Convergence Of Finite Element Solutions Research Articles

Convergence of finite element solution of stochastic Burgers equation

<abstract><p>We explore the numerical approximation of the stochastic Burgers equation driven by fractional Brownian motion with Hurst index $ H\in(1/4, 1/2) $ and $ H\in(1/2, 1) $, respectively. The spatial and temporal regularity properties for the solution are obtained. The given problem is discretized in time with the implicit Euler scheme and in space with the standard finite element method. We obtain the strong convergence of semidiscrete and fully discrete schemes, performing the error estimates on a subset $ \Omega_{k, h} $ of the sample space $ \Omega $ with the Gronwall argument being used to overcome the difficulties, caused by the subtle interplay of the nonlinear convection term. Numerical examples confirm our theoretical findings.</p></abstract>

FE SOLUTIONS OF NEAR-TIP FIELD FOR MODE-I CRACKS WITH RESIDUAL SURFACE TENSION

This paper presents finite element solutions of a near-tip elastic field of a straight, nano-scale crack in a two-dimensional, linear elastic, whole plane subjected to mode-I crack-face loads. The mathematical model is formulated using a continuum-based theory of classical linear elasticity together with Gurtin-Murdoch surface model to capture the role of the residual surface tension present on the crack-face material layer. The formulation finally yields a second-order, integrodifferential equation governing the crack opening displacement. A weighted residual technique along with the regularization procedure is applied to establish a weakly singular weak-form equation with the involved kernel of O(ln )r . Galerkin strategy and the finite element procedure are then employed to discretize the weak-form equation. Various types of element shape functions, generated by standard C0 -elements, standard C1 -elements, and special elements with built-in crack-tip functions, are considered. A proper quadrature rule is selected to efficiently and accurately evaluate both nearly and weakly singular double line integrals over pairs of elements resulting from the discretization and the solution of a dense system of linear algebraic equations is obtained using an efficient indirect solver. The rate of convergence of finite element solutions is fully investigated and such information is then used to conclude the influence of the residual surface tension on the behavior of the near-tip field.

Convergence of finite element solutions of stochastic partial integro-differential equations driven by white noise

Numerical approximation of a stochastic partial integro-differential equation driven by a space-time white noise is studied by truncating a series representation of the noise, with finite element method for spatial discretization and convolution quadrature for time discretization. Sharp-order convergence of the numerical solutions is proved up to a logarithmic factor. Numerical examples are provided to support the theoretical analysis.

An accurate numerical solution for the singular velocity field near the maximum friction surface in plane strain extrusion

In the case of rigid perfectly plastic material, the velocity field is singular in the vicinity of maximum friction surfaces (the equivalent strain rate approaches infinity in the vicinity of such surfaces). This causes significant difficulties with the convergence of finite element solutions. On the other hand, an accurate description of the velocity field near frictional interfaces is crucial for predicting the generation of a narrow fine grain layer that frequently appears near such interfaces. The present paper provides an accurate numerical solution for the singular velocity field in the vicinity of the maximum friction surface in plane strain extrusion. The singularity in the velocity field is represented by means of the strain rate intensity factor. The numerical approach is based on the method of characteristics. The strain rate intensity factor is determined by means of simple formulae once the solution for the radii of curvature of characteristic lines and velocities has been found. It is shown that the distribution of the strain rate intensity factor along the friction surface is discontinuous. Comparison with an analytic solution for material flow through an infinite wedge-shaped die is made and a high accuracy of the numerical solution is demonstrated. On the other hand, it is shown that the analytic solution is not accurate enough to find the strain rate intensity factor even for rather long dies.

Finite element methods for coefficient identification in an elliptic equation

We study the problem of identifying the spatially varying diffusion coefficient in the boundary value problems for the elliptic equation in , on and on , , , when the solution is imprecisely given by with and . The finite element method is applied to a convex energy functional with Tikhonov regularization for solving this coefficient identification problem. We show the -convergence of finite element solutions to the unique minimum norm solution of the identification problem. Furthermore, convergence rates of the method are established under certain source conditions.

Determination of local mesh parameters in finite element problems

Two basic cases of convergence of finite element solution in the static problems of the theory of elasticity are considered. Based on Saint-Venant’s principle, the sufficiency of finite element mesh refinement for local regions with stress concentration is determined. An algorithm is presented for evaluation of point-by-point convergence of solution for local mesh refinement. To illustrate the method, the results of computation of a cylindrical three-layered composite shell under local loads are given. For stresses reaching the maximum value in the loaded area, the mesh density parameters yielding an initial approximation are found.

