convex variational problemsnon–smooth elastodynamicstrialitypolaritynon–conservative Hamilton systemscomplementarity

Variational principles of asymmetric elasticity theory of finite deformation

In this paper, based on the finite deformation S-R decomposition theorem, the definition of the body moment is renewed as the sum of its internal and external. The expression of the increment rate of the deformation energy is derived and the physical meaning is clarified. The power variational principle and the complementary power variational principle for finite deformation mechanics are supplemented and perfected.

Micromechanical modelling of textile composites using variational principles

The ever increasing use of textile reinforced composite materials for structural applications makes the development of automated and computationally efficient analysis tools a must. In this paper, a micromechanical method for predicting the thermo-mechanical behaviour of textile reinforced (twodimensional woven and braided fabrics) composites is presented. The model considers the shape and type of the yarns, yarn interactions and crimp as well as the matrix distribution. This modelling scheme consists of a multilevel automated geometric decomposition of a representative volume element (unit cell) into smaller elements (block and micro cells) containing yarn and matrix parts. This way, the problem of stress analysis for the whole unit cell is split into a number of subproblems at each level of the decomposition scheme. This top to bottom decomposition is followed by a bottom to top homogenisation scheme in which internal stresses (sub-unit cell level) are linked to external ones (unit cell level). This procedure results in the calculation of the stress-strain field and the prediction of the elastic properties of the composite. To develop this method the complementary variational principle was used. A FORTRAN software programme called TEXCOMP-CEM, has been developed to automate calculation and turn the method into a practical design tool. This program computes stiffness, micro-stress fields and first cell failure. Results obtained are comparable to those obtained experimentally and from finite element modelling. Transactions on Engineering Sciences vol 21, © 1998 WIT Press, www.witpress.com, ISSN 1743-3533

Micro-Stress Analysis of Woven Fabric Composites by Multilevel Decomposition

A new micromechanics model is presented to predict the local stresses within woven fabric composites under three-dimensional mechanical loading. The model accounts for fabric geometry, yam interactions and matrix distribution. First, by constructing a multilevel decomposition scheme, the composite unit-cell is split up into matrix and yam cells. Then, by a multistep homogenization procedure, a link is established between the external loading and the internal stresses. The principle idea lies in the interpretation that the "stress concentration factors" can be computed at each step by applying the complementary variational principle. We achieve a straightforward analytical stress model for woven fabric composites which has not yet been reported. In addition to modeling the stress fields, the three-dimensional stiffness matrix is computed in this paper using the "stress concentration factors." The proposed modeling technique yields numerical stiffness results comparable to those obtained by three-dimensional finite element modeling.

An a posteriori error estimate for finite element approximations of a singularly perturbed advection-diffusion problem

In this paper the author presents an a posteriori error estimator for approximations of the solution to an advection-diffusion equation with a non-constant, vector-valued diffusion coefficient ϵ in a conforming finite element space. Based on the complementary variational principle, we show that the error of an approximate solution in an associated energy norm is bounded by the sum of the weighted L 2- norms of solutions to a set of independent complementary variational problems, each defined on only one element of the partition. This error bound guarantees the over-estimation of the true error and does not depend unfavourably on ϵ as ∥ ϵ∥ ∞ goes to zero. Although the original equation is a non-self-adjoint problem, the strong form of each local variational problem is always a Poisson equation with Neumann boundary conditions. The approximation of these local problems is then discussed and it is shown that, omitting a higher order term, the finite element solutions of these local complementary variational problems provide a computable upper error bound for the original finite element approximation in the energy norm. Numerical results, presented to validate the theoretical results, show that the computed error bounds are tight for a wide range of values of ϵ and always over-estimate the true errors.

Calculation of the complementary variational modified bounds on the phonon thermal conductivity

The complementary variational principle is used to derive suitable expressions for the first modified bounds on the phonon thermal conductivity. The variational trial function is taken to depend on a series of variational parameters. It is shown that the relations needed to select one decomposition of the collision operator in preference to the others are independent of the number of variational parameters. The convergence of the method is investigated both analytically and numerically when the number of variational parameters is increased. All types of phonon interactions are considered and the results agree quantitatively with the experimental data for Ge and LiF over a wide temperature range.

An analysis of electromagnetic waves by a complementary finite element method

This paper describes an analysis of electromagnetic waves in a waveguide using the finite element method based on the complementary variational principle. The vector Helmholtz equation is represented in dual forms, in which the solenoidal and irrotational transverse fields are chosen to be unknowns, and they are approximated by the edge and facet elements, respectively. The dual representations are then shown to provide the lower bounds of the eigenfrequencies of a waveguide, with the aid of the complementary variational principle. >

Indeterminacy of first order stress functions and the stress- and rotation-based formulation of linear elasticity

Following from the indeterminacy of the nonsymmetric tensor of first order stress functions, satisfaction of the linear three-dimensional translational equilibrium equations can be achieved by the use of only six non-zero first order stress functions instead of nine. Utilizing this fact, redundant variable approximations in assumed nonsymmetric stress-based procedures can be avoided on the one hand, and, on the other, independent Euler equations and natural boundary conditions of mixed variational principles using nonsymmetric stresses as independent variables can be obtained. In this paper, the independent governing equations of linear elasticity in terms of stresses and rotations are derived using Fraeijs de Veubeke's complementary energy-based mixed variational principle.

