Calculation of stresses in the finite elements of deformed constructions

Under study is the problem of deformation of a curved rod in the form of a circular arc. Using the previously developed version of the functions of the form of a curvilinear finite element, we construct a solution that differs slightly from the exact one with respect to displacements even for few elements; however, the bending moment is calculated with a greater error. As a result of the direct integration of the equations of the problem for this rod, there are constructed some modified functions of the form from which an “exact” stiffness matrix is calculated. These functions yield the construction of the functions of the form with a parameter and the reason is clarified why the calculation of the force factors by differentiation of such functions fails to be exact. Also, we demonstrate a possible nonuniqueness of the obtained results for the stresses under the same errors in the stiffness matrix.     

Nonconforming finite elements and the cascadic multi-grid method

Summary. We derive sufficient conditions under which the cascadic multi-grid method applied to nonconforming finite element discretizations yields an optimal solver. Key ingredients are optimal error estimates of such discretizations, which we therefore study in detail. We derive a new, efficient modified Morley finite element method. Optimal cascadic multi-grid methods are obtained for problems of second, and using a new smoother, of fourth order as well as for the Stokes problem. Received February 12, 1998 / Revised version received January 9, 2001 / Published online September 19, 2001     

Optimally accurate Petrov-Galerkin method of finite elements

We perform analysis for a finite elements method applied to the singular self-adjoint problem. This method uses continuous piecewise polynomial spaces for the trial and the test spaces. We fit the trial polynomial space by piecewise exponentials and we apply so exponentially fitted Galerkin method to singular self-adjoint problem by approximating driving terms by Lagrange piecewise polynomials, linear, quadratic and cubic. We measure the erroe in max norm. We show that method is optimal of the first order in the error estimate. We also give numerical results for the Galerkin approximation.     

On the super-approximation property of Galerkin's method with finite elements

Summary For Galerkin's method with finite elements as trial functions for strongly elliptic operator equations in the Hilbert scaleH ^t the super-approximation property and the optimal convergence rate are obtained by using the Aubin-Nitsche lemma. This applies in particular to spline collocation methods for a wide class of pseudodifferential equations.Dedicated to the memory of Professor Lothar Collatz     

Hybrid stress analysis of boundary and finite elements by a super-element method

This paper is concerned with the hybrid method of boundary element and finite element techniques by means of an “external-super-element” function of the commercial finite element method code . The proposed super-element method preserves the modelling simplicity of the boundary element method and the generality of . Two- and three-dimensional elastostatic analyses are performed to demonstrate the accuracy of this method as well as its applicability to practical problems.     

Full 3D finite element Cosserat formulation with application in layered structures

In this paper, a full three-dimensional (3D) finite element Cosserat formulation is developed within the principles of continuum mechanics in the small deformation framework. The developed finite element formulation is general; however, the proposed constitutive laws incorporate the effect of the internal length parameter of 3D layered continua. The extension of the existing two-dimensional (2D) Cosserat formulation to the 3D framework is novel and is consistent with plate theory which can be considered as the 3D version of beam theory. The results demonstrate a high level of consistency with the analytical solutions predicted by plate theory as well as predictions by alternative numerical techniques such as the discrete element method.     

Approximation by quadrilateral finite elements

We consider the approximation properties of finite element spaces on quadrilateral meshes. The finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element, which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element. It is known that for affine isomorphisms, a necessary and sufficient condition for approximation of order in and order in is that the given space of functions on the reference element contain all polynomial functions of total degree at most . In the case of bilinear isomorphisms, it is known that the same estimates hold if the function space contains all polynomial functions of separate degree . We show, by means of a counterexample, that this latter condition is also necessary. As applications, we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements.

     

The coupling method of finite elements and boundary elements for radiation problem

The paper presents the variational formulation and well posedness of the coupling method of finite elements and boundary elements for radiation problem. The convergence and optimal error estimate for the approximate solution and numerical experiment are provided.This research was supported in part by the Institute for Mathematics and its applications with funds provided by NSF, USA.     

