Weak Variational Formulation Research Articles

A Posteriori Error Estimation for the Optimal Control of Time-Periodic Eddy Current Problems

Abstract This work presents the multiharmonic analysis and derivation of functional type a posteriori estimates of a distributed eddy current optimal control problem and its state equation in a time-periodic setting. The existence and uniqueness of the solution of a weak space-time variational formulation for the optimality system and the forward problem are proved by deriving inf-sup and sup-sup conditions. Using the inf-sup and sup-sup conditions, we derive guaranteed, sharp and fully computable bounds of the approximation error for the optimal control problem and the forward problem in the functional type a posteriori estimation framework. We present here the first computational results on the derived estimates.

RETRACTED: Mixed convective heat transfer in a power-law fluid in a square enclosure: Higher order finite element solutions

Incorporation of momentum gradients produced due to inertial motion of the lid along with the presence of temperature differences in the configuration make the physical problem more significant. The joint variation of momentum and thermal diffusion in diversified natural liquids is recognized as mixed convection. Valuable attention has been received by such a phenomenon in different areas of science and technology such as in wind current–based solar receivers, electronic instruments, control of emergency shutdown in reactors, thermal exchangers, oceanic currents, control of atmospheric pollution, and so on. So, the main focus is to contemplate hydrothermal characteristics of a power-law fluid contained in a square cavity with the movement of the upper lid and being thermally adiabatic. The other extremities are considered to be at rest, and the base wall is prescribed with uniform/non-uniform temperature distributions. The governing formulation of the problem is handled by executing a finite element approach. Hybrid meshing is performed for domain discretization, and weak variational formulation is utilized for formulation discretization. Second-degree polynomials are employed as the interpolation function, providing information about velocity and temperature distributions at boundary and intermediate nodes. The system of finalized non-linear equations is resolved by using the Paradiso software. The results for velocity and temperature distributions are attained comparatively for uniformly and non-uniformly heated profiles. The kinetic energy and average Nusselt number are also computed against flow concerning variables. From the attained graphical and tabular data, it is deduced that by increasing the Reynolds number, inertial forces dominate over buoyancy forces and the effect of lid movement is prominent on flow characteristics. It is also inferred that for the shear thickening case and for all values of the Reynolds number, the average Nusselt number shows a constant behavior.

MULTI-SCALE NUMERICAL APPROACH TO THE POLYMER FILLING PROCESS IN THE WELD LINE REGION

In this paper, a multi-scale coupling mathematical model is suggested for simulating the polymer filling process in the weld line region on a micro scale. The model considers two aspects: one is the coupling model based on stresses in the whole cavity region; the other is the multi-scale coupling model of continuum mechanics (CM) and the molecular dynamics (MD) in a weldline region. A weak variational formulation is constructed for the finite element method (FEM), which is coupled with the Verlet algorithm based on the domain decomposition technique. Meanwhile, an overlap region is designed so that the FEM and the MD simulations are consistent with each other. The molecular backbone orientation of the whole cavity is illustrated and the position of the weld line is determined by the characteristics of the molecular backbone orientation. Finally, the properties of the polymer chain in the weld line region are studied conformationally and dynamically. The conformational changes and movement process elucidate that the polymer chains undertake stretching, entangling and orientating. Moreover, the effect of the number of chains and melt temperature on the spatial properties of chain conformation are investigated.

Rotational self-friction problem of elastic rods

The aim of this paper is to extend the modeling of a hyperelastic rod undergoing large displacements with tangential self-friction to their modeling with rotational self-friction. The discontinuity of contact force into a contact region not known in advance with taking into account the effects of friction in this problem type leads to serious difficulties in modelization, mathematical and numerical analysis. In this paper, we present an accurate modeling of rotational and tangential self-friction with Coulomb’s law and describe also an augmented Lagrangian method to present a weak variational formulation approach of this problem. We then use the minimization method of the total energy to present an existence result of solution for the nonlinear penalized formulation. Finally, we give the linearization and the finite-element discretization of the weak variational formulation that can be useful for a numerical implementation.

