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Variational Principle (variational + principle)

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Engineering	82%

Selected Abstracts

Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2010
C. Miehe
Abstract The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase-field. In this paper, we outline a thermodynamically consistent framework for phase-field models of crack propagation in elastic solids, develop incremental variational principles and consider their numerical implementations by multi-field finite element methods. We start our investigation with an intuitive and descriptive derivation of a regularized crack surface functional that ,-converges for vanishing length-scale parameter to a sharp crack topology functional. This functional provides the basis for the definition of suitable convex dissipation functions that govern the evolution of the crack phase-field. Here, we propose alternative rate-independent and viscous over-force models that ensure the local growth of the phase-field. Next, we define an energy storage function whose positive tensile part degrades with increasing phase-field. With these constitutive functionals at hand, we derive the coupled balances of quasi-static stress equilibrium and gradient-type phase-field evolution in the solid from the argument of virtual power. Here, we consider a canonical two-field setting for rate-independent response and a time-regularized three-field formulation with viscous over-force response. It is then shown that these balances follow as the Euler equations of incremental variational principles that govern the multi-field problems. These principles make the proposed formulation extremely compact and provide a perfect base for the finite element implementation, including features such as the symmetry of the monolithic tangent matrices. We demonstrate the performance of the proposed phase-field formulations of fracture by means of representative numerical examples. Copyright © 2010 John Wiley & Sons, Ltd. [source]

Variational principles for symmetric bilinear forms

MATHEMATISCHE NACHRICHTEN, Issue 6 2008
Jeffrey Danciger
Abstract Every compact symmetric bilinear form B on a complex Hilbert space produces, via an antilinear representing operator, a real spectrum consisting of a sequence decreasing to zero. We show that the most natural analog of Courant's minimax principle for B detects only the evenly indexed eigenvalues in this spectrum. We explain this phenomenon, analyze the extremal objects, and apply this general framework to the Friedrichs operator of a planar domain and to Toeplitz operators and their compressions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

EOF using the Ritz method: Application to superelliptic microchannels

ELECTROPHORESIS, Issue 18 2007
Chang Yi Wang
Abstract An efficient Ritz method is developed from the variational principle to solve the Poisson,Boltzmann equation under the Debye,Hückel approximation for studying the EOF in microchannels. The method is applied to the family of superelliptic cross sections which includes the elliptic channel and the rectangular channel as limiting cases. Several accurate tables presented are useful for design of electroosmotic channels, especially rectangular channels with rounded corners. It is shown how the flow rate Q is a sophisticated consequence of the nondimensional electrokinetic width K, the aspect ratio b as well as the superelliptic exponent n. [source]

Optimal stress recovery points for higher-order bar elements by Prathap's best-fit method

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 8 2009
S. Rajendran
Abstract Barlow was the first to propose a method to predict optimal stress recovery points in finite elements (FEs). Prathap proposed an alternative method that is based on the variational principle. The optimal points predicted by Prathap, called Prathap points in this paper, have been reported in the literature for linear, quadratic and cubic elements. Prathap points turn out to be the same as Barlow points for linear and quadratic bar elements but different for cubic bar element. Nevertheless, for all the three elements, Prathap points coincide with the reduced Gaussian integration points. In this paper, an alternative implementation of Prathap's best-fit method is used to compute Prathap points for higher-order (viz., 4,10th order) bar elements. The effectiveness of Prathap points as points of accurate stress recovery is verified by actual FE analysis for typical bar problems. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Energy-adjustable mechanism of the combined hybrid finite element method and improvement of Zienkiewicz's plate-element

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2005
Xiao-ping Xie
Abstract The combined hybrid finite element method for plate bending problems allows arbitrary combinations of deflection interpolation and bending moment approximations. A novel expression of the approach discloses the energy-adjustable mechanism of the hybrid variational principle to enhance accuracy and stability of displacement-based finite element models. For a given displacement approximation, appropriate choices of the bending moment mode and the combination parameter , , (0,1) can lead to accurate energy approximation which generally yields numerically high accuracy of the displacement and bending moment approximations. By virtue of this mechanism, improvement of Zienkiewicz's triangular plate-element is discussed. The deflection is approximated by Zienkiewicz incomplete cubic interpolation. And three kinds of bending moments approximations are considered: a 3-parameter constant mode, a 5-parameter incomplete linear mode, and a 9-parameter linear mode. Since the parameters of the assumed bending moments modes can be eliminated at an element level, the computational cost of the combined hybrid counterparts of Zienkiewicz's triangle are as same as that of Zienkiewicz's triangle. Numerical experiments show that the combined hybrid versions can attain high accuracy at coarse meshes. Copyright © 2005 John Wiley & Sons, Ltd. [source]