On the symmetric boundary element method and the symmetric coupling of boundary elements and finite elements

The finite element method and the boundary element method areamong the most frequently applied tools in the numerical treatmentof partial differential equations. However, their propertiesappear to be complementary: while the boundary element methodis appropriate for the most important linear partial differentialequations with constant coefficients in bounded or unboundeddomains, the finite element method seems to be more appropriatefor inhomogeneous or even nonlinear problems. but is somehowrestricted to bounded domains. The symmetric coupling of thetwo methods inherits the advantages of both methods. This paper treats the symmetric coupling of finite elementsand boundary elements for a model transmission problem in twoand three dimensions where we have two domains: a bounded domainwith nonlinear, even plastic material behaviour, is surroundedby an unbounded, exterior, domain with isotropic homogeneouslinear elastic material. Practically. the coupling is performedsuch that the boundary element method contributes a macro-element,like a large finite element, within a standard finite elementanalysis program. Emphasis is on two-dimensional problems wherethe approach using the Poincaré-Steklov operator seemsto be impossible at first glance. E-mail: cc{at}numerik.uni-kiel.de     

A constructive method for deriving finite elements of nodal type

Summary In this paper, we propose an algorithm to derive nodal methods corresponding to various two and three-dimensional nonconforming and mixed finite elements. We show that this algorithm can be used to obtain several classical schemes as well as some more recently developed schemes, and that it leads to a simple proof of unisolvence for these methods. Finally we use our method to obtain a three dimensional nodal scheme of BDM type.     

Numerical solution of the 1 + 2 dimensional Fisher's equation by finite elements and the Galerkin method

In Fisher's equation, the mechanism of logistic growth and linear diffusion are combined in order to model the spreading and proliferation of population, e.g., in ecological contexts. A Galerkin Finite Element method in two space dimensions is presented, which discretises a 1 + 2 dimensional version of this partial differential equation, and thus, providing a system of ordinary differential equations (ODEs) whose numerical solutions approximate those of the Fisher equation. By using a particular type of form functions, the off-diagonal elements of the matrix on the left-hand side of the ODE system become negligibly small, which makes a multiplication with the inverse of this matrix unnecessary, and therefore, leads to a simpler and faster computer program with less memory and storage requirements. It can, therefore, be considered a borderline method between finite elements and finite differences. A simple growth model for coral reefs demonstrates the program's adaptability to practical applications.     

Listing structure of a representation of slae in the method of finite elements

The question about construction and application of a listing structure of storage of systems of linear algebraic equations (SLAE) is considered, which appear in the method of finite elements. A complex of programs of automatic construction of a listing structure of data is developed. A standard direct method is used for solving SLAE that needs the profile structure of storage of elements of the matrix. The transition from the listing structure to the profile one is realized sufficiently simply. The complex of programs developed is applied to solution of the Poisson equation by the method of finite elements.Translated from Dinamicheskie Sistemy, No. 8, pp. 95–99, 1989.     

A multi-mesh finite element method for Lagrange elements of arbitrary degree

We consider within a finite element approach the usage of different adaptively refined meshes for different variables in systems of nonlinear, time-depended PDEs. To resolve different solution behaviors of these variables, the meshes can be independently adapted. The resulting linear systems are usually much smaller, when compared to the usage of a single mesh, and the overall computational runtime can be more than halved in such cases. Our multi-mesh method works for Lagrange finite elements of arbitrary degree and is independent of the spatial dimension. The approach is well defined, and can be implemented in existing adaptive finite element codes with minimal effort. We show computational examples in 2D and 3D ranging from dendritic growth to solid–solid phase-transitions. A further application comes from fluid dynamics where we demonstrate the applicability of the approach for solving the incompressible Navier–Stokes equations with Lagrange finite elements of the same order for velocity and pressure. The approach thus provides an easy way to implement alternative to stabilized finite element schemes, if Lagrange finite elements of the same order are required.     

Discontinuous enrichment in finite elements with a partition of unity method

A technique is presented to model arbitrary discontinuities in the finite element framework by locally enriching a displacement-based approximation through a partition of unity method. This technique allows discontinuities to be represented independently of element boundaries. The method is applied to fracture mechanics, in which crack discontinuities are represented using both a jump function and the asymptotic near-tip fields. As specific examples, we consider cracks and crack growth in two-dimensional elasticity and Mindlin–Reissner plates. A domain form of the J-integral is also derived to extract the moment intensity factors. The accuracy and utility of the method is also discussed.