A Structure-Preserving Parametric Finite Element Method for Surface Diffusion

We propose a structure-preserving parametric finite element method (SP-PFEM) for discretizing the surface diffusion of a closed curve in two dimensions (2D) or surface in three dimensions (3D). Here the "structure-preserving" refers to preserving the two fundamental geometric structures of the surface diffusion flow: (i) the conservation of the area/volume enclosed by the closed curve/surface, and (ii) the decrease of the perimeter/total surface area of the curve/surface. For simplicity of notations, we begin with the surface diffusion of a closed curve in 2D and present a weak (variational) formulation of the governing equation. Then we discretize the variational formulation by using the backward Euler method in time and piecewise linear parametric finite elements in space, with a proper approximation of the unit normal vector by using the information of the curves at the current and next time step. The constructed numerical method is shown to preserve the two geometric structures and also enjoys the good property of asymptotic equal mesh distribution. The proposed SP-PFEM is "weakly" implicit (or almost semi-implicit) and the nonlinear system at each time step can be solved very efficiently and accurately by the Newton's iterative method. The SP-PFEM is then extended to discretize the surface diffusion of a closed surface in 3D. Extensive numerical results, including convergence tests, structure-preserving property and asymptotic equal mesh distribution, are reported to demonstrate the accuracy and efficiency of the proposed SP-PFEM for simulating surface diffusion in 2D and 3D.

Modeling of the crystallization process of a polymer plate taking into account large deformations

The issues of modeling the thermomechanical behavior of a polymer material during cooling accompanied by crystallization are considered. The proposed approach is applied to the problem of deformation upon cooling of an infinite plate made of low-pressure polyethylene during crystallization. Mathematical formulations of thermal kinetic and thermomechanical boundary value problems are presented. The results of the numerical solution of a coupled heat-kinetic problem, including the non-stationary heat conduction equation and the crystallization kinetics equation obtained under the assumption that the characteristics of the material depend on temperature are given. When solving the boundary value problem of thermomechanics, the previously obtained phenomenological nonlinear constitutive relations are used, which continuously describe the behavior of a polymer material in a wide temperature range, including the range of phase transformations. The construction of constitutive relations was carried out using the Peng-Landel potential. A weak variational formulation of the boundary value problem, constructed using the Galerkin method, is presented. A linearization procedure is carried out, which makes it possible to reduce the solution of an initially nonlinear boundary value problem to solving a sequence of linear problems with relatively small increments of the displacement vector components. In this case, the linearization of geometric relations is performed by superimposing small increments of deformations on finite ones, linearization of the constitutive relations - expansion in a Taylor series with subsequent retention of linear terms under the assumption that the increment of the components of the strain tensor is small. An approach is demonstrated that naturally takes into account the small increments of temperature and structural deformations arising in the material. Some aspects of the numerical implementation of the created algorithm based on the finite element technology of constructing a discrete analogue of the problem under consideration are analyzed. The results of solving the problem under discussion in a linear formulation under the assumption of small deformations and in a nonlinear formulation are compared.

Априорные оценки локального разрывного метода Галеркина на разнесенных сетках для решения уравнения параболического типа в рамках однородной задачи Дирихле

Представлены априорные оценки точности решения однородной краевой задачи для параболического уравнения с помощью локального метода Галеркина с разрывными базисными функциями на разнесенных сетках. Дискретизация по пространству строится с помощью обращения к смешанной конечно-элементной формулировке. Производные второго порядка не могут быть согласованы напрямую в слабой вариационной формулировке, используя пространство разрывных функций. Для понижения порядка компоненты вектора потока рассматриваются как вспомогательные неизвестные искомого уравнения второго порядка. Аппроксимация строится на разнесенных сетках. Основная сетка состоит из треугольников, двойственная сетка состоит из медианных контрольных объемов вокруг узлов треугольной сетки. Аппроксимация искомой функции строится на ячейках основной сетки, в то время как аппроксимация вспомогательных неизвестных строится на ячейках двойственной сетки. Для вычисления потоков на границе между элементами используется стабилизирующий параметр. При этом поток искомой функции не зависит от вспомогательных функций, в то время как поток вспомогательных величин зависит от искомой функции. Для решения поставленной задачи в работе формулируются и доказываются необходимые леммы. В результате сформулирована и доказана основная теорема, результатом которой являются априорные оценки при решении параболического уравнения с помощью метода Галеркина с разрывными базисными функциями. Основную роль при анализе сходимости играет оценка для отрицательной нормы градиента. В работе для стабилизирующего параметра порядка $1$ показано, что порядок сходимости будет $k+{1}/{2}$, а в случае использования стабилизирующего параметра порядка $h^{-1}$ порядок сходимости увеличивается до $k+1$, когда в качестве базиса используются полиномы степени не выше $k$.

On the pressure Poisson equation for the Stokes system

AbstractFor the numerical solution of the pressure Poisson equation, we consider an ultra–weak variational formulation and a related finite element method of Galerkin–Petrov type. While this allows a piecewise constant approximation of the pressure, the test functions have to be sufficiently smooth – in particular, we use modified B–splines of second order to guarantee conformity and stability, and optimal convergence. This novel formulation allows the computation of the pressure from either a given, measured, or precomputed velocity field as it appears in many applications in fluid mechanics.

