Unsymmetric Formulation Research Articles

A Hybrid CM-BEM Formulation for Solving Large-Scale 3D Eddy-Current Problems Based on ℋ-Matrices and Randomized Singular Value Decomposition for BEM Matrix Compression

We present a novel a,v-q hybrid method for solving large-scale time-harmonic eddy-current problems. This method combines a hybrid unsymmetric formulation based on the cell method and the boundary element method with a hierarchical matrix-compression technique based on randomized singular value decomposition. The main advantage is that the memory requirements are strongly reduced compared to the corresponding hybrid method without matrix compression while retaining the same robust solution strategy consisting of a simple construction of the preconditioner. In addition, the matrix-compression accuracy and efficiency are enhanced compared to traditional compression methods, such as adaptive cross approximation. The numerical results show that the proposed hybrid approach can also be effectively used to analyze large-scale eddy-current problems of engineering interest.

Two 8‐node quadrilateral unsymmetric elements with different incompatible modes immune to severe distortion

AbstractMost unsymmetric quadrilateral elements have demonstrated a high tolerance to mesh distortion in pure bending problems involving coarse meshes. However, they have difficulty achieving acceptable results in shearing‐dominated problems, and the element's performance is affected by the degree of mesh distortion. In this paper, two 8‐node quadrilateral elements are developed by introducing two different incompatible modes into the unsymmetric formulation. The unsymmetric framework with two different sets of interpolation functions for displacement fields provides the basis for the immunity property. The incompatible displacement mode and the enhanced assumed strain guarantee the completeness of the displacement field interpolation polynomials, and the satisfaction of ‐orthogonality condition ensures the accuracy of the simulations. The resulting elements, respectively, denoted as UQ10 and UQ10/E4, can provide exact solutions for both pure and linear bending problems and exhibit excellent insensitivity in shear‐dominated tests. Furthermore, they are invariant, rapidly converging, and free of various locking problems.

4‐node unsymmetric quadrilateral membrane element with drilling DOFs insensitive to severe mesh‐distortion

SummaryThe unsymmetric finite element method is a promising technique to produce distortion‐immune finite elements. In this work, a simple but robust 4‐node 12‐DOF unsymmetric quadrilateral membrane element is formulated. The test function of this new element is determined by a concise isoparametric‐based displacement field that is enriched by the Allman‐type drilling degrees of freedom. Meanwhile, a rational stress field, instead of the displacement one in the original unsymmetric formulation, is directly adopted to be the element's trial function. This stress field is obtained based on the analytical solutions of the plane stress/strain problem and the quasi‐conforming technique. Thus, it can a priori satisfy related governing equations. Numerical tests show that the presented new unsymmetric element, named as US‐Q4θ, exhibits excellent capabilities in predicting results of both displacement and stress, in most cases, superior to other existing 4‐node element models. In particular, it can still work very well in severely distorted meshes even when the element shape deteriorates into concave quadrangle or degenerated triangle.

An unsymmetric 4-node, 8-DOF plane membrane element perfectly breaking through MacNeal's theorem

Summary Among numerous finite element techniques, few models can perfectly (without any numerical problems) break through MacNeal's theorem: any 4-node, 8-DOF membrane element will either lock in in-plane bending or fail to pass a C0 patch test when the element's shape is an isosceles trapezoid. In this paper, a 4-node plane quadrilateral membrane element is developed following the unsymmetric formulation concept, which means two different sets of interpolation functions for displacement fields are simultaneously used. The first set employs the shape functions of the traditional 4-node bilinear isoparametric element, while the second set adopts a novel composite coordinate interpolation scheme with analytical trail function method, in which the Cartesian coordinates (x,y) and the second form of quadrilateral area coordinates (QACM-II) (S,T) are applied together. The resulting element US-ATFQ4 exhibits amazing performance in rigorous numerical tests. It is insensitive to various serious mesh distortions, free of trapezoidal locking, and can satisfy both the classical first-order patch test and the second-order patch test for pure bending. Furthermore, because of usage of the second form of quadrilateral area coordinates (QACM-II), the new element provides the invariance for the coordinate rotation. It seems that the behaviors of the present model are beyond the well-known contradiction defined by MacNeal's theorem. Copyright © 2015 John Wiley & Sons, Ltd.

Time-harmonic solution for acousto-elastic interaction with controllability and spectral elements

The classical way of solving the time-harmonic linear acousto-elastic wave problem is to discretize the equations with finite elements or finite differences. This approach leads to large-scale indefinite complex-valued linear systems. For these kinds of systems, it is difficult to construct efficient iterative solution methods. That is why we use an alternative approach and solve the time-harmonic problem by controlling the solution of the corresponding time dependent wave equation. In this paper, we use an unsymmetric formulation, where fluid-structure interaction is modeled as a coupling between pressure and displacement. The coupled problem is discretized in space domain with spectral elements and in time domain with central finite differences. After discretization, exact controllability problem is reformulated as a least-squares problem, which is solved by the conjugate gradient method.