A large time incremental finite element method for finite deformation problem,

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 11 2001
Y. Liu
Abstract Based on the process optimal control variational principle, some new ideas for finite deformation analysis using large increment are proposed. Combined with hyperelastic,plastic constitutive equation, the governing equations and the corresponding numerical algorithm are formulated. The proposed approaches are validated with the application to the analysis for finite deformation involving contact and friction. Copyright © 2001 John Wiley & Sons, Ltd. [source]

Splitting elastic modulus finite element method and its application

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2001
Dang Faning
Abstract To establish the best precision FEM, the proportion of potential and complementary energy in the functional of the variational principles must be changeable. A new kind of variational principle in linear theory of solid mechanics, called the splitting elastic modulus variational principle, is introduced. Its distinctive feature is that the functional contains one arbitrary additional parameter, called splitting factor; the proportion of potential and complementary energy in the functional can be changed by the splitting factor. Finite element method, which is based on the new principle, is established. It is called splitting modulus FEM, its stiffness can be adjusted by properly selecting the splitting factors, some ill-conditioned problem can be conquered by it. The methods to choose the splitting factors, reduce the condition number of stiffness matrix and improve the precision of solutions are also discussed. The reason why the new method can transform the ill-conditioned problems into well-conditioned ones is analysed finally. Copyright © 2001 John Wiley & Sons, Ltd. [source]

Lie-Poisson integrators: A Hamiltonian, variational approach

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 13 2010
Zhanhua Ma
Abstract In this paper we present a systematic and general method for developing variational integrators for Lie-Poisson Hamiltonian systems living in a finite-dimensional space ,,*, the dual of Lie algebra associated with a Lie group G. These integrators are essentially different discretized versions of the Lie-Poisson variational principle, or a modified Lie-Poisson variational principle proposed in this paper. We present three different integrators, including symplectic, variational Lie-Poisson integrators on G×,,* and on ,,×,,*, as well as an integrator on ,,* that is symplectic under certain conditions on the Hamiltonian. Examples of applications include simulations of free rigid body rotation and the dynamics of N point vortices on a sphere. Simulation results verify that some of these variational Lie-Poisson integrators are good candidates for geometric simulation of those two Lie-Poisson Hamiltonian systems. Copyright © 2009 John Wiley & Sons, Ltd. [source]

On the optimum support size in meshfree methods: A variational adaptivity approach with maximum-entropy approximants

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 7 2010
Adrian Rosolen
Abstract We present a method for the automatic adaption of the support size of meshfree basis functions in the context of the numerical approximation of boundary value problems stemming from a minimum principle. The method is based on a variational approach, and the central idea is that the variational principle selects both the discretized physical fields and the discretization parameters, here those defining the support size of each basis function. We consider local maximum-entropy approximation schemes, which exhibit smooth basis functions with respect to both space and the discretization parameters (the node location and the locality parameters). We illustrate by the Poisson, linear and non-linear elasticity problems the effectivity of the method, which produces very accurate solutions with very coarse discretizations and finds unexpected patterns of the support size of the shape functions. Copyright © 2009 John Wiley & Sons, Ltd. [source]

An adaptive stabilization strategy for enhanced strain methods in non-linear elasticity

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2010
Alex Ten Eyck
Abstract This paper proposes and analyzes an adaptive stabilization strategy for enhanced strain (ES) methods applied to quasistatic non-linear elasticity problems. The approach is formulated for any type of enhancements or material models, and it is distinguished by the fact that the stabilization term is solution dependent. The stabilization strategy is first constructed for general linearized elasticity problems, and then extended to the non-linear elastic regime via an incremental variational principle. A heuristic choice of the stabilization parameters is proposed, which in the numerical examples proved to provide stable approximations for a large range of deformations, different problems and material models. We also provide explicit lower bounds for the stabilization parameters that guarantee that the method will be stable. These are not advocated, since they are generally larger than the ones based on heuristics, and hence prone to deteriorate the locking-free behavior of ES methods. Numerical examples with two different non-linear elastic models in thin geometries and incompressible situations show that the method remains stable and locking free over a large range of deformations. Finally, the method is strongly based on earlier developments for discontinuous Galerkin methods, and hence throughout the paper we offer a perspective about the similarities between the two. Copyright © 2009 John Wiley & Sons, Ltd. [source]

