Mixed Variational Method Research Articles

The unified nonlocal peridynamics-based phase-field damage theory

The peridynamic correspondence model is a promising and attractive candidate for the modeling of localized failure in solids, in that it can incorporate many local classical damage constitutive models through introducing a nonlocal averaged deformation gradient. However, the zero-energy mode and the limited bond breaking criterion greatly restrict the potential applications of this correspondence model. To address these two problems, a unified nonlocal peridynamics-based phase-field damage theory is proposed within the framework of thermodynamics. Firstly, the unified correspondence principle for the displacement field is developed on the basis of the energy compensation method in such a way that a new peridynamic deformation gradient, shape matrix and force state are derived and redefined. And then, the proposed principle is applied to derive the nonlocal phase-field damage constitutive model that the general nonlocal phase-field flux, phase-field flow state and phase-field internal force are defined. Moreover, we propose a mixed variational derivative method to obtain the coupled equilibrium governing equation and present the general linearization approach of the proposed peridynamics-based phase-field model (PD-PFM), where the double states for the coupled displacement and phase fields are derived in detail. It will be found that PD-PFM can not only resolve the zero-energy mode existed in the conventional peridynamic correspondence model, but also provide a rational criterion for the breakage of the peridynamic bond. Some representative numerical examples including mixed mode fracture of single-material media and interface fracture of ceramic coating systems are presented for the validation of PD-PFM. The satisfactory results show both quantitative and qualitative agreement with the available experiment.

A mixed variational inequality method for solving Signorini problems

This paper presents a new iterative method for solving Signorini problems. Boundary integral equations are established by discretizing the boundary of the problem, and the system equations related only to the unknown variables on the boundary of the problem are derived. Then, the Signorini boundary conditions of the problem are expressed as equivalent variational inequalities. By considering a simultaneous system of equations and variational inequalities, the Signorini problem is reduced to a mixed variational inequality problem. Finally, a compatibility iteration algorithm for solving mixed variational inequalities is designed based on the projection-contraction algorithm. Several examples are tested to demonstrate the accuracy and effectiveness of the proposed algorithm.

Band-structure calculation of SH-waves in 1D hypersonic nano-sized phononic crystals with deformable interfaces

Nanoscopic interface deformations between the layers in a superlattice have a great impact on its phononic band-structure. This work aims to study the SH-wave band-structure in one-dimensional (1D) multilayer hypersonic nano-sized phononic crystals with consideration of local deformations at the interfaces of the layers. At this scale, the traditional theory of elasticity is unable to capture the realistic forbidden and permitted frequencies. Thus, in light of surface/interface elastodynamics theory for nanostructures with deformable interfaces, a mixed variational method which accounts for the notion of interface elasticity, deformability, and inertia is introduced in the present work. For illustration of the capability of the present theory the phononic band-structures of: (1) HfO2–ZrO2 multilayer stack, and (2) ultrathin Pd-layer stack are calculated and compared with the corresponding results when the interface deformations are ignored. By matching the band-structure of HfO2–ZrO2 multilayer stack obtained in the literature based on ab-initio calculations with that of the present theory, the appurtenant interface parameters are predicted. For ultrathin Pd-layer stack, the experimental value of the interface compliance between the layers which is available in the literature is used. The effect of the nanoscopic interface deformability on the band gap and negative refraction of HfO2–ZrO2 multilayer stack is addressed. The noted effect on the band gap of Pd-layer stack is also studied. The presented formulation provides a fast and accurate method for predicting the wave propagation properties in hypersonic phononic crystals and hence opens a pathway in designing advanced metamaterials performing in gigahertz or terahertz frequency ranges.

Calculation of reflection and transmission coefficients for waves in multilayered piezoelectric structures using the mixed variable method.

The reflection and transmission (R/T) behavior of elastic waves in an anisotropic multilayered piezoelectric structure bounded by two homogeneous half-spaces is studied using the mixed variable method. The mixed variable method, which is based on the mixed energy matrix, takes the displacement vector at one end of the structure and the stress vector at other end of the structure as the basic variables. The wave equation for a homogeneous piezoelectric layer is transformed into a first-order state equation in the Hamiltonian system by introducing a dual vector. The general solution of the state equation can be expressed in terms of the eigenvalues and eigenvectors of the complex Hamiltonian matrix. A mixed energy matrix is applied to establish the relationship between the generalized displacement and stress vectors on the upper and lower interfaces of a layer. By an efficient recursive algorithm, the global mixed energy matrix is formed for an arbitrarily anisotropic multilayered piezoelectric structure. The R/T coefficients of the waves in an anisotropic multilayered piezoelectric structure are derived by the global mixed energy matrix. Numerical examples are provided to show the robustness of the mixed variable method. The effects of the incident angles, wavenumbers, and critical angles on the R/T coefficients are discussed.

