Shear locking reduction in eight-noded tri-linear solid finite elements

A new quadrature rule for eight-noded solid finite elements is suggested. The formulation reduces those locking problems associated with the classical full and selectively-reduced Gauss integration schemes and it is based on an element local coordinate system where the shear strain components are averaged in groups of four integration points.This work is mainly interesting in contexts of plasticity and explicit time integration, where low order elements often are preferable.     

A note on integration over finite elements

In the application of the finite element method for structural engineering problems, the evaluation of element stiffness matrices involves the computation of an integral over each element. This note shows that these integrals can be evaluated by a mapping, from the integral over a single element, with the same material properties, in all cases where the sides of the elements are linear in the coordinate system chosen. This brings about a remarkable saving in computation time where numerical integration is used.     

Residual energy balancing technique in the generation of plate bending finite elements

The residual shear balancing technique of Refs [1 and 2] is discussed and applied to the generation of the stiffness matrix of a 12 degree-of-freedom rectangular plate bending element. Numerical results both in statics and dynamics are compared with those obtained with elements of similar size.     

Iterative solution of systems of linear equations arising in the context of stochastic finite elements

The particular properties of systems of linear equations arising in the context of the Stochastic Finite Element Method (SFEM) motivate the customization of existing iterative solution algorithms. The implementation described in this paper has aimed at optimizing data management, MAT-VEC operations and preconditioning strategies. It turns out that SFEM-systems can be solved with much less effort than their size suggests. The main idea is based on the fact that the full system matrix consists of few, relatively small submatrices with identical dimensions and sparsity pattern. This makes it very efficient to perform matrix–vector multiplications at the submatrix level and to avoid the assembly of the full coefficient matrix.     

Field redistribution in finite elements—a mathematical alternative to reduced integration

A general method for “redistributing” high order finite element fields into lower order fields is presented. Occasionally consistent finite element procedures lead to abnormal solutions. The heuristic method of reduced integration has been proposed as a cure and good results have actually been achieved. However, the method of field redistribution presented here is consistent with a variational principle and may for this reason be more appealing. The method is applied to a simple shear flexible beam element and it is shown that in this special case the resultant element stiffness matrix is identical to the one obtained by employing reduced integration.     

Time domain electromagnetic scattering using finite elements and perfectly matched layers

We consider a model for the interrogation of a planar Debye medium by a non-harmonic microwave pulse from an antenna source in free space, and we compute the reflected solution using finite elements in the spatial variables and finite differences in the time variable. Perfectly matched layers (PMLs) and an absorbing boundary condition are used to damp waves interacting with artificial boundaries imposed to allow finite computational domains. We present simulation results showing that numerical reflections from interfaces at PML boundaries can be controlled.     

Time integration errors and some new functionals for the dynamics of a free mass

We study the numerical integration of the Poisson second-order ordinary differential equation which describes, for instance, the dynamics of a free mass. Classical integration algorithms, when applied to such an equation, furnish solutions affected by a significant “drift” error, apparently not studied so far. In the first part of this work we define measures of such a drift. We then proceed to illustrate how to construct both classical and extended functionals for the equation of motion of a free mass with given initial conditions. These tools allow both the derivation of new variationally-based time integration algorithms for this problem, and, in some cases, the theoretical isolation of the source of the drift. While we prove that this particular error is unavoidable in any algorithmic solution of this problem, we also provide some new time integration algorithms, extensions at little added cost of classical methods, which permit to substantially improve numerical predictions.     

Supervisory control synthesis for deterministic context free specification languages

This paper describes two steps in the generalization of supervisory control theory to situations where the specification is modeled by a deterministic context free language (DCFL). First, it summarizes a conceptual iterative algorithm from Schneider et al. (2014) solving the supervisory control problem for language models. This algorithm involves two basic iterative functions. Second, the main part of this paper presents an implementable algorithm realizing one of these functions, namely the calculation of the largest controllable marked sublanguage of a given DCFL. This algorithm least restrictively removes controllability problems in a deterministic pushdown automaton realizing this DCFL.     

Lower bound of tracking performance with finite control energy and channel energy constraint

This paper studies the tracking performance of the single-input single-output (SISO), finite dimensional, linear and time-invariant (LTI) system over an additive white Gaussian noise (AWGN) channel with finite control energy and channel input energy constraint. A new performance index is proposed which is minimized over all stabilizing two-degree-of-freedom controllers. The explicit expressions of the lower bound of the tracking performance and the minimum of signal-to-noise required are obtained. The results show that the lower bound is correlated to the unstable pole, nonminimum phase zero and the channel scaling factor. Finally, one example is given to validate the conclusions by adopting the special inner-outer factorization.     

