Dual Finite Element Method Research Articles

Postbuckling analysis of a nonlinear beam with axial functionally graded material

The postbuckling analysis of a modified nonlinear beam composed of axial functionally graded material (FGM) is investigated by a canonical dual finite element method (CD-FEM). The governing equation of the axial FGM nonlinear beam is derived through a variational method. The CD-FEM is adopted to find the nonconvex postbuckling configurations of the beam according to Gao’s triality theory. Using duality transition, the original potential energy functional becomes a functional of deformation and dual stress fields. By variation of the mixed complementary energy, the coupling equations are derived to find deformation and dual stress fields. In FEM, matrices of a beam element depend on the gradient of material property (elastic modulus). To obtain general forms of matrices of a beam element, the graded elastic modulus is approximated by piecewise linear functions with respect to axial position. Numerical examples are presented to show the effects of graded elasticity on the postbuckling configurations of the beam.

Post-buckling solutions of hyper-elastic beam by canonical dual finite element method

The post-buckling problem of a large deformed beam is analyzed using the canonical dual finite element method (CD-FEM). The feature of this method is to choose correctly the canonical dual stress so that the original non-convex potential energy functional is reformulated in a mixed complementary energy form with both displacement and stress fields, and a pure complementary energy is explicitly formulated in finite dimensional space. Based on the canonical duality theory and the associated triality theorem, a primal–dual algorithm is proposed, which can be used to find all possible solutions of this non-convex post-buckling problem. Numerical results show that the global maximum of the pure-complementary energy leads to a stable buckled configuration of the beam, while the local extrema of the pure-complementary energy present unstable deformation states. We discovered that the unstable buckled state is very sensitive to the number of total elements and the external loads. Theoretical results are verified through numerical examples and some interesting phenomena in post-bifurcation of this large deformed beam are observed.

Canonical dual finite element method for solving post-buckling problems of a large deformation elastic beam

This paper presents a canonical dual mixed finite element method for the post-buckling analysis of planar beams with large elastic deformations. The mathematical beam model employed in the present work was introduced by Gao in 1996, and is governed by a fourth-order non-linear differential equation. The total potential energy associated with this model is a non-convex functional and can be used to study both the pre- and the post-buckling responses of the beams. Using the so-called canonical duality theory, this non-convex primal variational problem is transformed into a dual problem. In a proper feasible space, the dual variational problem corresponds to a globally concave maximization problem. A mixed finite element method involving both the transverse displacement field and the stress field as approximate element functions is derived from the dual variational problem and used to compute global optimal solutions. Numerical applications are illustrated by several problems with different boundary conditions.

Domain decomposition methods for the neutron diffusion problem

The neutronic simulation of a nuclear reactor core is performed using the neutron transport equation, and leads to an eigenvalue problem in the steady-state case. Among the deterministic resolution methods, simplified transport ( S P N ) or diffusion approximations are often used. The MINOS solver developed at CEA Saclay uses a mixed dual finite element method for the resolution of these problems, and has shown his efficiency. In order to take into account the heterogeneities of the geometry, a very fine mesh is generally required, and leads to expensive calculations for industrial applications. In order to take advantage of parallel computers, and to reduce the computing time and the local memory requirement, we propose here two domain decomposition methods based on the MINOS solver. The first approach is a component mode synthesis method on overlapping subdomains: several eigenmodes solutions of a local problem on each subdomain are taken as basis functions used for the resolution of the global problem on the whole domain. The second approach is an iterative method based on a non-overlapping domain decomposition with Robin interface conditions. At each iteration, we solve the problem on each subdomain with the interface conditions given by the solutions on the adjacent subdomains estimated at the previous iteration. Numerical results on parallel computers are presented for the diffusion model on realistic 2D and 3D cores.

On the Complementarity of Dual Formulations on Dual Meshes

The finite element method (FEM) using Whitney elements and the finite integral technique (FIT) using dual meshes work both with the metric free global variables. Though the mechanisms of two methods are very different, they both exhibit the analogy to the circuit network. Hence, there exists a similarity between the dual FEM formulations and the dual FIT formulations. This paper analyzes the complementarity of the dual formulations of both methods on dual meshes and emphasizes the complementary energy bounds through numerical experiments. The study supports the idea that the energy bounds are due to the geometry discretization on dual meshes.