Extension of unsymmetric finite elements US‐QUAD8 and US‐HEXA20 for geometric nonlinear analyses

Purpose – This paper aims to present an extension of two recently published elements (which are based on Petrov‐Galerkin formulation) to geometric nonlinear (GNL) problems.Design/methodology/approach – Two different sets of shape functions, namely isoparametric and metric, suitably chosen to satisfy the necessary compatibility and completeness conditions, are used as test and trial functions, respectively. Total Lagrangian formulation is used for the implementation of the element.Findings – In implementing the unsymmetric formulation for nonlinear problems, the deformation gradient tensor can be evaluated invariably using either isoparametric or metric shape functions. The developed elements are found to exhibit improved performance in the presence of mesh distortions.Research limitations/implications – The numerical problems in this paper involve linear elastic materials.Originality/value – Extension of US‐QUAD8 and US‐HEXA20 for GNL problems is new.

A novel unsymmetric 8‐node plane element immune to mesh distortion under a quadratic displacement field

AbstractAn 8‐node quadrilateral plane finite element is developed based on a novel unsymmetric formulation which is characterized by the use of two sets of shape functions, viz., the compatibility enforcing shape functions and completeness enforcing shape functions. The former are chosen to satisfy exactly the minimum inter‐ as well as intra‐element displacement continuity requirements, while the latter are chosen to satisfy all the (linear and higher order) completeness requirements so as to reproduce exactly a quadratic displacement field. Numerical results from test problems reveal that the new element is indeed capable of reproducing exactly a complete quadratic displacement field under all types of admissible mesh distortions. In this respect, the proposed 8‐node unsymmetric element emerges to be better than the existing symmetric QUAD8, QUAD8/9, QUAD9, QUAD12 and QUAD16 elements, and matches the performance of the quartic element, QUAD25. For test problems involving a cubic or higher order displacement field, the proposed element yields a solution accuracy that is comparable to or better than that of QUAD8, QUAD8/9 and QUAD9 elements. Furthermore, the element maintains a good accuracy even with the reduced 2× 2 numerical integration. Copyright © 2003 John Wiley & Sons, Ltd.

Dynamic modelling of skeletal systems with arbitrary mass distributions

AbstractThis study develops and illustrates a dynamic model for vibration analysis of rod and beam elements with arbitrary mass distributions. This arbitrary mass model represents the element shape function using a frequency dependent series that becomes exact as the number of terms in the series approaches infinity. The arbitrary mass model yields the natural frequencies and modes of elements with mass distributions which vary as continuous functions of the spanwise co‐ordinate.The arbitrary mass model, implemented using the unsymmetric formulation for vibration analysis, is illustrated in examples and compared with other dynamic models which use stepped mass representations to model the arbitrary mass distributions. Studies show that the arbitrary mass model is more computationally efficient than the stepped mass representation models, particularly when accuracy requirements are high.

The unsymmetric formulation for vibration analysis of frames

The unsymmetric formulation is a dynamic model for computing numerically exact free vibration frequencies and modes of structural systems. Based on exact representation of element inertia forces and a frequency independent stiffness matrix, this formulation obtains discretization error-free solutions. Previous studies have shown that in modeling the vibration response of trusses, the unsymmetric formulation is more computationally efficient than the conventional finite element formulation, particularly when accuracy requirements are high. This study extends the development of the unsymmetric formulation to more general vibration problems. The unsymmetric formulation is developed and illustrated here for the vibration analysis of beams and frames with uniform and/or lumped mass distributions. The computational efficiency of the formulation is studied and compared to that of the exact displacement and conventional finite element models. Results from numerical examples suggest that the unsymmetric formulation provides more accurate free vibration modes and frequencies of frames in less computational time than do other dynamic models.

A note on rotational energy levels in nuclei

It is shown that the kinetic energy T of any system of interacting particles may be written in a “canonical form”, which displays the dependence of T on the total angular momentum. A formula is given for the zero order moments of inertia associated with collective rotation of the system. The formalism is applied to nuclei, and for purposes of comparison, to molecules. In addition, it is shown that an unsymmetric formulation of the many particle problem, in which only the nucleons of a “core” participate in the definition of the body fixed axes, leads to a Hamiltonian which is related by a simple averaging process with the “unified” model of Bohr and Mottelson. Finally, a connection is established between the “unified” model of Bohr-Mottelson and the “cranking model” of Inglis.