A New Variational Formulation for a Kind of Reaction-diffusion Problem in Broken Sobolev Space

In this paper, a new variational formulation for a reaction-diffusion problem in broken Sobolev space is proposed. And the new formulation in the broken Sobolev space will be proved that it is well-posed and equivalent to the standard Galerkin variational formulation. The method will be helpful to easily solve the original partial differential equation numerically. And the method is novel and interesting, which can be used to deal with some complicated problem, such as the low regularity problem, the differential-integral problem and so on.

A MIXED VARIATIONAL FORMULATION FOR LARGE DEFORMATION ANALYSIS OF PLATES

A mixed vuriational formulation for large deformation analysis of plates is introduced. In this formulation the equilibrium ami compatibility equations are satisfied identically by means of stress functions and displacement components, respectively, and the constitu,lye equations are satisfied in a least square sense. An example is solved and the results are compared with those available in the literature.Further, the functional is particularized for buckling analysis of plates and a simple example is solved to illustrate the theory.

ERROR ESTIMATION FOR NUMERICAL METHODS USING THE ULTRA WEAK VARIATIONAL FORMULATION IN MODEL OF NEAR FIELD SCATTERING PROBLEM

In this paper, we investigate the use of ultra weak variational formulation to solve a wave scattering problem in near field optics. In order to capture the sub-scale features of waves, we utilize evanescent wave functions together with plane wave functions to approximate the local properties of the field. We analyze the global convergence and give an error estimation of the method. Numerical examples are also presented to demonstrate the effectiveness of the strategy.

Variational Formulation for Guided and Leaky Modes in Multilayer Dielectric Waveguides

The guided and leaky modes of a planar dielectric waveguide are eigensolutions of a singular Sturm-Liouville problem. The modes are the roots of a characteristicfunction which can be found with several methods that have been introduced in thepast. However, the evaluation of the characteristic function suffers from numericalinstabilities, and hence it is often difficult to find all modes in a given range. Here anew variational formulation is introduced, which, after discretization, leads either to aquadratic or a quartic eigenvalue problem. The modes can be computed with standardsoftware for polynomial eigenproblems. Numerical examples show that the method isnumerically stable and guarantees a complete set of solutions.

TheUltraWeakVariational FormulationUsing Bessel Basis Functions

We investigate the ultra weak variational formulation(UWVF)of the 2-D Helmholtz equation using a new choice of basis functions.Traditionally the UWVF basis functions are chosen to be plane waves.Here,we instead use first kind Bessel functions.We compare the performance of the two bases.Moreover,we show that it is possible to use coupled plane wave and Bessel bases in the same mesh.As test cases we shall consider propagating plane and evanescent waves in a rectangular domain and a singular 2-D Helmholtz problem in an L-shaped domain.

EVOLUTION FILTRATION PROBLEMS WITH SEAWATER INTRUSION: TWO-PHASE FLOW DUAL MIXED VARIATIONAL ANALYSIS

Tow-phase flow mixed variational formulations of evolution filtration problems with seawater intrusion are analyzed. A dual mixed fractional flow velocity-pressure model is considered with an air-fresh water and a fresh water-seawater characterization. For analysis and computational purposes, spatial decompositions based on nonoverlapping multidomains, above and below the sea level, are variationally introduced with internal boundary fluxes dualized as weak transmission constraints. Further, parallel augmented and exactly penalized duality algorithms, and proximation semi-implicit time marching schemes, are established and analyzed.

VARIATIONAL PRINCIPLES FOR NONLOCAL CONTINUUM MODEL OF ORTHOTROPIC GRAPHENE SHEETS EMBEDDED IN AN ELASTIC MEDIUM

Equations governing the vibrations and buckling of multilayered orthotropic graphene sheets can be expressed as a system of n partial differential equations where n refers to the number of sheets. This description is based on the continuum model of the graphene sheets which can also take the small scale effects into account by employing a nonlocal theory. In the present article a variational principle is derived for the nonlocal elastic theory of rectangular graphene sheets embedded in an elastic medium and undergo- ing transverse vibrations. Moreover the graphene sheets are subject to biaxial compression. Rayleigh quotients are obtained for the frequencies of freely vibrating graphene sheets and for the buckling load. The influence of small scale effects on the frequencies and the buckling load can be observed qualiatively from the expressions of the Rayleigh quotients. Elastic medium is modeled as a combination of Winkler and Pasternak foundations acting on the top and bottom layers of the mutilayered nano-structure. Natural boundary con- ditions of the problem are derived using the variational principle formulated in the study. It is observed that free boundaries lead to coupled boundary conditions due to nonlocal theory used in the continuum formulation while the local （classical） elasticity theory leads to uncoupled boundary conditions. The mathematical methods used in the study involve calculus of variations and the semi-inverse method for deriving the variational integrals.