Comparisons of three kinds of plane wave methods for the Helmholtz equation and time-harmonic Maxwell equations with complex wave numbers

In this paper we are concerned with some plane wave discretization methods of the Helmholtz equation and time-harmonic Maxwell equations with complex wave numbers. We design two new variants of the variational theory of complex rays and the ultra weak variational formulation for the discretization of these types of equations, respectively. The well posedness of the approximate solutions generated by the two methods is derived. Moreover, based on the PWLS–LSFE method introduced in Hu and Yuan (2018), we extend these two methods (VTCR method and UWVF method) combined with local spectral element to discretize nonhomogeneous Helmholtz equation and Maxwell’s equations. The numerical results show that the resulting approximate solution generated by the UWVF method is clearly more accurate than that generated by the VTCR method.

Mixed Convolved Action for Thermoelasticity

In the present work, a variational formulation for fully coupled thermoelasticity is developed in the context of the mixed convolved action. This weak variational formulation recovers all the governing differential equations along with proper initial and boundary conditions as its Euler–Lagrange equations. Thus, it encapsulates the entire description of thermoelastic problems. In addition to theoretical significance, it provides sound basis for the development of novel computational methods involving unified space-time finite element approach. The simplest unified space-time finite element method is also developed here with representative examples for its viability.

Finite element computation of magnetohydrodynamic nanofluid convection from an oscillating inclined plate with radiative flux, heat source and variable temperature effects

This work describes finite element computations for radiative magnetohydrodynamic convective Newtonian nanofluid flow from an oscillating inclined porous plate with variable temperature. Heat source/sink and buoyancy effects are included in the mathematical model. The problem is formulated by employing Tiwari–Das nanofluid model, and two water-based nanofluids, copper and alumina, with spherical shaped metal nanoparticles are considered. The Brinkman and Maxwell–Garnetts models are used for the dynamic viscosity and effective thermal conductivity of the nanofluids, respectively. An algebraic flux model, the Rosseland diffusion approximation, is adopted to simulate thermal radiative flux effects. The dimensionless, coupled governing partial differential equations are numerically solved via the finite element method with weak variational formulation by imposing initial and boundary conditions with a weighted residual scheme. A grid independence study is also conducted. The finite element solutions are reduced to known previous solutions in some limiting cases of this investigation and are found to be in good agreement with published work. This investigation is relevant to electromagnetic nano-material manufacturing processes operating at high temperatures where radiation heat transfer is significant.

The ultra weak variational formulation for the modified mild-slope equation

• The Ultra Weak Variational Formulation (UWVF) has been used successfully on the modified mild slope equation. • The UWVF was improved by using the main direction with the ray tracing method. • The UWVF solves some of the difficulties that arise when Finite Element Method is applied. This paper is devoted to evaluate a new alternative numerical method to capture accurately the diffraction-refraction process of waves in coastal areas. The modified mild-slope equation (MMSE) has been used to predict the water wave transformation when waves approach the shoreline. We perform numerical simulations in order to illustrate the efficiency of the ultra weak variational formulation (UWVF) method in comparison with the finite elements method (FEM). The UWVF method uses plane wave solutions on each element and has been shown to reduce the computational complexity at high wave numbers. We also present an alternative method to seek the angle of attack of the wave front on the domain boundary and show that the UWVF method reproduces effectively the numerical experimental data. We compare the FEM and the UWVF method for the MMSE and show that the UWVF method solves some of the difficulties that arise when the FEM is applied.

Performance analysis of the ultra weak variational formulation to compute electromagnetic fields on nonuniform meshes

AbstractThe purpose of this work is to apply a previously developed ultra weak variational formulation (UWVF) to the solution of the Helmholtz equation in electromagnetic scattering problems. The variational form is based on discontinuous Galerkin with plane wave basis functions in 2‐dimensional finite element mesh. We present a general strategy, which allows to adjust the number of wave directions based on both the element size and the local conditioning number. This provides a cost‐effective mechanism for application of UWVF using nonuniform meshes. Through simulation of scattering of a dielectric‐coated circular cylinder, it is shown that the process significantly improves the UWVF performance.

Extension of the variational theory of complex rays to orthotropic shallow shell structures

Nowadays, the interest of aerospace and automotive industries on virtual testing of medium-frequency vibrational behavior of shallow shell structures is growing. The development of software capable of predicting the vibrational response in such frequency range is still an open question because classical methods (i.e., FEM, SEA) are not fully suitable for the medium-frequency bandwidth. In this context the Variational Theory of Complex Rays (VTCR) is taking place as an ad-hoc technique to address medium-frequency problems. It is a Trefftz method based on a weak variational formulation. It allows great flexibility because any shape function that satisfies the governing equations can be used. This work further develops such theory. In particular, orthotropic materials are introduced in the VTCR formulation for shallow shell structures. A significant numerical example is proposed to show the strategy.