Timoshenko Beams and the Hamiltonian System

The significance of the transition from Lagrangian system to Hamiltonian system lies in that it has entered the form of symplectic geometry from the traditional Euclidean geometry and broken through the traditional concept, so that the dual mixed variable method has entered into the vast field of applied mechanics. Thus the Hamiltonian system method is very significant for engineering mechanics system and even mathematical physics method. It provides a powerful tool for discussing this kind of problems and improves the solution of elasticity to a new platform.

The Eigensolution Expansion in Viscoelastic Materials

By using the eigensolution expansion method, the problem can be transformed into a Hamiltonian matrix. In symplectic system, the study of elasticity takes the primal variable and its dual variable as the basic variable, which makes the separated variable method be implemented smoothly, thus forming a unique direct method. The significance of the transition from Lagrangian system to Hamiltonian system lies in the transition from the traditional Euclidean geometry to the symplectic geometry, and the introduction of dual mixed variable method into the broad field of applied mechanics, which provides a powerful tool for the discussion of mechanical problems. At the same time, this method, marked by the all state vector, is more in line with and adapts to the development of computer. The trend is very important to the engineering mechanics system, the mathematical physics method and the research of other subjects.

Nearly-constrained transversely isotropic linear elasticity: energetically consistent anisotropic deformation modes for mixed finite element formulations

Strong anisotropies and/or near-incompressibility properties introduce internal constraints in material deformation. Numerical simulations comprising such a constrained behaviour show an overstiff structural response, referred to as element locking. Implementations based on mixed variational methods can heal locking but available solutions in the state-of-the-art are still non-optimal for anisotropic materials. This paper addresses this issue, by proposing a novel decomposition of anisotropic deformation modes on the basis of kinematic and energy requirements. Theoretical results exploit the Walpole’s formalism. The proposed kinematic split allows to introduce a new class of variational principles, referred to as energetically decoupled, for nearly-constrained transversely isotropic materials in linear elasticity. Low-order mixed finite element models are thus derived for treating near-inextensibility and/or near-incompressibility. Two-dimensional benchmark tests reproducing pure-bending and Cook’s membrane problems are conducted. Numerical results show that the accuracy of energetically decoupled formulations is high and robust with respect to variations of material properties, while the accuracy of non-energetically decoupled formulations is more sensitive.

Stability analysis of the mixed variable method and its application in wave reflection and transmission in multilayered anisotropic structures

This paper focuses on the reflection and transmission (R/T) problem of elastic waves in multilayered anisotropic structures. Stability analysis of the mixed variable method (MVM) for computing the R/T coefficients of elastic waves in multilayered anisotropic structures is presented. For this purpose, a detailed comparison of the MVM with the other two widely used methods, namely the transfer matrix method (TMM) and the stiffness matrix method (SMM), is made. Although the TMM, the SMM and the MVM are mathematically equivalent, they are quite different in numerical stability. The theoretical analysis shows that the MVM is unconditionally stable for arbitrary wavenumber–thickness products, whereas the TMM and the SMM may become unstable for large or small wavenumber–thickness products, respectively. This conclusion is numerically verified by various examples. Finally, the R/T coefficients of elastic waves in generally anisotropic multilayered structures bounded by two semi-infinite spaces are calculated using the MVM for a quasi-longitudinal or quasi-transverse wave incidence, and the effects of incident angles and wavenumber–thickness products on the R/T coefficients are discussed in detail through an example.

A Finite Element Formulation for Crack Problem in Couple-Stress Elasticity

In this research, we investigated the crack behavior in the Mode I fracture in a plane strain state, considering the size effects and under the couple-stress theory using finite element method (FEM). First, using the eigenfunction expansion technique, we provided the stress, couple-stress, displacement, strain, and curvature fields at the crack-tip. Then, using the FEM, fundamental parameters that determine the stress and couple-stress fields at the crack-tip (i.e., stress intensity factor and couple-stress intensity factor) were obtained. In the formulation of FEM, we used a mixed variational method, in which displacement and rotation are considered as independent variables, and their kinematic constraints are applied using Lagrange multipliers. Based on this formulation, it was possible to implement the classical quarter-point elements for the crack-tip. The results show that under the framework of the couple-stress theory, energy release rate can still be considered as the fundamental measure and determinant parameter for the crack behavior, but the stress intensity factor alone cannot appropriately describe the crack behavior.