On Frequency Sensitivity and Mode Orthogonality of Flexible Robotic Manipulators

This paper presents sensitivity analysis of vibration frequencies of flexible manipulators with respect to variations of systems parameters such as rotational inertia of hub,and mass,moment,and side of tip load.Both Euler-Bernoulli and Timoshenko dynamical models of flexible manipulators are discussed.By using variational method,sensitivity indices are obtained with explicit expressions for measuring the sensitivity of frequencies.Based on variational formulations,a novel method for deriving the orthogonal relations among vibration modal shape functions of flexible manipulators is introduced.With this method,the orthogonal relations can be derived easily without invoking the tedious process of differentiation and integration by part,as commonly used in their derivation.

Free vibration of membrane/bounded incompressible fluid

Vibration of a circular membrane in contact with a fluid has extensive applications in industry. The natural vibration frequencies for the asymmetric free vibra- tion of a circular membrane in contact with a bounded incompressible fluid are derived in this paper. Considering small oscillations induced by the membrane vibration in an incompressible and inviscid fluid, the velocity potential function is used to describe the fluid field. Two approaches are used to derive the free vibration frequencies of the sys- tem, which include a variational formulation and an approximate solution employing the Rayleigh quotient method. A good correlation is found between free vibration frequencies evaluated by these methods. Finally, the effects of the fluid depth, the mass density, and the radial tension on the free vibration frequencies of the coupled system are investigated.

ANALYSIS OF AUGMENTED THREE-FIELD MACRO-HYBRID MIXED FINITE ELEMENT SCHEMES

On the basis of composition duality principles, augmented three-field macrohybrid mixed variational problems and finite element schemes are analyzed. The compatibility condition adopted here, for compositional dualization, is the coupling operator surjectivity, property that expresses in a general operator sense the Ladysenskaja-Babulka-Brezzi inf-sup condition. Variational macro-hybridization is performed under the assumption of decomposable primal and dual spaces relative to nonoverlapping domain decompositions. Then, through compositional dualization macro-hybrid mixed problems are obtained, with internal boundary dual traces as Lagrange multipliers. Also, ＂mass＂ preconditioned aug- mentation of three-field formulations are derived, stabilizing macro-hybrid mixed finite element schemes and rendering possible speed up of rates of convergence. Dual mixed incompressible Darcy flow problems illustrate the theory throughout the paper.

Reiterated homogenization of a laminate with imperfect contact:gain-enhancement of effective properties

A family of one-dimensional(1D) elliptic boundary-value problems with periodic and rapidly-oscillating piecewise-smooth coefficients is considered. The coefficients depend on the local or fast variables corresponding to two different structural scales. A finite number of imperfect contact conditions are analyzed at each of the scales. The reiterated homogenization method(RHM) is used to construct a formal asymptotic solution. The homogenized problem, the local problems, and the corresponding effective coefficients are obtained. A variational formulation is derived to obtain an estimate to prove the proximity between the solutions of the original problem and the homogenized problem. Numerical computations are used to illustrate both the convergence of the solutions and the gain of the effective properties of a three-scale heterogeneous 1D laminate with respect to their two-scale counterparts. The theoretical and practical ideas exposed here could be used to mathematically model multidimensional problems involving multiscale composite materials with imperfect contact at the interfaces.

Neural Network-Based Variational Methods for Solving Quadratic Porous Medium Equations in High Dimensions

In this paper,we propose and study neural network-based methods for solutions of high-dimensional quadratic porous medium equation(QPME).Three variational formulations of this nonlinear PDE are presented:a strong formulation and two weak formulations.For the strong formulation,the solution is directly parameterized with a neural network and optimized by minimizing the PDEresidual.It can be proved that the convergence of the optimization problem guarantees the convergence of the approximate solution in the L^(1)sense.Theweak formulations are derived following(Brenier in Examples of hidden convexity in nonlinear PDEs,2020)which characterizes the very weak solutions of QPME.Specifically speaking,the solutions are represented with intermediate functions who are parameterized with neural networks and are trained to optimize the weak formulations.Extensive numerical tests are further carried out to investigate the pros and cons of each formulation in low and high dimensions.This is an initial exploration made along the line of solving high-dimensional nonlinear PDEs with neural network-based methods,which we hope can provide some useful experience for future investigations.

求带隔板贮箱中液体线性Stokes-Joukowski势的变分区域剖分技术

液体晃动引起的力和力矩会影响贮箱及其相关结构的动力学行为.为了能够使用解析方法研究旋转(例如俯仰)激励下贮箱中液体晃动,往往需要已知解析表达的液体线性Stokes-Joukowski势.本文提出了一种变分区域剖分技术,用以获得带隔板刚性贮箱中液体的解析近似线性Stokes-Joukowski势.该技术包括3步:(1)根据隔板位置(例如使用隔板作为子域的部分边界)引入人工界商,将静流体区域剖分为简单子域.同时引入不同子域中流体的人工密度或人工界面上的辅助法向速度函数;(2)将每个子域的线性Stokes-Joukowski势解表示为一类系数待定的调和函数的线性组合,并将人工界面上的辅助法向流体速度函数表示为系数待定的Fourier级数;(3)用Trefftz方法和提出的变分公式求解待定系数.本文得到的半解析线性Stokes-Joukowski势与文献或有限元法(FEM)给出的结果吻合良好.为了验证其在非线性液体晃动问题研究中的适用性,将其应用于俯仰激励下的二维部分充液带T形隔板矩形贮箱.本文得到的半解析结果与CFD(计算流体动力学)软件或文献给出的结果符合良好.