Numerical convergence of finite element solutions of nonrational B-spline element and absolute nodal coordinate formulation

In this investigation, numerical convergence of finite element solutions obtained using the B-spline approach and the absolute nodal coordinate formulation (ANCF) is discussed. Furthermore, equivalence of the two formulations with different orders of polynomials and degrees of continuity is demonstrated by several numerical examples. The degree of continuity can be easily controlled in B-spline elements by changing knot multiplicities, while continuity conditions associated with higher order derivatives need to be imposed to achieve C2 and higher continuities in ANCF elements. In order to compare element performances of the third and quartic B-spline and ANCF elements, the three-node quartic ANCF beam element is developed. It is demonstrated in several numerical examples that use of B-spline and ANCF elements with same orders and continuities leads to identical results. Furthermore, effects of polynomial orders and continuities on the accuracy and numerical convergence are demonstrated.

Finite Element Approximations of the Ericksen–Leslie Model for Nematic Liquid Crystal Flow

The Ericksen-Leslie model describes dynamics of low molar-mass nematic liquid crystals, where the spatiotemporal distribution of defects defining texture is represented by the director unit vector field ${\bf d}: \Omega_T \rightarrow \mathbb{S}^2$. It consists of the Navier-Stokes equations with an extra viscous stress tensor, and a convective harmonic map heat flow equation to govern the dynamics of the director field. Two fully discrete finite element methods, for a regularized system using the Ginzburg-Landau regularization and for the limiting Ericksen-Leslie model, are proposed, and well-posedness and related discrete energy laws are established. For the regularized model, unconditional convergence of finite element solutions towards weak solutions of the continuum model as well as convergence towards measure-valued solutions of the limiting Ericksen-Leslie model are verified when the mesh parameters and the regularization parameter successively tend to zero. Computational experiments are also presented to show the importance of balancing numerical and regularization parameters, to compare regularized and direct approaches, and to show numerically the finite-time formation, annihilation, and evolution of point defects.

On the convergence of finite element solutions to the interface problem for the Stokes system

The Stokes system with a discontinuous coefficient (Stokes interface problem) and its finite element approximations are considered. We firstly show a general error estimate. To derive explicit convergence rates, we introduce some appropriate assumptions on the regularity of exact solutions and on a geometric condition for the triangulation. We mainly deal with the MINI element approximation and then consider P1-iso-P2/P1 element approximation. Results are expected to give an instructive remark in numerical analysis for two-phase flow problems.

Natural Vibration of Pack-Assembled Liquid-Propellant Carrier Rockets

A mathematical model has been constructed for numerical calculation of the natural vibration spectrum of pack-assembled liquid-propellant carrier rockets. The dynamic model of the mechanical-hydraulic system is represented by elements with reduced piecewise-uniform (beam-type) distributed and localized parameters of stiffness and inertial characteristics, while the vibrating masses of liquid are described by using mechanical oscillators. The paper gives the results of calculations of natural frequencies and modes of vibration for a design variant of the carrier rocket for the initial flight stage. We study the influence of taking into account the rotary inertia and shear strain of the beam-type elements on the calculated values of natural frequencies. We analyze also the convergence of finite-element solutions.

Adaptive finite element analysis of fractal interfaces in contact problems

Structures involving interfaces with fractal geometry are referred here as a sequence of classical interfaces problems, which result from the consideration of the fractal interfaces as the unique “fixed point” or the “deterministic attractor” of a given Iterated Function System (IFS). On the interface, unilateral contact conditions are assumed to hold. The approximations of the fractal interfaces are combined with a penalty regularization based on the minimization of the potential energy, after some appropriate transformations are performed. For this type of contact problems there often result singular points on the interfaces which lead to possible stress concentrations. Further-more, the convergence of finite element solution under a sufficient discretization can not be determined from the outset. An adaptive finite element strategy appears to be suitable for such kind of contact problems in that it possesses the properties of adjusting automatically the mesh sizes both in the interior of the bodies and on the contact zone. In this spirit, both the goals of exactly determining the real contact areas, and of enhancing the accuracy of finite element solution (meanwhile consuming reasonable computational costs) may be achieved. The error estimator based on the residual stress analysis is discussed. Numerical examples illustrate the validity and effectiveness of the method proposed in this paper.