A reduced integration solid-shell finite element based on the EAS and the ANS concept,Geometrically linear problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2009
Marco Schwarze
Abstract In this paper a new reduced integration eight-node solid-shell finite element is presented. The enhanced assumed strain (EAS) concept based on the Hu,Washizu variational principle requires only one EAS degree-of-freedom to cure volumetric and Poisson thickness locking. One key point of the derivation is the Taylor expansion of the inverse Jacobian with respect to the element center, which closely approximates the element shape and allows us to implement the assumed natural strain (ANS) concept to eliminate the curvature thickness and the transverse shear locking. The second crucial point is a combined Taylor expansion of the compatible strain with respect to the center of the element and the normal through the element center leading to an efficient and locking-free hourglass stabilization without rank deficiency. Hence, the element requires only a single integration point in the shell plane and at least two integration points in thickness direction. The formulation fulfills both the membrane and the bending patch test exactly, which has, to the authors' knowledge, not yet been achieved for reduced integration eight-node solid-shell elements in the literature. Owing to the three-dimensional modeling of the structure, fully three-dimensional material models can be implemented without additional assumptions. Copyright © 2009 John Wiley & Sons, Ltd. [source]

A finite element formulation for thermoelastic damping analysis

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 6 2009
Enrico Serra
Abstract We present a finite element formulation based on a weak form of the boundary value problem for fully coupled thermoelasticity. The thermoelastic damping is calculated from the irreversible flow of entropy due to the thermal fluxes that have originated from the volumetric strain variations. Within our weak formulation we define a dissipation function that can be integrated over an oscillation period to evaluate the thermoelastic damping. We show the physical meaning of this dissipation function in the framework of the well-known Biot's variational principle of thermoelasticity. The coupled finite element equations are derived by considering harmonic small variations of displacement and temperature with respect to the thermodynamic equilibrium state. In the finite element formulation two elements are considered: the first is a new 8-node thermoelastic element based on the Reissner,Mindlin plate theory, which can be used for modeling thin or moderately thick structures, while the second is a standard three-dimensional 20-node iso-parametric thermoelastic element, which is suitable to model massive structures. For the 8-node element the dissipation along the plate thickness has been taken into account by introducing a through-the-thickness dependence of the temperature shape function. With this assumption the unknowns and the computational effort are minimized. Comparisons with analytical results for thin beams are shown to illustrate the performances of those coupled-field elements. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Parallel asynchronous variational integrators

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2007
Kedar G. Kale
Abstract This paper presents a scalable parallel variational time integration algorithm for nonlinear elastodynamics with the distinguishing feature of allowing each element in the mesh to have a possibly different time step. Furthermore, the algorithm is obtained from a discrete variational principle, and hence it is termed parallel asynchronous variational integrator (PAVI). The underlying variational structure grants it outstanding conservation properties. Based on a domain decomposition strategy, PAVI combines a careful scheduling of computations with fully asynchronous communications to provide a very efficient methodology for finite element models with even mild distributions of time step sizes. Numerical tests are shown to illustrate PAVI's performance on both slow and fast networks, showing scalability properties similar to the best parallel explicit synchronous algorithms, with lower execution time. Finally, a numerical example in which PAVI needs ,100 times less computing than an explicit synchronous algorithm is shown. Copyright © 2006 John Wiley & Sons, Ltd. [source]

Optimal solid shell element for large deformable composite structures with piezoelectric layers and active vibration control