Sensitivity of structural response in context of linear and non-linear buckling analysis with solid shell finite elements

The paper is concerned with the sensitivity analysis of structural responses in context of linear and non-linear stability phenomena like buckling and snapping. The structural analysis covering these stability phenomena is summarised. Design sensitivity information for a solid shell finite element is derived. The mixed formulation is based on the Hu-Washizu variational functional. Geometrical non-linearities are taken into account with linear elastic material behaviour. Sensitivities are derived analytically for responses of linear and non-linear buckling analysis with discrete finite element matrices. Numerical examples demonstrate the shape optimisation maximising the smallest eigenvalue of the linear buckling analysis and the directly computed critical load scales at bifurcation and limit points of non-linear buckling analysis, respectively. Analytically derived gradients are verified using the finite difference approach.     

Membrane locking and assumed strain shell elements

An explanation of membrane locking behaviour in shell elements and also the use of reduced and selective integration is described. To overcome the conflict between the locking and mechanism problems the author proposed the degenerated shell elements with assumed transverse shear and membrane strains. The location of sampling points for the assumed strain fields is given in the present work. In the formulation of the new elements, assumed transverse shear strains in the natural coordinate system are used to overcome the shear locking problem. Also, assumed membrane strains in the orthogonal curvilinear coordinate system are applied to avoid membrane locking behaviour. Some numerical tests are presented to illustrate the good performance of the assumed strain shell elements.     

Shear and membrane locking in curved C0 elements

Locking phenomena in C⁰ curved finite elements are studied for displacement, hybrid-stress and mixed formulations. It is shown that for a curved beam element, shear and membrane locking are interrelated and either shear or membrane underintegration can alleviate it. However, reduced shear integration tends to diminish the membrane-flexural coupling which characterizes curved elements. Locking can also be expected in certain types of mixed formulations, the hybrid-stress formulations avoids locking for beams (but not for shells). Methods for avoiding locking are explored and alternatives evaluated.     

Numerical integration in the finite element method

The theoretical predictions of [1] concerning the needed accuracy in the numerical integration of curvilinear (isoparametric) finite elements is confirmed experimentally. The theoretical arguments and numerical results arrived at here suggest a way to lump the mass matrix with no accuracy loss. In the finite element analysis of almost inextentional shells and nearly incompressible materials where the lack of zero energy modes severely impedes convergence, numerical integration has the beneficial effect of reducing the rank of some stiffness matrices assuring thereby the presence of discrete zero modes.     

A Fourier analysis of spurious mechanisms and locking in the finite element method

A Fourier analysis technique is used to examine spurious mechanisms and the element locking phenomena engendered by different finite element discretizations. It is shown that these phenomena can be identified by examining the uncoupled discrete Fourier operators and corresponding characteristic equations and are caused, in terms of discrete filter concepts, by either a spurious mode carrier or by violating the unlocking condition specified in the paper. The analysis is performed for two simple problems: wave propagation in a bar and vibration of a Timoshenko beam which, when discretized by linear shape functions, succinctly manifest the two phenomena as being directly traceable to specific components of the finite element discretizations.     

Implementation and analysis of multigrid schemes with finite elements for elliptic optimal control problems

The detailed implementation and analysis of a finite element multigrid scheme for the solution of elliptic optimal control problems is presented. A particular focus is in the definition of smoothing strategies for the case of constrained control problems. For this setting, convergence of the multigrid scheme is discussed based on the BPX framework. Results of numerical experiments are reported to illustrate and validate the optimal efficiency and robustness of the performance of the present multigrid strategy.     

Neural classification of finite elements

The work deals with a comparative performance of finite elements, making use of their formulation as vectors (or patterns) in a multi-dimensional space of proper attributes. Since the attributes control the performance, elements defined by similar patterns and related to the same class should show similar behavior. The pattern classification may be carried out with the help a self-organizing feature map of Kohonen with the patterns corresponding to the input space. These networks learn both the distribution and topology of a set of input space. At the end of the learning process, the neurons become selectively tuned to classes of input patterns, thus specifying “family relationships” among the elements. The work makes use of the four attributes: the element dimensionality, its number of nodes, maximum degree of interpolation polynomials and number of degrees of freedom per node, though a more general characterization is also possible.     

A priori identification of shear locking and stiffening in triangular mindlin elements

An a priori, explicit algebraic procedure for identifying shear locking and excessive solution stiffening in thin shear-deformable plates is explored. Only element-level solutions of element Kirchhoff modes are required to establish the nodal degree-of-freedom constraints. These constraints clearly identify whatever kinematic stiffening might exist. The methodology is demonstrated by the use of a conforming, three-node Mindlin element. Several discretizations of square plates are examined. The results are compared, and fully confirmed, with the corresponding numerical solutions. Examples of alternate discretizations, which alleviate the locking effect, are also presented.