Slope constrained topology optimization

The problem of minimum compliance topology optimization of an elastic continuum is considered. A general continuous density–energy relation is assumed, including variable thickness sheet models and artificial power laws. To ensure existence of solutions, the design set is restricted by enforcing pointwise bounds on the density slopes. A finite element discretization procedure is described, and a proof of convergence of finite element solutions to exact solutions is given, as well as numerical examples obtained by a continuation/SLP (sequential linear programming) method. The convergence proof implies that checkerboard patterns and other numerical anomalies will not be present, or at least, that they can be made arbitrarily weak. © 1998 John Wiley & Sons, Ltd.

On an alleged philosophers' stone concerning variational principles with subsidiary conditions

AbstractIn general, the Lagrange multiplier method (LMM) is used to incorporate subsidiary conditions into variational principles. The Lagrange multipliers represent an additional field of independent variables. The attempt to satisfy subsidiary conditions without employing additional independent unknowns has led to the development of simplified variational principles (SVP). They are characterized by expressing Lagrange multipliers in terms of original field variables by means of the Euler‐Lagrange equations for the multipliers, providing their physical interpretation.In the first part of the theoretical investigation, systems with infinitely many degrees‐of‐freedom are studied. It is shown that the Euler‐Lagrange equations of a LMM based on a modification of the principle of minimum of potential energy (PMIPE) do not hold unconditionally, that is, for arbitrary subsidiary conditions, for the corresponding SVP. The second part of the theoretical investigation is concerned with systems with a finite number of degrees‐of‐freedom. The finite element method (FEM) is employed to discuss and compare the characteristics of the LMM and of the corresponding SVP. Contrary to the former, the latter are found to be problem‐dependent. Several shortcomings of the SVP are listed, including the possibility of obtaining an infinite sequence of singular coefficient matrices in the process of a systematic mesh refinement. Consequently, convergence of finite element solutions to the true solution in the limit of finite element representations is not guaranteed. The theoretical findings are corroborated by the results of a detailed numerical study.

Convergence of finite element solutions in viscous flow problems

Convergence of finite element solutions in viscous flow problems

Finite Element Analysis of Some Critical Fast Assemblies

The finite element method is applied in multigroup formalism to the analysis of some fast and intermediate spherical critical systems. The approximation scheme is based on a self-adjoint variational principle associated with the formally self-adjoint form of the monoenergetic transport equation. A number of experimental critical assemblies are analyzed using unmodified and modified Hansen--Roach 16-group cross sections. Comparison of the results with those obtained by S/sub N/l calculations indicates that high accuracy is obtained by low-order finite element techniques. Several optional strategies are proposed which may further accelerate the convergence of finite element solutions. (auth)

A note on convergence of finite element solutions

AbstractThe convergence of nodal displacements obtained by the finite element method is investigated for special element patterns in plane elasticity with the aid of the differential equation derived from the equilibrium condition at each node. The convergence of strains is also discussed. It is shown that averaging is an effective tool for obtaining approximations to strain for some kinds of mesh arrangement. It is also shown that the approach becomes inaccurate along force boundaries.

Comments on the convergence of finite element solutions

(Uc ~ UJ/(Ui tfco) oc i/x This result indicates that there is a possibility to define the proportionality constant such that the flow will be selfpreserving. Applying the transformations reported for the stagnant external conditions did not yield the necessary transformation. It is necessary, also, to postulate that further transformation is possible, and that the proportionality constant is a function of the external fluid velocity such that /3 = 1 when Um ' = 0. By analyzing the data reported in Ref. 9 we obtain 0 = 1 + Q.2(Um/Ui). Combining this result with Boynton's correlation we obtain an empirical result with good approximation.

Kinds of convergence and improved convergence of conforming finite element solutions in plate bending

By use of a variational principle, a mathematical means of deriving the convergence criteria for the trial displacement functions employed in the finite element method for plate bending is presented. The nature of the convergence of the finite element solution for a displacement field satisfying required convergence criteria is discussed. On the basis of uniform convergence of the approximate displacement to its exact value, the authors propose a practical method of improving the convergence of approximate finite element solutions and a scheme of selecting the finite element size such that the resulting error stays within prescribed accuracy.