An accurate modeling of piezoelectric smart beams with first-order shear deformation theory

A finite element model for piezoelectric smart beam in extension mode based on First-order Shear Deformation Theory(FSDT)with an appropriate through-thickness distribution of electric potential is presented.Accuracy of piezoelectric finite element formulations depends on the selection of assumed mechanical and electrical fields.Most of the conventional FSDT-based piezoelectric beam formulations available in the literature use linear through-thickness distribution of electric potential which is actually nonlinear.Here,a novel quadratic profile of the through-thickness electric potential is proposed to include the nonlinear effects.The results obtained show that the accuracy of conventional formulations with linear through-thickness potential approximation is affected by the material configuration,especially when the piezoelectric material dominates the beam cross section.It is shown that the present formulation gives the same level of accuracy for all regimes of material configurations in the beam cross section.Also,a modified form of the FSDT displacements is employed,which utilizes the shear angle as a degree of freedom instead of section rotation.Such a FSDT displacement field shows improved performance compared to the conventional field.The present formulation is validated by comparing the results with ANSYS 2D simulation.The comparison of results proves the improved efficiency and accuracy of the present formulation over the conventional formulations.

Modelling and Analysis of a Class of Metal–Forming Problems

A class of steady-state metal-forming problems,with rigid-plastic,incompressible,strain-rate dependent material model and nonlocal Coulomb’s friction,is considered.Primal,mixed and penalty variational formulations,containing variational inequalities with nonlinear and nondifferentiable terms,are derived and studied.Existence,uniqueness and convergence results are obtained and shortly presented.A priori finite element error estimates are derived and an algorithm,combining the finite element and secant-modulus methods,is utilized to solve an illustrative extrusion problem.

Phase-field modeling of fracture for quasi-brittle materials

This paper addresses the modeling of fracture in quasi-brittle materials using a phase-field approach to the description of crack topol-ogy.Within the computational mechanics community,several studies have treated the issue of modeling fracture using phase fields.Most of these studies have used an approach that implies the lack of a damage threshold.We herein explore an alternative model that includes a damage threshold and study how it compares with the most popular approach.The formulation is systematically explained within a rigorous variational framework.Subsequently,we present the corresponding three-dimensional finite element discretization that leads to a straightforward numerical implementation.Benchmark simulations in two dimensions and three dimensions are then presented.The results show that while an elastic stage and a damage threshold are ensured by the present model,good agreement with the results reported in the literature can be obtained,where such features are generally absent.

Spectral Element Discretization of the Stokes Equations in Deformed Axisymmetric Geometries

In this paper,we study the numerical solution of the Stokes system in deformed axisymmetric geometries.In the azimuthal direction the discretization is carried out by using truncated Fourier series,thus reducing the dimension of the problem.The resulting two-dimensional problems are discretized using the spectral element method which is based on the variational formulation in primitive variables.The meridian domain is subdivided into elements,in each of which the solution is approximated by truncated polynomial series.The results of numerical experiments for several geometries are presented.

A Solver for Helmholtz System Generated by the Discretization of Wave Shape Functions

An interesting discretization method for Helmholtz equations was introduced in B.Despres[1].This method is based on the ultra weak variational formulation(UWVF)and the wave shape functions,which are exact solutions of the governing Helmholtz equation.In this paper we are concerned with fast solver for the system generated by the method in[1].We propose a new preconditioner for such system,which can be viewed as a combination between a coarse solver and the block diagonal preconditioner introduced in[13].In our numerical experiments,this preconditioner is applied to solve both two-dimensional and three-dimensional Helmholtz equations,and the numerical results illustrate that the new preconditioner is much more efficient than the original block diagonal preconditioner.

Spurious Solutions in the Multiband Effective Mass Theory Applied to Low Dimensional Nanostructures

In this paper we analyze a long standing problem of the appearance of spurious,non-physical solutions arising in the application of the effective mass theory to low dimensional nanostructures.The theory results in a system of coupled eigenvalue PDEs that is usually supplemented by interface boundary conditions that can be derived from a variational formulation of the problem.We analyze such a system for the envelope functions and show that a failure to restrict their Fourier expansion coeffi-cients to small k components would lead to the appearance of non-physical solutions.We survey the existing methodologies to eliminate this difficulty and propose a simple and effective solution.This solution is demonstrated on an example of a two-band model for both bulk materials and low-dimensional nanostructures.Finally,based on the above requirement of small k,we derive a model for nanostructures with cylindrical symmetry and apply the developed model to the analysis of quantum dots using an eight-band model.

Characteristic Local Discontinuous Galerkin Methods for Incompressible Navier-Stokes Equations

By combining the characteristicmethod and the local discontinuous Galerkin method with carefully constructing numerical fluxes,variational formulations are established for time-dependent incompressible Navier-Stokes equations in R^(2).The nonlinear stability is proved for the proposed symmetric variational formulation.Moreover,for general triangulations the priori estimates for the L^(2)−norm of the errors in both velocity and pressure are derived.Some numerical experiments are performed to verify theoretical results.