Nonlinear analysis of Reissner's plate by the variational approaches and boundary element methods

The purpose of the paper is twofold, first to suggest a general method, called the second variation-convex analysis mixed method, for analysis of nonlinear functionals. As an application of the approach, a strictly dual complementary variational principle of Reissner plates with the local form 1 and the existing criterion of variational solution have been studied. The study shows that the principle and the criterion 1 can be improved into a global form by using the novel approach. In contrast with existing methods (Jin 1 and Gao and Stang 2), this method will be able to analyze general nonlinear problems, 1 rather than geometrically nonlinear ones. 2 Our second purpose is to present a new approach to the derivation of the exact boundary integral equation for the analysis of nonlinear Reissner plates and to the derivation of the criterion for the solution of the boundary integral equation. Subsequently, the boundary and the domain of the plate are discretized to solve the nonlinear problems. All unknown variables are at the boundary. Numerical results are presented to illustrate the method and demonstrate its effectiveness and accuracy.

Use of variational methods to establish and increase the ranges of application of analytic approximate solutions of the Poisson—Boltzmann equation for a charged microcapillary

In order to solve the transport equations for a microcapillary pore, it is necessary to deal with the Poisson—Boltzmann equation. Here, a variational method is proposed in order to obtain analytic solutions. The procedure starts with a known analytic solution for an extreme range of electric potentials inside the pore. Then, the solutions for both very low and very high potentials are slightly modified in order to have a uniparametric family of solutions. By using the appropriate complementary variational principle, one of the possible solutions is chosen. Of course this allows us to show how close to the respective ranges of the analytic solutions we are for any concentration, pore radius, and zeta potential, which are the actual magnitudes characterizing the porous structure, the solution, and their interactions. The parameter in question is a function of the pore radius—in terms of the Debye length—and the zeta potential. Then we get two analytic solutions, whose accuracy can be tested by comparing them with the numerically obtained solution. This is done for a general binary electrolyte.

Quantitative error estimate in modeling the laminar stationary flow of a Bingham fluid

We present a quantitative a posteriori error estimate when the linear laminar stationary flow of a Bingham fluid in a cylindrical pipe is modeled as a Newtonian fluid. The estimate is proved by using a complementary variational principle for the Bingham fluid flow. Examples show that the estimate is effective.

A regularization procedure for a simplified friction problem

A simplified friction problem can be described by minimizing a non-differentiable functional. The non-differentiability of the functional causes difficulty when one solves the minimization problem numerically. We consider a regularization procedure to overcome the difficulty. The convergence of the procedure is proved. An a posteriori quantitative error estimate is established by employing a complementary variational principle for the simplified friction problem.

The Hamiltonian system and the spline function

This paper examines the inherent connection between the generalized Hamiltonian system and the spline function. By considering the generalized canonical multipoint boundary value (GCMBV) problem, a multipoint complementary variational principle for generalized Hamiltonian system is provided, and the sufficient and necessary condition for solvability of the GCMBV problem is obtained. With the aid of Hamiltonian equations, a new class of vector spline functions with good variational and approximation properties called H-spline are defined, it includes the general vector spline as a special one. Finally, the H-spline interpolation problems are also discussed.

Anticandical antibodies in the vaginal discharge and sera of patients with vaginal candidiasis

The purpose of the paper is twofold, first to suggest a general method, called the second variation-convex analysis mixed method, for analysis of nonlinear functionals. As an application of the approach, a strictly dual complementary variational principle of Reissner plates with the local form1 and the existing criterion of variational solution have been studied. The study shows that the principle and the criterion1 can be improved into a global form by using the novel approach. In contrast with existing methods (Jin1 and Gao and Stang2), this method will be able to analyze general nonlinear problems,1 rather than geometrically nonlinear ones.2 Our second purpose is to present a new approach to the derivation of the exact boundary integral equation for the analysis of nonlinear Reissner plates and to the derivation of the criterion for the solution of the boundary integral equation. Subsequently, the boundary and the domain of the plate are discretized to solve the nonlinear problems. All unknown variables are at the boundary. Numerical results are presented to illustrate the method and demonstrate its effectiveness and accuracy.

COMPLEMENTARY ENERGY PRINCIPLES FOR ELASTO-PERFECT PLASTICITY

A real complementary energy variational principle for the theory of elasto-perfect plasticity is established with the aid of convex analysis. Based on the property of plastic superpotential in this principle, various variational principles are constructed, and an interesting penalty-duality variational principle is proved, which affords an efficient algorithm for engineering problems.