Combining Plane Wave Expansion and Variational Techniques for Fast Phononic Computations

In this paper the salient features of the Plane Wave Expansion (PWE) method and the mixed variational technique are combined for the fast eigenvalue computations of arbitrarily complex phononic unit cells. This is done by expanding the material properties in a Fourier expansion, as is the case with PWE. The required matrix elements in the variational scheme are identified as the discrete Fourier transform coefficients of material properties, thus obviating the need for any explicit integration. The process allows us to provide succinct and closed form expressions for all the matrices involved in the mixed variational method. The scheme proposed here preserves both the simplicity of expression which is inherent in the PWE method and the superior convergence properties of the mixed variational scheme. We present numerical results and comment upon the convergence and stability of the current method. We show that the current representation renders the results of the method stable over the entire range of the expansion terms as allowed by the spatial discretization. When compared with a zero order numerical integration scheme, the present method results in greater computational accuracy of all eigenvalues. A higher order numerical integration scheme comes close to the accuracy of the present method but only with significantly more computational expense.

Reissner mixed variational theorem-based finite cylindrical layer methods for the static analysis of functionally graded piezoelectric circular hollow cylinders under electro-mechanical loads

ABSTRACTA unified formulation of Reissner's mixed variational theorem-based finite cylindrical layer methods is developed for the static analysis of simply-supported, multilayered functionally graded piezoelectric material (FGPM) circular hollow cylinders. The material properties of the cylinder are assumed to obey an exponent-law exponentially varying through the thickness coordinate of this. The trigonometric functions and Lagrange polynomials are used to interpolate the in-surface and thickness variations of the primary variables of each individual layer, respectively. The coupled electro-elastic effects on the static behaviors of multilayered FGPM cylinders are closely examined.

A powerful method to analyze of photonic crystals: mixed variational method

Photonic crystal has drawn much attention because of its application in molding the flow of light, which can be used in optical communication, optical storage and computing. In theory, plane wave expansion method, finite difference time domain (FDTD) method and transfer matrix method are widely used methods to study photonic crystal, and each of them has its own advantages and disadvantages.Here, a new method i.e. mixed variational method is introduced to study the photonic crystals, which is from the work of anti-plane shear waves in periodic layered elastic composites. The calculations of this method are direct and require no iteration, which accurately and efficiently produce the entire band structure of the composite and other field characteristics. Moreover, the composite cell in this method may consist of any number of units of any variable permittivity and permeability.Firstly, based on the variational principle, the Lagrangian density of electro-magnetic field is obtained. Then through the surface integral of the Lagrangian density in the unit cell, the Lagrangian is acquired. The first variation of Lagrangian with respect to electric field and magnetic field yields a set of Euler-Lagrange equations. Approximate solutions in explicit series expressions subject to the Bloch periodicity are substituted into the above equations. Minimization of Lagrangian with respect to the electric field and magnetic field results in an eigenvalue problem, and to solve it, the band structure of the composite is yielded. Electrical field, magnetic field, group velocity and energy flux density are also calculated. Secondly, we use the above method to study a two dimensional air-rod unit cell system. Bandgaps with respect to different structural parameters are plotted, which are the same as the results from the plane wave expansion method and FDTD method. In theory, the entire band structure can be calculated with our method. There are more gaps for TE wave than for TM case. By constant frequency contours, it is shown that there is a gap between the first and the second pass band for TE wave, however, there is no a gap for the corresponding TM wave. The directions of group velocity for the first and the second bands are shown in the contours. Electrical field, magnetic field and energy flux in cells illustrate the energy distribution, and the energy-flux directions and the group-velocity directions are also essentially the same. Lastly, we apply this mixed variational method to one-dimensional media-air slab and three dimensional sphere-air structure. The obtained band results accord with those reported previously former, which demonstrates that our method is universal and correct.In the present work, a mixed variational approach is proposed to produce the entire band structure of the composite for unit cells with any arbitrary properties. Explicit expressions are developed for the band, electrical field, magnetic field, group velocity and energy flux.