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 15 2005
X. G. Tan
Abstract In this paper, we present an optimal low-order accurate piezoelectric solid-shell element formulation to model active composite shell structures that can undergo large deformation and large overall motion. This element has only displacement and electric degrees of freedom (dofs), with no rotational dofs, and an optimal number of enhancing assumed strain (EAS) parameters to pass the patch tests (both membrane and out-of-plane bending). The combination of the present optimal piezoelectric solid-shell element and the optimal solid-shell element previously developed allows for efficient and accurate analyses of large deformable composite multilayer shell structures with piezoelectric layers. To make the 3-D analysis of active composite shells containing discrete piezoelectric sensors and actuators even more efficient, the composite solid-shell element is further developed here. Based on the mixed Fraeijs de Veubeke,Hu,Washizu (FHW) variational principle, the in-plane and out-of-plane bending behaviours are improved via a new and efficient enhancement of the strain tensor. Shear-locking and curvature thickness locking are resolved effectively by using the assumed natural strain (ANS) method. We also present an optimal-control design for vibration suppression of a large deformable structure based on the general finite element approach. The linear-quadratic regulator control scheme with output feedback is used as a control law on the basis of the state space model of the system. Numerical examples involving static analyses and dynamic analyses of active shell structures having a large range of element aspect ratios are presented. Active vibration control of a composite multilayer shell with distributed piezoelectric sensors and actuators is performed to test the present element and the control design procedure. Copyright © 2005 John Wiley & Sons, Ltd. [source]

A rational approach to mass matrix diagonalization in two-dimensional elastodynamics

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 15 2004
E. A. Paraskevopoulos
Abstract A variationally consistent methodology is presented, which yields diagonal mass matrices in two-dimensional elastodynamic problems. The proposed approach avoids ad hoc procedures and applies to arbitrary quadrilateral and triangular finite elements. As a starting point, a modified variational principle in elastodynamics is used. The time derivatives of displacements, the velocities, and the momentum type variables are assumed as independent variables and are approximated using piecewise linear or constant functions and combinations of piecewise constant polynomials and Dirac distributions. It is proved that the proposed methodology ensures consistency and stability. Copyright © 2004 John Wiley & Sons, Ltd. [source]

A variational r -adaption and shape-optimization method for finite-deformation elasticity

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2004
P. Thoutireddy
Abstract This paper is concerned with the formulation of a variational r -adaption method for finite-deformation elastostatic problems. The distinguishing characteristic of the method is that the variational principle simultaneously supplies the solution, the optimal mesh and, in problems of shape optimization, the equilibrium shapes of the system. This is accomplished by minimizing the energy functional with respect to the nodal field values as well as with respect to the triangulation of the domain of analysis. Energy minimization with respect to the referential nodal positions has the effect of equilibrating the energetic or configurational forces acting on the nodes. We derive general expressions for the configurational forces for isoparametric elements and non-linear, possibly anisotropic, materials under general loading. We illustrate the versatility and convergence characteristics of the method by way of selected numerical tests and applications, including the problem of a semi-infinite crack in linear and non-linear elastic bodies; and the optimization of the shape of elastic inclusions. Copyright © 2004 John Wiley & Sons, Ltd. [source]

Finite element formulation for modelling large deformations in elasto-viscoplastic polycrystals

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 14 2004
Karel Matou
Abstract Anisotropic, elasto-viscoplastic behaviour in polycrystalline materials is modelled using a new, updated Lagrangian formulation based on a three-field form of the Hu-Washizu variational principle to create a stable finite element method in the context of nearly incompressible behaviour. The meso-scale is characterized by a representative volume element, which contains grains governed by single crystal behaviour. A new, fully implicit, two-level, backward Euler integration scheme together with an efficient finite element formulation, including consistent linearization, is presented. The proposed finite element model is capable of predicting non-homogeneous meso-fields, which, for example, may impact subsequent recrystallization. Finally, simple deformations involving an aluminium alloy are considered in order to demonstrate the algorithm. Copyright © 2004 John Wiley & Sons, Ltd. [source]

The modeling and numerical analysis of wrinkled membranes

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2003
Hongli Ding
Abstract In this paper three fundamental issues regarding modeling and analysis of wrinkled membranes are addressed. First, a new membrane model with viable Young's modulus and Poisson's ratio is proposed, which physically characterizes stress relaxation phenomena in membrane wrinkling, and expresses taut, wrinkled and slack states of a membrane in a systematic manner. Second, a parametric variational principle is developed for the new membrane model. Third, by the variational principle, the original membrane problem is converted to a non-linear complementarity problem in mathematical programming. A parametric finite element discretization and a smoothing Newton method are then used for numerical solution. The proposed membrane model and numerical method are capable of delivering convergent results for membranes with a mixture of wrinkled and slack regions, without iteration of membrane stresses. Three numerical examples are provided. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Efficient mixed Timoshenko,Mindlin shell elements

