Spurious Kinematic Modes Research Articles

A direct numerical method to evaluate the geometric stability of arbitrary spatial structures

Each structural designer has to ensure that the load-bearing structure to be designed is geometrically stable; i.e. the structure does not exhibit any spurious kinematic modes. Existing methods evaluating this requirement are either indirect and thus computationally inefficient, or restricted to 2- or 3-dimensional pin-jointed structures or to 2-dimensional general structures. In order to close this gap, a new method is proposed that is able to handle arbitrary spatial structures, consisting of arbitrarily shaped 1-, 2- and 3-dimensional bodies (e.g. truss, beam, plate, shell, solid), which are arbitrarily connected to each other in reference to their types (e.g. rigid, pin-jointed, torsional joint, shear force and bending connection) and their acting spots (at points, along a line or over a plane). The same holds for the support. The proposed method is based on a kinematic approach, where only translational degrees of freedom are considered. It has been proved that 8 different types of kinematic constraints are required to evaluate the geometric stability of a general spatial structure; all these constraints are stated explicitly. With the resulting constraint matrix, the geometric stability can be evaluated, and all existing spurious kinematic modes can be calculated. The new method can simply be added as an extra module to the existing structural analysis software.

On the detection, visualization and characterization of spurious kinematic modes in arbitrary plane structures

A direct and computationally efficient numerical method for the detection, visualization and characterization of spurious kinematic modes in arbitrary plane structures is presented. The proposed method is based on a pure kinematic approach. Compared to alternative approaches, like the displacement method, where kinematic modes can be indirectly detected by solving the eigenvalue problem for the related stiffness matrix, the proposed method is direct and computationally less expensive. In contrast to existing kinematic approaches, which are restricted to pin-jointed trusses, the members of the structures here can be both arbitrarily shaped (e.g. line, triangle, rectangle etc.) and arbitrarily connected with each other (e.g. pin-jointed, rigid, shear force and bending connections etc.). Also arbitrary supports can be considered. The method is able to detect and visualize all existing spurious kinematic modes of a structure. Additionally, analytical formulas have been derived which allow an explicit calculation of the position of both all instant centers of rotation and all relative centers of rotation of a structure exhibiting one or more kinematic modes. In order to keep the total number of degrees of freedom as small as possible, only translational and no rotational dofs have been introduced. The proposed method can simply be added as an extra module to existing structural analysis software.

The stability of stars of simplicial hybrid equilibrium finite elements for solid mechanics

Summary In this paper, we define the spurious kinematic modes of hybrid equilibrium 2D and 3D simplicial elements of general degree and present the results of studies on the stability of star patches of such elements. The approach used in these studies is based on first establishing the kinematic properties of a pair of elements that share an interface. This approach is introduced by its application to star patches of hybrid equilibrium triangular plate elements for modelling membrane and plate bending problems and then generalised to 3D continua. It is then shown how the existence of spurious kinematic modes depends on the topological and geometrical properties of a patch, as well as on the degree of the polynomial approximation functions of stress and displacement. Copyright © 2015 John Wiley & Sons, Ltd.

Improved Bilinear Degenerated Shell Element

The development of a new four node 24 degree of freedom bilinear degenerated shell element is presented for the analysis of shell structures. The present finite element formulation considers the assumed covariant transverse shear strains to avoid the shear locking problem and the assumed covariant membrane strains, which are separated from covariant in-plane strains, to overcome the membrane locking problem. The formulation also includes the deviation of the normal torsional rotation of the mid surface in the governing equation. This element is free from serious shear and membrane locking problems and undesirable spurious kinematic deformation modes. The element is tested for rigid body modes and distorted edges to meet the patch test requirements. The versatility and accuracy of this new degenerated shell element is demonstrated by solving several numerical examples for thick and thin plates.

