Lead field computation for the electrocardiographic inverse problem--finite elements versus boundary elements

In order to be able to solve the inverse problem of electrocardiography, the lead field matrix (transfer matrix) has to be calculated. The two methods applied for computing this matrix, which are compared in this study, are the boundary element method (BEM) and the finite element method (FEM). The performance of both methods using a spherical model was investigated. For a comparable discretization level, the BEM yields smaller relative errors compared to analytical solutions. The BEM needs less computation time, but a larger amount of memory. Inversely calculated myocardial activation times using either the FEM or BEM computed lead field matrices give similar activation time patterns. The FEM, however, is also capable of considering anisotropic conductivities. This property might have an impact for future development, when also individual myocardial fiber architecture can be considered in the inverse formulation.     

Three-dimensional finite element analysis of interlaminar stresses in thick composite laminates

The three-dimensional finite element computer program has been developed to investigate interlaminar stresses in thick composite laminates. The finite element analysis is based on displacement formulation employing curved isoparametric 16-node elements. By using substructure technique, the program developed is capable of handling any composite laminates which consist of any number of orthotropic laminae and any orientations. In this paper, solid laminates and laminates with a circular hole were taken to study interlaminar stresses at the straight edge and the curved edge, respectively. Various solid laminates such as [45_n/0_n − 45_n/90_n]_s, [45/0/ − 45/90]_ns, and [45/0/ − 45/90]_sn (n = 1˜4) were analyzed. Also, [45/0/ − 45/90]_sn laminates with a circular hole were studied for n = 1 ˜ 20. The effect of laminate thickness and stacking sequence on the interlaminar stresses near the free edge was investigated. Interlaminar stresses were governed by stacking sequence rather than laminate thickness. The boundary layer width did not increase with laminate thickness but with the number of plies in the repeating unit.     

Recovery of consistent stresses for compatible finite elements

In displacement based finite element models, stresses deduced directly from the constitutive relationship can show local erratic behaviour. This occurs in problems involving initial stresses or strains, or varying rigidities over the element domain, when local stresses do not meet a specific consistency requirement. In this context, an integrated procedure for recovering consistent stresses, that is stresses ridded of spurious outcomes, is proposed. The procedure is developed within a general weighted residual approach, suitably specialized for the purpose. The relationship between the proposed procedure and those based on the Hu–Washizu formulation is also elucidated. For illustration purpose, some numerical tests are included.     

An efficient algorithm for computation of stresses due to strain measurements

An efficient and accurate algorithm for computation of stresses corresponding to measurements of strains, in materials obeying the von Mises yield criterion and perfect plasticity, is presented. The relevant formulation and computer implementation is described. Through a numerical example the accuracy, efficiency and robustness of the algorithm is verified. The listing of the source code is added for completeness.     

New approach in the determination of interlaminar shear stresses from the results of MSC/NASTRAN

A finite difference technique is applied to the strains and curvatures obtained from MSC/NASTRAN thin plate solutions to determine their derivatives. These quantities are then incorporated into classical thin plate theory to calculate interlaminar shear stresses. Several analyses are performed comparing the interlaminar shear stresses calculated by this method with those calculated theoretically and using MSC/NASTRAN beam theory approach. In all cases, the interlaminar shear stresses calculated by this method compare favorably with theoretical values. In addition, the interlaminar shear stresses are determined using MSC/NASTRAN beam theory approach and the method presented in this paper for a simply supported plate with a 30/-30/-30/30 layup subject to a uniform pressure. The technique presented in this paper can also be applied to other finite element programs.     

网格工作流中转移概率的计算方法研究

网格工作流的转移概率指的是工作流中一个活动发生之后另外一个活动发生的概率或者该活动再次发生的概率。转移概率广泛地应用在工作流的分析中,如过程重组、工作流服务质量管理等。基于一种网格工作流模型GPEL,分析了计算转移概率的不同的场景,并讨论了这些场景中转移概率的计算方法。     

An approximate semi-analytical method for prediction of interlaminar shear stresses in an arbitrarily laminated thick plate

An approximate semi-analytical method for determination of interlaminar shear stress distribution through the thickness of an arbitrarily laminated thick plate has been presented. The method is based on the assumptions of transverse inextensibility and layerwise constant shear angle theory (LCST) and utilizes an assumed quadratic displacement potential energy based finite element method (FEM). Centroid of the triangular surface has been proved, from a rigorous methematical point of view (Aubin-Nitsche theory), to be the point of exceptional accuracy for the interlaminar shear stresses. Numerical results indicate close agreement with the available three-dimensional elasticity theory solutions. A comparison between the present theory and that due to an assumed stress hybrid FEM suggests that the (normal) traction-free-edge condition is not satisfied in the latter approach. Furthermore, the present paper is the first to present the results for interlaminar shear stresses in a two-layer thick square plate of balanced unsymmetric angle-ply construction. A comparison with the recently proposed Equilibrium Method (EM) indicates the superiority of the present method, because the latter assures faster convergence as well as simultaneous vanishing of the transverse shear stresses on both the exposed surfaces of the laminate. Superiority of the present method over the EM, in the case of a symmetric laminate, is limited to faster convergence alone. It has also been demonstrated that the combination of the present method and the reduced (quadratic order) numerical integration scheme yields convergence of the interlaminar shear stresses almost as rapidly as that of the nodal displacements, in the case of a thin plate.     