Refraction characteristics of phononic crystals

Some of the most interesting refraction properties of phononic crystals are revealed by examining the anti-plane shear waves in doubly periodic elastic composites with unit cells containing rectangular and/or elliptical multi-inclusions. The corresponding band structure, group velocity, and energy–flux vector are calculated using a powerful mixed variational method that accurately and efficiently yields all the field quantities over multiple frequency pass-bands. The background matrix and the inclusions can be anisotropic, each having distinct elastic moduli and mass densities. Equifrequency contours and energy–flux vectors are readily calculated as functions of the wave-vector components. By superimposing the energy–flux vectors on equifrequency contours in the plane of the wave-vector components, and supplementing this with a three-dimensional graph of the corresponding frequency surface, a wealth of information is extracted essentially at a glance. This way it is shown that a composite with even a simple square unit cell containing a central circular inclusion can display negative or positive energy and phase velocity refractions, or simply performs a harmonic vibration (standing wave), depending on the frequency and the wave-vector. Moreover, that the same composite when interfaced with a suitable homogeneous solid can display: (1) negative refraction with negative phase velocity refraction; (2) negative refraction with positive phase velocity refraction; (3) positive refraction with negative phase velocity refraction; (4) positive refraction with positive phase velocity refraction; or even (5) complete reflection with no energy transmission, depending on the frequency, and direction and the wavelength of the plane-wave that is incident from the homogeneous solid to the interface. For elliptical and rectangular inclusion geometries, analytical expressions are given for the key calculation quantities. Expressions for displacement, velocity, linear momentum, strain, and stress components, as well as the energy–flux and group velocity components are given in series form. The general results are illustrated for rectangular unit cells, one with two and the other with four inclusions, although any number of inclusions can be considered. The energy–flux and the accompanying phase velocity refractions at an interface with a homogeneous solid are demonstrated. Finally, by comparing the results of the present solution method with those obtained using the Rayleigh quotient and, for the layered case, with the exact solutions, the remarkable accuracy and the convergence rate of the present solution method are demonstrated. Anomalous refractive characteristics of phononic crystals are revealed using anti-plane shear waves in doubly periodic elastic composites. It is shown that negative or positive refraction can be accompanied by negative or positive phase velocity refraction, and/or the crystal can have complete refraction with no energy transmission, all depending on the wavelength and frequency of the plane wave that is incident from a homogeneous solid to its interface with the composite. A powerful computational tool is proposed which applies to one-, two-, three-dimensional phononic crystals yielding results with great accuracy.

Under Uniformly Distributed Load are Fixed on one Side (Simple) on the Edge Simply Supported and Fixed Half Bending of Rectangular Plates

In this paper mixed variable method is generalized to solve the problem of bending of the rectangular plate with one side is fixed (simple) on the edge simply supported and fixed half under the action of a uniformly distributed load. Gives the surface deflection equation and a chart, the chart can be directly applied to engineering practice.

The Triangular Bending Plate with Three Sides Clamped under the Action of a Concentrated Load

In this paper mixed variable method is generalized to solve the problem of bending of the triangular plate with three sides clamped under the action of a concentrated load. Gives the surface deflection equation and a chart, the chart can be directly applied to engineering practice.

Provably unconditionally stable, second-order time-accurate, mixed variational methods for phase-field models

We introduce provably unconditionally stable mixed variational methods for phase-field models. Our formulation is based on a mixed finite element method for space discretization and a new second-order accurate time integration algorithm. The fully-discrete formulation inherits the main characteristics of conserved phase dynamics, namely, mass conservation and nonlinear stability with respect to the free energy. We illustrate the theory with the Cahn–Hilliard equation, but our method may be applied to other phase-field models. We also propose an adaptive time-stepping version of the new time integration method. We present some numerical examples that show the accuracy, stability and robustness of the new method.

Resolution of Operator Singularities via the Mixed-Variable Method

This paper applies a modern method of singularity resolution in algebraic geometry to resolving singularities of integral operators in Fourier analysis. This is achieved by introducing a method of mixed variables that is equivalent to changing coordinates for integral operators. We decompose the integral operator into dyadic pieces via monomial transforms and the mixed-variable method so as to obtain its sharp estimates on different domains. These sharp estimates can be written in an elegant form in terms of continued fractions.

Adaptive Variational Multiscale Methods for Multiphase Flow in Porous Media

Porous media are characterized by strongly heterogeneous properties on all scales of observation. This has motivated a sustained research effort in upscaling methods and lately multiscale methods. However, due to the lack of scale separation for realistic problems, static upscaling and multiscale methods may not be suitable in general, and adaptive approaches are needed. In this paper, we outline the application of an adaptive variational multiscale method for two-phase flow in porous media. We give the symmetric and nonsymmetric forms of the mixed variational multiscale method and suggest a framework for adaptively selecting the support of the fine scale Green's operators in space and time. The applicability of our framework is validated through a set of numerical comparisons.

My Challenge in the Development of a Mixed Variational Method in Solid Mechanics

My Challenge in the Development of a Mixed Variational Method in Solid Mechanics

Mixed finite element methods for the Signorini problem with friction

In this article, we propose and study different mixed variational methods in order to approximate the Signorini problem with friction using finite elements. The discretized normal and tangential constraints at the contact interface are expressed by using either continuous piecewise linear or piecewise constant Lagrange multipliers in the saddle−point formulation. A priori error estimates are established and several numerical examples corresponding to the different choices of the discretized normal and tangential constraints are carried out. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006