A traction‐based equilibrium finite element free from spurious kinematic modes for linear elasticity problems

SUMMARYThis paper presents equilibrium elements for dual analysis. A traction‐based equilibrium element is proposed in which tractions of an element instead of stresses are chosen as DOFs, and therefore, the interelement continuity and the Neumann boundary balance are directly satisfied. To be solvable, equilibrated tractions with respect to the space of rigid body motion are required for each element. As a result, spurious kinematic modes that may inflict troubles on stress‐based equilibrium elements do not appear in the element because only equilibrium constraints on tractions are required. An admissible stress field is eventually constructed in terms of the equilibrated tractions for the element, and hence, equilibrium finite element procedures can proceed. The element is also generalized to accommodate non‐zero body forces, nonlinear boundary tractions and curved Neumann boundaries. Numerical tests including a single equilibrium element, error estimation of a cantilever beam and an infinite plate with a circular hole are conducted, displaying excellent convergence and effectiveness of the element for error estimation. Copyright © 2014 John Wiley & Sons, Ltd.

Restraining approach for the spurious kinematic modes in hybrid equilibrium element

The present paper proposes a rigorous approach for the elimination of spurious kinematic modes in hybrid equilibrium elements, for three well known mesh patches. The approach is based on the identification of the dependent equations in the set of inter-element and boundary equilibrium equations of the sides involved in the spurious kinematic mode. Then the kinematic variables related to the dependent equations are reciprocally constrained and, by application of master slave elimination method, the set of inter-element equilibrium equations is reduced to full rank. The elastic solutions produced by means of the proposed approach verify the homogeneous, the inter-element and the boundary equilibrium equations. Hybrid stress formulation is developed in a rigorous mathematical setting. The results of linear elastic analysis obtained by the proposed approach and by classical displacement based method are compared for some structural examples.

A general degree hybrid equilibrium finite element for Kirchhoff plates

SUMMARYWe present a family of hybrid equilibrium finite elements for the quasistatic linear elastic analysis of plates governed by Kirchhoff theory. The moments are approximated by self‐balanced polynomial fields of general degree, and in order to impose strong codiffusivity, the normal boundary rotations are approximated with complete polynomials of the same degree, whereas the transverse deflections use polynomials one degree lower. Furthermore, it is also necessary to include an independent approximation of the vertex translations. We show that the triangular form of this element is stable, that is, free from spurious kinematic modes, and the formulation that we present allows these elements to be used as a standard displacement element. Examples of computed values and convergence of the solutions are presented, which demonstrate the performance of these elements. Copyright © 2013 John Wiley & Sons, Ltd.

Pure equilibrium tetrahedral finite elements for global error estimation by dual analysis

AbstractThis study presents a general procedure of creating pure equilibrium tetrahedral finite elements for use under the elastostatic hypothesis. These pure equilibrium elements are of the Fraeijs de Veubeke type and the degree of the polynomial approximation functions of their internal stress field is the parameter generating this new elements family. The spurious kinematic modes (SKM), inherent in the equilibrium approach, are eliminated at the element level by converting each tetrahedron into a super‐element defined as an assembly of four tetrahedral primitive elements. A mathematical discussion on the number of SKM of the primitive elements as well as their elimination by the super‐element technique has been carried out. The development of first and second degree elements is presented here in detail and their efficiency in the frame of global error estimation by dual analysis is emphasized by two numerical applications. The main attribute of the error estimation by dual analysis is that it provides an upper bound on the global discretization error. Copyright © 2009 John Wiley & Sons, Ltd.

The stability of stars of triangular equilibrium plate elements

AbstractEquilibrium models for finite element analyses are becoming increasingly important in complementary roles to those from conventional conforming models, but when formulating equilibrium models questions of stability, or admissibility of loads, are of major concern. This paper addresses these questions in the context of flat plates modelled with triangular hybrid elements involving membrane and/or flexural actions. Patches of elements that share a common vertex are considered, and such patches are termed stars. Stars may be used in global analyses as assemblies of elements forming macro‐elements, or in local analyses. The conditions for stability, or the existence and number of spurious kinematic modes, are determined in a general algebraic procedure for any degree of the interpolation polynomials and for any geometric configuration. The procedure involves the determination of the rank of a compatibility matrix by its transformation to row echelon form. Examples are presented to illustrate some of the characteristics of spurious kinematic modes when they exist in stars with open or closed links. Copyright © 2008 John Wiley & Sons, Ltd.