Variational theory of complex rays applied to shell structures: in-plane inertia, quasi-symmetric ray distribution, and orthotropic materials

Recently, interest of aerospace and automotive industries on medium-frequency vibrational behavior of composite shell structures has grown due to their high specific stiffness and fatigue resistance. Conventional methods such as the finite element method and the statistical energy analysis are not suitable for the medium-frequency bandwidth. Conversely, the variational theory of complex rays (VTCR) is taking place as an ad-hoc technique to tackle such frequency band. It is a Trefftz method based on a weak variational formulation. Equilibrium equations are met using exact solutions as shape functions. The variational problem imposes boundary conditions in weak form. The present paper extends VTCR to orthotropic shell structures. Moreover, several new enhancements are introduced. Now, we use a quasi-symmetric ray distribution which can greatly reduce computational costs, and addresses in-plane inertia which was neglected in previous works. Some relevant numerical examples are presented to show the strategy and results are compared with a FEM reference to study performances.

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.

Implementation of an interior point source in the ultra weak variational formulation through source extraction

The Ultra Weak Variational Formulation (UWVF) is a powerful numerical method for the approximation of acoustic, elastic and electromagnetic waves in the time-harmonic regime. The use of Trefftz-type basis functions incorporates the known wave-like behaviour of the solution in the discrete space, allowing large reductions in the required number of degrees of freedom for a given accuracy, when compared to standard finite element methods. However, the UWVF is not well disposed to the accurate approximation of singular sources in the interior of the computational domain. We propose an adjustment to the UWVF for seismic imaging applications, which we call the Source Extraction UWVF. Differing fields are solved for in subdomains around the source, and matched on the inter-domain boundaries. Numerical results are presented for a domain of constant wavenumber and for a domain of varying sound speed in a model used for seismic imaging.

Dispersion analysis of plane wave discontinuous Galerkin methods

SUMMARYThe plane wave DG (PWDG) method for the Helmholtz equation was introduced and analyzed in [GITTELSON, C., HIPTMAIR, R., AND PERUGIA, I. Plane wave discontinuous Galerkin methods: analysis of the h‐version. Math. Model. Numer. Anal. 43 (2009), 297–331] as a generalization of the so‐called ultra‐weak variational formulation, see [O. CESSENAT AND B. DESPRéS, Application of an ultra weak variational formulation of elliptic PDEs to the two‐dimensional Helmholtz equation, SIAM J. Numer. Anal., 35 (1998), pp. 255–299]. The method relies on Trefftz‐type local trial spaces spanned by plane waves of different directions, and links cells of the mesh through numerical fluxes in the spirit of DG methods.We conduct a partly empirical dispersion analysis of the method in a discrete translation‐invariant setting by studying the mismatch of wave numbers of discrete and continuous plane waves traveling in the same direction. We find agreement of the wave numbers for directions represented in the local trial spaces. For other directions, the PWDG methods turn out to incur both phase and amplitude errors. This manifests itself as a pollution effect haunting the h‐version of the method. Our dispersion analysis allows a quantitative prediction of the strength of this effect and its dependence on the wave number and number of plane waves. Copyright © 2014 John Wiley & Sons, Ltd.

A Plane-Wave Least-Squares Method for Time-Harmonic Maxwell's Equations in Absorbing Media

In this paper we discuss numerical methods for solving time-harmonic Maxwell equations in three-dimensional absorbing media. We propose a new variational problem on the space spanned by the plane-wave basis functions for the discretization of these types of equations. The variational problem is derived mathematically from a minimization problem, and the resulting stiffness matrix is Hermitian positive definite. A simple domain decomposition preconditioner is introduced for the system generated by the proposed variational formula. Numerical results indicate that the approximate solution generated by the proposed discretization method is clearly more accurate than that generated by the ultra weak variational formulation method, and the proposed preconditioner is effective.

The ultra weak variational formulation of thin clamped plate problems

We develop a new numerical scheme for a fourth order elliptic partial differential equation based on Kirchhoffʼs thin plate theory. In particular we extend the ultra weak variational formulation (UWVF) to thin plate problems with clamped plate boundary conditions. The UWVF uses a finite element mesh and non-polynomial basis functions. After deriving the new method we then prove L2 norm convergence on the boundary. Finally we investigate numerically the feasibility of the UWVF for both homogeneous and inhomogeneous problems and show examples of p- and h-convergence.