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2002
G. M. Kulikov
Abstract The precise representation of rigid body motions in the displacement patterns of curved Timoshenko,Mindlin (TM) shell elements is considered. This consideration requires the development of the strain,displacement relationships of the TM shell theory with regard to their consistency with the rigid body motions. For this purpose a refined TM theory of multilayered anisotropic shells is elaborated. The effects of transverse shear deformation and bending-extension coupling are included. The fundamental unknowns consist of five displacements and eight strains of the face surfaces of the shell, and eight stress resultants. On the basis of this theory the simple and efficient mixed models are developed. The elemental arrays are derived using the Hu,Washizu mixed variational principle. Numerical results are presented to demonstrate the high accuracy and effectiveness of the developed 4-node shell elements and to compare their performance with other finite elements reported in the literature. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Micromechanics of fibre glass composites at elevated temperatures

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2002
Alexander Tesar
Abstract The micromechanical assessment of ultimate response of moulded fibre glass (MFG) composites used in structural engineering and acting at elevated temperatures is treated in the present paper. The advanced crystal simulation model of the MFG-material at elevated temperatures, based on Washizu's variational principle, is presented. The numerical treatment of non-linear problems possibly appearing is made using the updated Lagrangian formulation of motion. Each step of iteration approaches the solution of the linear problem and the feasibility of the parallel processing FETM-simulation approach is established. Some numerical and experimental results are presented in order to demonstrate the efficiency of procedures suggested. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Dual-mixed p and hp finite elements for elastic membrane problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2002
E. BertótiArticle first published online: 26 OCT 200
Abstract A complementary energy-based, dimensionally reduced plate model using a two-field dual,mixed variational principle of non-symmetric stresses and rotations is derived. Both the membrane and bending equilibrium equations, expressed in terms of non-symmetric mid-surface stress components, are satisfied a priori introducing first-order stress functions. It is pointed out that (i) the membrane-, shear- and bending energies of the plate written in terms of first-order stress functions are decoupled, (ii) although unmodified 3-D constitutive equations are applied, the energy parts do not contain the 1/(1-2,) term for isotropic, linearly elastic materials. These facts mean that the finite element formulation based on the present plate model should be free from shear locking when the thickness tends to zero and free from incompressibility locking when the Poisson ratio , converges to 0.5, irrespective of low-order h -, or higher-order p elements are used. Curvilinear dual-mixed hp finite elements with higher-order stress approximation and continuous surface tractions are developed and presented for the membrane (2-D elasticity) problem. In this case the formulation requires the approximation of three independent variables: two components of a first-order stress function vector and a scalar rotation. Numerical performance of three quadrilateral dual,mixed elements is presented and compared to displacement-based hp finite elements when the Poisson ratio converges to the incompressible limit of 0.5. The numerical results show that, as expected, the dual,mixed elements developed in this paper are free from locking in the energy norm as well as in the stress computations, for both h - and p -extensions. Copyright © 2001 John Wiley & Sons, Ltd. [source]

FEM simulation of turbulent flow in a turbine blade passage with dynamical fluid,structure interaction

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2009
Lixiang Zhang
Abstract Results are described from a combined mathematical modeling and numerical iteration schemes of flow and vibration. We consider the coupling numerical simulations of both turbulent flow and structure vibration induced by flow. The methodology used is based on the stabilized finite element formulations with time integration. A fully coupled model of flow and flow-induced structure vibration was established using a hydride generalized variational principle of fluid and solid dynamics. The spatial discretization of this coupling model is based on the finite element interpolating formulations for the fluid and solid structure, while the different time integration schemes are respectively used for fluid and solid structure to obtain a stabilized algorithm. For fluid and solid dynamics, Hughes' predictor multi-corrector algorithm and the Newmark method are monolithically used to realize a monolithic solution of the fully coupled model. The numerical convergence is ensured for small deformation vibrating problems of the structure by using different time steps for fluid and solid, respectively. The established model and the associated numerical methodology developed in the paper were then applied to simulate two different flows. The first one is the lid-driven square cavity flow with different Reynolds numbers of 1000, 400 and 100 and the second is the turbulent flows in a 3-D turbine blade passage with dynamical fluid,structure interaction. Good agreement between numerical simulations and measurements of pressure and vibration acceleration indicates that the finite element method formulations developed in this paper are appropriate to deal with the flow under investigation. Copyright © 2009 John Wiley & Sons, Ltd. [source]