An algorithm for the computation of the integral of the state transition matrix

An efficient algorithm for the computation of the integral of the state transition matrix over the interval [0, T_{1}] is developed, It is assumed that there exists someT < T_{1}such that the state transition matrix and the integral of the state transition matrix can be computed as accurately as required over the interval [0, T].     

Geometrically nonlinear formulation for the axi-symmetric transition finite elements

This paper presents a geometrically nonlinear formulation for the axi-symmetric transition finite elements using total lagrangian approach. The basic element is formulated using properties of the axi-symmetric solids and the axi-symmetric shells. A novel feature of the formulation presented here is that the restriction on the magnitude of the rotations for the shell nodes of the transition element is eliminated. This is accomplished by retaining true nonlinear functions of nodal rotations in the definition of the element displacement field. Such transition elements are essential for geometrically nonlinear applications requiring both axi-symmetric solids and the axi-symmetric shells. They ensure proper connection of the axi-symmetric solid portion of the structure to the shell like portion of the structure. It is shown that the selection of different stress and strain components at the integration points does not effect the overall linear response of the element. However, in the geometrically nonlinear formulation, it is necessary to select appropriate components of the stresses and the strains at the integration point for accurate and converging element behavior. Numerical examples are presented to demonstrate such characteristics of the transition elements.     

Construction of transition finite elements for the plane triangular family

A method of constructing mixed order strain triangles from a high order strain triangle is presented for the family of plane triangular elements. The necessary transformation matrices are given for all members of the triangular family that can be formed from the quadratic strain triangle. These transformation matrices are universal numerical matrices which are independent of the size and shape of the triangles, and are readily applicable to field problems also. A FORTRAN program listing which generates stiffness matrices for these mixed order strain triangles is provided for the user.     

A symbolic-numerical algorithm for the computation of matrix elements in the parametric eigenvalue problem

A symbolic-numerical algorithm for the computation of the matrix elements in the parametric eigenvalue problem to a prescribed accuracy is presented. A procedure for calculating the oblate angular spheroidal functions that depend on a parameter is discussed. This procedure also yields the corresponding eigenvalues and the matrix elements (integrals of the eigenfunctions multiplied by their derivatives with respect to the parameter). The efficiency of the algorithm is confirmed by the computation of the eigenvalues, eigenfunctions, and the matrix elements and by the comparison with the known data and the asymptotic expansions for small and large values of the parameter. The algorithm is implemented as a package of programs in Maple-Fortran and is used for the reduction of a singular two-dimensional boundary value problem for the elliptic second-order partial differential equation to a regular boundary value problem for a system of second-order ordinary differential equations using the Kantorovich method.     

Recursive computation of matrix elements in the numerical renormalization group

The numerical renormalization group is an efficient method to diagonalize model Hamiltonians describing correlated orbitals coupled to conduction states. While only the resulting eigenvalues are needed to calculate the thermodynamical properties for such models, matrix elements of Fermi operators must be evaluated before excitation and transport properties can be computed. The traditional procedure to calculate matrix elements is typically as expensive as the diagonalization of the model Hamiltonian. Here, we present a substantially faster alternative that demands much less memory, yields equally accurate matrix elements and is easier to code.     

A method for the computation of self-sustained oscillations in systems with piecewise linear elements

In this paper a study of the self-sustained oscillations in systems which contain a nonlinear element of the piecewise linear type with or without memory is presented. A method is developed by which one may systematically detect possible symmetric limit cycles, and subsequently identify the oscillations completely. Vector-matrix techniques are employed to transform certain points between regions of the phase space. The boundaries of the regions of phase space correspond to the break points of the piecewise linear characteristic. A set of equations which determine points on a possible closed trajectory are derived using the transformations. Certain switching conditions which are necessary for the existence of an oscillation are applied to these equations. From the solution of the resulting equations, points on the closed trajectory are determined. Using the values of the coordinates of these points, one computes the waveform of the oscillation. Applications of the method to control systems with various configurations are discussed in theory and illustrative examples are presented.     