Hybrid equilibrium hexahedral elements and super‐elements

AbstractEquilibrated solutions, locally satisfying all the equilibrium conditions, may be obtained by using a special case of the hybrid finite element formulation. Unlike simplicial super‐elements in 2D and 3D, which are free from spurious kinematic modes, and the quadrilateral super‐element with diagonal subdivision, which has exactly one internal spurious kinematic mode, no hexahedral super‐element free of external spurious kinematic modes is yet known as far as the author is aware. Copyright © 2006 John Wiley & Sons, Ltd.

Determination of coefficients of the crack tip asymptotic field by fractal hybrid finite elements

This paper applies the fractal finite element method (FFEM) together with 9-node Lagrangian hybrid elements to the calculation of linear elastic crack tip fields. An explicit stabilization scheme is employed to suppress the spurious kinematic modes of the sub-integrated Lagrangian element. An extensive convergence study has been conducted to examine the effects of the similarity ratio, reduced integration and the type of elements on the accuracy and stability of the numerical solutions. It is concluded that (i) a similarity ratio close to unity should be used to construct the fractal mesh, (ii) sub-integrated Lagrangian elements with occasionally unstable behaviour should not be used, and (iii) the good accuracy (with differences less than 0.5% with existing available solutions) and stability over a wide range of numerical tests support the use of fractal hybrid finite elements for determining crack tip asymptotic fields.

A triangular hybrid equilibrium plate element of general degree

Hybrid stress-based finite elements with side displacement fields have been used to generate equilibrium models having the property of equilibrium in a strong form. This paper establishes the static and kinematic characteristics of a flat triangular hybrid equilibrium element with both membrane and plate bending actions of general polynomial degree p. The principal characteristics concern the existence of hyperstatic stress fields and spurious kinematic modes. The former are shown to exist for p>3, and their significance to finite element analysis is reviewed. Knowledge of the latter is crucial to the determination of the stability of a mesh of triangular elements, and to the choice of procedure adopted for the solution of the system of equations. Both types of characteristic are dependent on p, and are established as regards their numbers and general algebraic forms. Graphical illustrations of these forms are included in the paper. Copyright © 2005 John Wiley & Sons, Ltd.

A partial assumed strain formulation for triangular solid shell element

An assumed strain formulation is based on the Hellinger–Reissner variational principle with two unknown fields; assumed displacements and independently assumed strains. It is effective to alleviate locking without triggering undesirable spurious kinematic modes if proper assumed strain field is carefully selected in the formulation. Since C 0 continuity does not require strains continuous across the element boundaries, the assumed strain field can be eliminated at an element level. However, elimination of the assumed strain field requires more operations including matrix inverse, where matrix size depends on the number of assumed strain parameters, at each element to obtain an element stiffness matrix. Therefore, small number of assumed strain parameters saves computation time. In this study a triangular solid element based on a partial assumed strain formulation is presented. The assumed strain field is divided into in-plane part and transverse part in the formulation. The transverse part of the assumed strain field is independently assumed to alleviate transverse shear locking. However, the in-plane part of the assumed strain field is replaced by displacement-dependent strain, which reduces the number of assumed strain parameters. Since the number of assumed strain parameters of the formulation is smaller than the conventional assumed strain formulation, the present formulation saves computation time for constructing stiffness matrix without sacrificing accuracy.

A hybrid stress-assumed transition element for solid-to-beam and plate-to-beam connections

This paper presents a new transition element for modelling solid-to-beam and plate-to-beam connections. The approach is based upon the hybrid stress method and the resulting transition element is hypostatic, i.e. the number of independent stress parameters is less than the number of independent displacements. As a consequence, there are further displacement modes called spurious kinematic modes which are eliminated by means of a suitable “penalty” method. The numerical performance of the proposed element is assessed by a number of relevant tests.

An assumed strain formulation of efficient solid triangular element for general shell analysis

An efficient assumed strain triangular solid element is developed for the analysis of plate and shell structures. The finite element formulation is based on the two-field assumed strain formulation with two independent fields of assumed displacement and assumed strain. The assumed strain field is carefully selected to alleviate the shear locking effect without triggering undesirable spurious kinematic modes. The curvilinear surface of shell structures is modelled with flat facet elements to obviate the membrane locking effect. The patch tests are successfully passed, and numerical test involving various example problems demonstrates the validity and efficiency of the present element. Copyright © 2000 John Wiley & Sons, Ltd.

