Jarkko Niiranen

A posteriori error estimates for the Morley plate bending element

A local a posteriori error indicator for the well known Morley element for the Kirchhoff plate bending problem is presented. The error indicator is proven to be both reliable and efficient. The technique applied is general and it is shown to have also other applications.

A REFINED ERROR ANALYSIS OF MITC PLATE ELEMENTS

We consider the Mixed Interpolated (Tensorial Components) finite element families for the Reissner–Mindlin plate model. For the case of a convex domain with clamped boundary conditions, we prove regularity results and derive new error estimates which are uniformly valid with respect to the thickness parameter.

A Family of

A new finite element formulation for the Kirchhoff plate model is presented. The method is a displacement formulation with the deflection and the rotation vector as unknowns, and it is based on ideas stemming from a stabilized method for the Reissner-Mindlin model [R. Stenberg, in Asymptotic Methods for Elastic Structures, P. Ciarlet, L. Trabucho, and J. M. Viano, eds., de Gruyter, Berlin, 1995] and a method to treat a free boundary [P. Destuynder and T. Nevers, RAIRO Model. Math. Anal. Numer., 22 (1988), pp. 217-242]. Optimal a priori and a posteriori error estimates are derived.

{C}^0

Abstract This article is devoted to isogeometric analysis of higher-order strain gradient elasticity by user element implementations within a commercial finite element software Abaqus. The sixth-order boundary value problems of four parameter second strain gradient-elastic bar and plane strain/stress models are formulated in a variational form within an H 3 Sobolev space setting. These formulations can be reduced to two parameter first strain gradient-elastic problems of H 2 variational forms. The implementations of the isogeometric C 2 - and C 1 -continuous Galerkin methods, for the second and first strain gradient elasticity, respectively, are verified by a series of benchmark problems. With the first benchmark problem, a clamped bar in static tension, the convergence properties of the method in the energy norm are shown to be optimal with respect to the NURBS order of the discretizations. For the second benchmark, a clamped bar in extensional free vibrations, the analytical frequencies are captured by the numerical results within the classical and the first strain gradient elasticity. With three examples for the plane stress/strain elasticity, the convergence properties are shown to be optimal, the stress fields of different models are compared to each other, and the differences between the eigenfrequencies and eigenmodes of the models are analyzed. The last example, the Kraus problem, analyses the stress concentration factors within the different models.

Finite Elements For Kirchhoff Plates I: Error Analysis

As a first step, variational formulations and governing equations with boundary conditions are derived for a pair of Euler–Bernoulli beam bending models following a simplified version of Mindlin’s strain gradient elasticity theory of form II. For both models, this leads to sixth-order boundary value problems with new types of boundary conditions that are given additional attributes singly and doubly, referring to a physically relevant distinguishing feature between free and prescribed curvature, respectively. Second, the variational formulations are analyzed with rigorous mathematical tools: the existence and uniqueness of weak solutions are established by proving continuity and ellipticity of the associated symmetric bilinear forms. This guarantees optimal convergence for conforming Galerkin discretization methods. Third, the variational analysis is extended to cover two other generalized beam models: another modification of the strain gradient elasticity theory and a modified version of the couple stress theory. A model comparison reveals essential differences and similarities in the physicality of these four closely related beam models: they demonstrate essentially two different kinds of parameter-dependent stiffening behavior, where one of these kinds (possessed by three models out of four) provides results in a very good agreement with the size effects of experimental tests. Finally, numerical results for isogeometric Galerkin discretizations with B-splines confirm the theoretical stability and convergence results. Influences of the gradient and thickness parameters connected to size effects, boundary layers and dispersion relations are studied thoroughly with a series of benchmark problems for statics and free vibrations. The size-dependency of the effective Young’s modulus is demonstrated for an auxetic cellular metamaterial ruled by bending-dominated deformation of cell struts.

Isogeometric analysis of higher-order gradient elasticity by user elements of a commercial finite element software

In the present contribution, isogeometric methods are used to analyze the statics of the plane strain and plane stress problems based on the theory of strain gradient elasticity. The adopted strain gradient elasticity models, in particular, include only one length scale parameter enriching the classical strain energy expression and resulting in fourth order partial differential equations instead of the corresponding second order ones based on the classical elasticity. The problems are discretized by an isogeometric NURBS based \( C^1\) continuous Galerkin method which is implemented as a user subroutine into a commercial software Abaqus. Computational results for benchmark problems, a square plate in tension and a Lame problem, demonstrate the applicability of the method and verify the implementation.

Variational formulations, model comparisons and numerical methods for Euler–Bernoulli micro- and nano-beam models

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Isogeometric Static Analysis of Gradient-Elastic Plane Strain/Stress Problems

In the present contribution, isogeometric methods are used to analyze the statics and dynamics of rods as well as plane strain and plane stress problems based on a simplified version of the form II of Mindlin’s strain gradient elasticity theory. The adopted strain gradient elasticity models, in particular, include only two length scale parameters enriching the classical energy expressions and resulting in fourth order partial differential equations instead of the corresponding second order ones based on the classical elasticity. The problems are discretized by an isogeometric non-uniform rational B-splines (NURBS) based \( C^{p-1} \) continuous Galerkin method. Computational results for benchmark problems demonstrate the applicability of the method and verify the implementation.

Modeling chemical reaction front propagation by using an isogeometric analysis

We present a posteriori error analysis for two finite element methods for the Kirchhoff plate bending model. The first method is a recently introduced C0-continuous family, while the second one is the classical nonconforming Morley element.

Isogeometric analysis of gradient-elastic 1D and 2D problems

The approximation of the deflection for the MITC plate elements [1, 2] is shown to be superconvergent with respect to a special interpolation operator [3]. This property holds in the H1-norm and the interpolation operator is closely related to the reduction operator used in the MITC methods. A part of the superconvergence result is, roughly speaking, that the vertex values obtained with the MITC methods are superconvergent. This may be an explanation why these methods have become so popular.