Geometrically non-linear formulation for the three dimensional solid-shell transition finite elements

This paper presents a geometrically non-linear formulation using total lagrangian approach for the solid-shell transition finite elements. Such transition finite elements are necessary in geometrically non-linear analysis of structures modelled with three dimensional solid elements and the curved shell elements. These elements are an essential connecting link between the solid elements and the shell elements. The element formulation presented here is derived using the properties of the three dimensional solid elements and the curved shell elements. No restrictions are imposed on the magnitude of the nodal rotations. Thus the element formulation is capable of handling large rotations between two successive load increments. The element properties are derived and presented in detail. Numerical examples are also presented to demonstrate their behavior, accuracy and applications in three dimensional stress analysis.

It is shown that the selection of different stress and strain components at the integration points do not effect the overall linear response of the element. However, in geometrically non-linear applications it may be necessary to select appropriate stress and the strain components at the integration points for stable and converging element behavior. Numerical examples illustrate various characteristics of the element.     

Effective computation of matrix elements between polynomial basis functions

Two methods of evaluating matrix elements of a function in a polynomial basis are considered: the expansion method, where the function is expanded in the basis and the integrals are evaluated analytically, and the numerical method, where the integration is performed directly using numerical quadrature. A reduced grid is proposed for the latter which makes use of the symmetry of the basis. Comparison of the two methods is presented in the context of evaluation of matrix elements in a non-direct product basis. If high accuracy of all matrix elements is required then the expansion method is the best choice. If however the accuracy of high order matrix elements is not important (as in variational ro-vibrational calculations where one is typically interested only in the lowest eigenstates), then the method based on the reduced grid offers sufficient accuracy and is much quicker than the expansion method.     

Fast computation of morphological operations with arbitrary structuring elements

This paper presents a general algorithm that performs basic mathematical morphology operations, like erosions and openings, with any arbitrary shaped structuring element in an efficient way. It is shown that our algorithm has a lower or equal complexity but better computing time than all comparable known methods.     

Three dimensional solid-shell transition finite elements for heat conduction

This paper presents a finite element formulation for a special class of finite elements referred to as ‘Solid-Shell Transition Finite Elements’ for three dimensional heat conduction. The solid-shell transition elements are necessary in applications requiring the use of both three dimensional solid elements and the curved shell elements. These elements permit transition from the solid portion of the structure to the shell portion of the structure. A novel feature of the formulation presented here is that nodel temperatures as well as nodal temperature gradients are retained as primary variables. The element geometry is defined in terms of coordinates of the nodes as well as the nodal point normals for the nodes lying on the middle surface of the element. The temperature field with the element is approximated in terms of element approximation functions, nodal temperatures and nodal temperature gradients. The properties of the transition element are then derived using the weak formulation (or the quadratic functional) of the Fourier heat conduction equation in the Cartesian coordinate system and the element temperature approximation. The formulation presented here permits linear temperature distribution in the element thickness direction.

Convective boundaries as well as distributed heat flux is permitted on all six faces of the elements. Furthermore, the element formulation also permits internal heat generation and orthotropic material behavior. Numerical examples are presented firstly to illustrate the accuracy of the formulation and secondly to demonstrate its usefulness in practical application. Numerical results are also compared with the theoretical solutions.     

A FORTRAN routine for computation of coupled bending-torsional dynamic stiffness matrix of beam elements

A FORTRAN subroutine for computing coupled bending-torsional dynamic stiffness matrix of a uniform beam element is developed. An annotated listing of the program is presented. The application of the subroutine is discussed with particular reference to an established algorithm. An illustrative example on the use of the subroutine is given with representative results.     

Adaptive refinement analysis using hybrid-stress transition elements

In this paper, 4-node to 7-node hybrid-stress transition elements are developed for automatic adaptive refinement analysis of plane elasticity problems. The displacement-based transition quadrilateral elements are first adopted and applied to refinement analysis using both full and reduced integration schemes. As the stress field over the displacement-based transition elements is not continuous, a more smooth stress pattern is desirable and could enhance the performance of the element. Indeed, continuous stress field of various orders can be easily introduced into a displacement-based element through a variational procedure based on the Hellinger–Reissner functional. Of the same kinematics and displacement pattern, the resulting hybrid-stress transition elements are more superior to the displacement-based elements in possessing a more continuous high quality stress field within the element. The hybrid-stress transition elements are tested with classical benchmark examples, and the results indicate that hybrid-stress transition elements are consistently more efficient than the displacement-based counterparts in adaptive refinement analysis. A more economical rank-deficient version of hybrid-stress transition elements is also available. While they are less expensive to evaluate, they enjoy a very similar convergence rate as the rank-sufficient hybrid-stress transition elements.