Geometrically nonlinear analysis of laminated composites by an improved degenerated shell element

The objective of this paper is to extend the use of the improved degenerated shell element to the linear and the large displacement analysis of plates and shells with laminated composites. In the formulation of the element stiffness, the combined use of three different techniques was made. This element is free of serious shear/membrane locking problems and undesirable compatible/commutable spurious kinematic deformation modes. The total Lagrangian approach has been utilized for the definition of the deformation and the solution to the nonlinear equilibrium equations is obtained by the Newton-Raphson method. The applicability and accuracy of this improved degenerated shell element in the analysis of laminated composite plates and shells are demonstrated by solving several numerical examples.

Curious convergence with hypostatic hybrid equilibrium models

The paper describes an unexpected type of convergence behaviour occurring for a single, variable degree, primitive-type equilibrium element. For this element the number of independent stress fields is less than the number of independent boundary displacement variables that do not correspond to rigid element modes of displacement. This leads to the conclusion that the element is hypostatic and that, in the absence of self-stressing modes, no convergence can occur. Such ‘conventional’ counting procedures do not, however, reveal the whole picture, and numerical determination of the rank of the coefficient matrix of the equilibrium equations shows that, in practice, self-stressing modes can and do exist in a model which would conventionally be described as hypostatic. The rank deficiency in the coefficient matrix is shown to be due to the fact that, upon transformation, independent stress fields do not necessarily lead to independent boundary tractions. Generalization to conventionally iso- and hyperstatic models demonstrates that, whenever the coefficient matrix is rank-deficient, spurious kinematic modes coexist with self-stressing modes. The problem which reveals the curious convergence characteristics for the primitive-type element is resolved using a macro-type element, and it is seen that, with the larger degree of hyperstaticity available to this element, strictly monotonic convergence characteristics are observed. © 1997 John Wiley & Sons, Ltd.

A GENERAL FORMULATION OF EQUILIBRIUM MACRO-ELEMENTS WITH CONTROL OF SPURIOUS KINEMATIC MODES: THE EXORCISM OF AN OLD CURSE

This paper illustrates a method whereby a family of robust equilibrium elements can be formulated in a general manner. The effects of spurious kinematic modes, present to some extent in all primitive equilibrium elements, are eliminated by judicious assembly into macro-equilibrium elements. These macroelements are formulated with sufficient generality so as to retain the polynomial degree of the stress field as a variable. Such a family of macro-elements is a new development, and results for polynomials of degree greater than two have not been seen before. The quality of results for macro-equilibrium elements with varying degrees of polynomial is demonstrated by numerical examples.

An effective four node degenerated shell element for geometrically nonlinear analysis

The development of a new four node degenerated shell element is presented for the analysis of the shell structures undergoing large deformations. In the formulation of the new element, the assumed covariant transverse shear strains are used to avoid the shear locking problem, and the assumed covariant membrane strains which are separated from the covariant inplane strains by mid-surface interpolation, are applied to eliminate the membrane locking problem and also to improve the membrane bending performance. This element is free of serious shear and membrane locking problems and undesired spurious kinematic deformation modes. An incremental total Lagrangian formulation is presented which allows the calculation of arbitrarily large displacements and rotations. The resulting nonlinear equilibrium equations are solved by the Newton-Raphson method combined with load or arc-length control. The versatility and accuracy of this new degenerated shell element is demonstrated by solving several numerical examples.

An efficient assumed strain element model with six DOF per node for geometrically non‐linear shells

AbstractThe present paper describes an assumed strain finite element model with six degrees of freedom per node designed for geometrically non‐linear shell analysis. An important feature of the present paper is the discussion on the spurious kinematic modes and the assumed strain field in the geometrically non‐linear setting. The kinematics of deformation is described by using vector components in contrast to the conventional formulation which requires the use of trigonometric functions of rotational angles. Accordingly, converged solutions can be obtained for load or displacement increments that are much larger than possible with the conventional formulation with rotational angles. In addition, a detailed study of the spurious kinematic modes and the choice of assumed strain field reveals that the same assumed strain field can be used for both geometrically linear and non‐linear cases to alleviate element locking while maintaining kinematic stability. It is strongly recommended that the element models, described in the present paper, be used instead of the conventional shell element models that employ rotational angles.