Representation of the Dirac equation and the variational principle

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 15 2006
J. Karwowski
Abstract The performance of the variational method applied to the Dirac equation depends on the selected representation of this equation in four-dimensional spinor space. This dependence is analyzed. It is pointed out that the representations of Weyl and of Biedenharn have several attractive features. The usefulness of the so-called kinetically balanced Dirac equation is also briefly discussed. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006 [source]

A variational characterization of rate-independent evolution

MATHEMATISCHE NACHRICHTEN, Issue 11 2009
Ulisse Stefanelli
Abstract Rate-independent evolution driven by non-convex potentials is by nature non-smooth and some weak solvability notions have been recently advanced. This note is intended to contribute to this discussion by proposing a variational characterization of rate-independent evolution based on a variational principle and a maximal dissipation criterion. The resulting novel solution notion is assessed in an elementary yet critical scalar case (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Ekeland's variational principle in locally complete spaces

MATHEMATISCHE NACHRICHTEN, Issue 1 2003
Qiu Jing-Hui
Abstract We extend Ekeland's variational principle to locally complete locally convex spaces. As an application of the extension, we obtain a drop theorem in locally convex spaces which improves the related known result. [source]

Theory and Numerics of Rate-Dependent Incremental Variational Formulations in Ferroelectricity

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2008
Daniele Rosato
This paper is concerned with macroscopic continuous and discrete variational formulations for domain switching effects at small strains, which occur in ferroelectric ceramics. The developed new three,dimensional model is thermodynamically,consistent and determined by two scalar,valued functions: the energy storage function (Helmholtz free energy) and the dissipation function, which is in particular rate,dependent. The constitutive model successfully reproduces the ferroelastic and the ferroelectric hysteresis as well as the butterfly hysteresis for ferroelectric ceramics. The rate,dependent character of the dissipation function allows us also to reproduce the experimentally observed rate dependency of the above mentioned hysteresis phenomena. An important aspect is the numerical implementation of the coupled problem. The discretization of the two,field problem appears, as a consequence of the proposed incremental variational principle, in a symmetric format. The performance of the proposed methods is demonstrated by means of a benchmark problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

A Poroelastic Mindlin-Plate

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003
Anke Busse Dipl.Ing
The numerical treatment of noise insulation of solid walls has been an object of scientific research for many years. The main noise source is the bending vibration of the walls usually modeled as plates. Generally, walls consist of porous material, for instance concrete or bricks. Therefore, a poroelastic plate theory is necessary. A theory of dynamic poroelasticity was developed by Biot using the solid displacements and the pore pressure as unknowns. After formulating the poroelastic theory for thick plates, Mindlin's theory, a variational principle for this poroelastic thick plate model is developed. This is the basis of a Finite Element formulation. [source]

A Note on a Nonlinear Model of a Piezoelectric Rod

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2003
R. Gausmann Dipl.-Ing.
If piezoceramics are excited by weak electric fields a nonlinear behavior can be observed, if the excitation frequency is close to a resonance frequency of the system. To derive a theoretical model nonlinear constitutive equations are used, to describe the longitudinal oscillations of a slender piezoceramic rod near the first resonance frequency. Hamilton's principle is used to receive a variational principle for the piezoelectric rod. Introducing a Rayleigh Ritz ansatz with the eigenfunctions of the linearized system to approximate the exact solution leads to nonlinear ordinary differential equations. These equations are approximated with the method of harmonic balance. Finally it is possible to calculate the amplitudes of the displacements numerically. As a result it is shown, that the Duffing type nonlinearities found in measurements can be described with this model. [source]

Analysis of laterally loaded piles with rectangular cross sections embedded in layered soil

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 7 2008
D. Basu
Abstract An analysis is developed to determine the response of laterally loaded rectangular piles in layered elastic media. The differential equations governing the displacements of the pile,soil system are derived using variational principles. Closed-form solutions of pile deflection, the slope of the deflected curve, the bending moment and the shear force profiles can be obtained by this method for the entire pile length. The input parameters needed for the analysis are the pile geometry and the elastic constants of the soil and pile. The new analysis allows insights into the lateral load response of square, rectangular and circular piles and how they compare. Copyright © 2007 John Wiley & Sons, Ltd. [source]