Multi-scale modelling and canonical dual finite element method in phase transitions of solids

This paper presents a multi-scale model in phase transitions of solid materials with both macro and micro effects. This model is governed by a semi-linear nonconvex partial differential equation which can be converted into a coupled quadratic mixed variational problem by the canonical dual transformation method. The extremality conditions of this variational problem are controlled by a triality theory, which reveals the multi-scale effects in phase transitions. Therefore, a potentially useful canonical dual finite element method is proposed for the first time to solve the nonconvex variational problems in multi-scale phase transitions of solids. Applications are illustrated. Results shown that the canonical duality theory developed by the first author in nonconvex mechanics can be used to model complicated physical phenomena and to solve certain difficult nonconvex variational problems in an easy way. The canonical dual finite element method brings some new insights into computational mechanics.

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

A Posteriori Error Estimates for Axisymmetric and Nonlinear Problems

We propose and examine the primal and dual finite element method for solving an axially symmetric elliptic problem with mixed boundary conditions. We derive an a posteriori error estimate and generalize the method used for a nonlinear elliptic problem. Finally, an a posteriori error estimate for a nonlinear parabolic problem based on the concept of hierarchical finite element basis functions is introduced.

Dual finite element methods in homogenization for elastic–plastic fibrous composite material

Abstract The problem of homogenization for a periodic, elastic–perfectly plastic, fiber reinforced, composite material is considered. The overall mechanical behavior of the material is described using the anisotropic model of elastic–plastic body with kinematic hardening. The appropriate initial–boundary value problem, set for one repeatable cell of the composite, is solved in order to find the effective constitutive relations. The cell problem is solved using the finite element method formulated in two dual forms: in displacements and in stresses. Stress functions are used in the latter formulation.

Direct dual finite element solution method for the determination of capacitance, conductance or inductance in static linear problems

Presents a direct solution method for the determination of the capacitance, conductance or inductance of static linear problems with the dual finite element method. The direct solution method provides an entirely energy‐based method for the calculation of these global parameters and differs from the conventional solution techniques in that no recourse is required to the intermediate determination of the potential distribution. The direct solution method can be seen as the reduction of a dual circuit parameter network for the finite element representation. Compares the dual‐circuit parameter description with another, well‐known variational method called “Tubes and Slices”. Shows that by aligning the finite element mesh with the approximate positions of the equipotentials and flux lines in the system, the dual finite element model can be simplified to the equivalent of Tubes and Slices model and that both methods can produce exactly the same results.

On the numerical solution of axisymmetric domain optimization problems by dual finite element method

AbstractShape optimization of an axisymmetric three‐dimensional domain with an elliptic boundary value state problem is solved. Since the cost functional is given in terms of the cogradient of the solution, a dual finite element method based on the minimum of complementary energy principle is used. © 1994 John Wiley & Sons, Inc.

Domain optimization in $3D$-axisymmetric elliptic problems by dual finite element method

An axisymmetric second order elliptic problem with mixed boundary conditions is considered. The shape of the domain has to be found so as to minimize a cost functional, which is given in terms of the cogradient of the solution. A new dual finite element method is used for approximate solutions. The existence of an optimal domain is proven and a convergence analysis presented.

Hybrid finite element techniques for oil recovery simulation

This paper is devoted to a new numerical simulation of two-phases incompressible fluid flow (2D and 3D). Velocity and pressure equations are solved by an hybrid dual finite element method (FEM) with suppression of the grid effect. A Van Leer upwind scheme is used for the saturation equation, in order to reduce the front dispersion. For equal computational cost this method is more accurate than the classical finite difference method (FDM) used in reservoir simulation.

Dual extremum principles for a class of electrostatic field interface problems

Dual extremum principles (DEP), based on Arthurs' canonical approach, are applied to solve a class of electrostatic field interface problems. To illustrate the formulation, a rectangular metal tank, half-filled with petroleum products, is considered. The electrostatic potential and field distributions inside the petroleum tank are calculated using the dual finite element method, the method of Kantoro-vich and Ritz procedure. The bounds for the system energy are computed through the dual finite element method.

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.

Shape optimization in two-dimensional elasticity by the dual finite element method

On considere deux problemes de minimisation de fonction de cout par rapport a la partie de la frontiere ou un corps elastique est fixe. Le probleme elastostatique est resolu par le principe de Castigliano et par les elements finis. On demontre l'existence d'une frontiere optimale et la convergence des approximations

Optimization of the domain in elliptic problems by the dual finite element method

An optimal part of the boundary of a plane domain for the Poisson equation with mixed boundary conditions is to be found. The cost functional is (i) the internal energy, (ii) the norm of the external flux through the unknown boundary. For the numerical solution of the state problem a dual variational formulation - in terms of the gradient of the solution - and spaces of divergence-free piecewise linear finite elements are used. The existence of an optimal domain and some convergence results are proved.

The density of solenoidal functions and the convergence of a dual finite element method

The density of solenoidal functions and the convergence of a dual finite element method