A 3-D,symmetric, finite element formulation of the Biot equations with application to acoustic wave propagation through an elastic porous medium

A weak solution of the coupled, acoustic-elastic, wave propagation problem for a flexible porous material is proposed for a 3-D continuum. Symmetry in the matrix equations; with respect to both volume, i.e. ‘porous frame’–‘pore fluid’, and surface, i.e. ‘porous frame/pore fluid’–‘non-porous media’, fluid–structure interaction; is ensured with only five unknowns per node; fluid pore pressure, fluid-displacement potential and three Cartesian components of the porous frame displacement field. Taking Biot's general theory as starting point, the discretized form of the equations is derived from a weighted residual statement, using a standard Galerkin approximation and iso-parametric interpolation of the dependent variables. The coupling integrals appearing along the boundary of the porous medium are derived for a number of different surface conditions. The primary application of the proposed symmetric 3-D finite element formulation is modelling of noise transmission in typical transportation vehicles, such as aircraft, cars, etc., where porous materials are used for both temperature and noise insulation purposes. As an example of an application of the implemented finite elements, the noise transmission through a double panel with porous filling and different boundary conditions at the two panel boundaries are analysed. © 1998 John Wiley & Sons, Ltd.     

Finite element calculation of energy release rate for 2-D rubbery material problems with non-conservative crack surface tractions

The energy release rate G for 2-D rubbery material problems with non-conservative crack surface tractions is calculated by a modified version of virtual crack extension method with finite element solutions. The formulation is demonstrated to be ‘patch-independent’ and therefore a complicated finite element model around the crack tip is not required.     

Mapped infinite elements for 3-D vector potential magnetic problems

Numerous engineering problems, especially those in electromagnetics, often require the treatment of the unbounded continua. Mapped infinite elements have been developed for the solution of 3-D magnetic vector potential equations in infinite domain that may be used in conjunction with the standard finite elements. The electromagnetic field equations are written in terms of the magnetic vector potential for the infinite domain, and 3-D mapped infinite eiement formulation based on these equations is presented in detail. A series of magnetostatics and eddy current problems are solved to demonstrate the validity and efficiency of the procedure. These numerical results indicate that the combined finite–infinite element procedure is computationally much more economical for the solution of unbounded electromagnetic problems, especially when using the vector potential formulation, as the number of system equations decreases substantially compared to the finite element only procedure. The present procedure shows promise for the treatment of large practical industrial 3-D eddy current problems with manageable computer resources.     

用二维等效电路模型研究大截面圆柱变幅杆的振动

由振动方程的近似解得到了具有大尺寸比的圆柱变幅杆的二维等效电路,用于带外载或不带外载的圆柱轴对称振动分析,以及尺寸比变化引起的频率变化评价。与有限元及一维模型进行了对比。发现该模型计算频率比一维模型更准确,与有限元模态分析相一致。进而由此确定了这些频率所对应的振动模态。计算发现轴向和周向水负载对应的圆柱振动频率接近,可用于超声变幅杆的优化。     

AUTOMATIC ADAPTIVE 3-D FINITE ELEMENT REFINEMENT USING DIFFERENT-ORDER TETRAHEDRAL ELEMENTS

Automatic refinement finite element analyses were carried out employing three different-order tetrahedral solid elements for the solution of 3-D stress analysis problems. Numerical results indicated that the adaptive refinement procedure could eliminate effectively the effect of singularities and the optimal convergence rate was achieved in all the examples tested. The preconditioned conjugate gradient technique was used for the solution of the large system of simultaneous equations. By interpolating the initial guess of the iteration solver from the previous converged solution, the number of iterations needed for the solution is lower than expected. Furthermore, when the mesh density distribution pattern has converged, it became even more efficient and independent of the number of degrees of freedom in the finite element mesh. The relative efficiency of the three different-order tetrahedral elements has also been compared in terms of storage and computational cost needed for achieving a certain accuracy. It is found that although the cubic T20 element can achieve the highest convergence rate, the T10 element is the most competitive and effective element in terms of storage and computational cost needed. © 1997 John Wiley & Sons, Ltd.     

The post-processing approach in the finite element method—part 1: Calculation of displacements,stresses and other higher derivatives of the displacements

This is the first in a series of three papers in which we discuss a method for ‘post-processing’ a finite element solution to obtain high accuracy approximations for displacements, stresses, stress intensity factors, etc. Rather than take the values of these quantities ‘directly’ from the finite element solution, we evaluate certain weighted averages of the solution over the entire region. These yield approximations are of the same order of accuracy as the strain energy. We obtain error estimates, and also present some numerical examples to illustrate the practical effectiveness of the technique. In the third paper of this series we address the matters of adaptive mesh selection and a posteriori error estimation.     

A FINITE ELEMENT FORMULATION FOR 3-D CONTINUA USING THE CO-ROTATIONAL TECHNIQUE

The co-rotational technique is described for the three-dimensional analysis of continua. The technique exploits the proven technology of the best continua elements for linear analysis which are embedded into a formulation that applies an element-attached local co-ordinate frame that continuously moves and rotates with the element. The geometric non-linearity is then incorporated via the rotation of this local system. The method uses similar procedures to those recently described for 2-D continua elements but introduces concepts from a more conventional ‘continuum mechanics’ approach. The general framework for the co-rotational procedure is kept. However, a much neater formulation is derived, which readily allows the extension from two to three dimensions.     

考虑三维圆柱铰间隙碰撞的空间机构柔性多体动力学分析方法

三维圆柱铰的动态性能对空间机构的运动精度和动力学特性有着重要的影响。基于多体动力学方法和刚体有限元方法,提出考虑三维圆柱铰的间隙碰撞和动态接触作用的空间机构柔性多体接触动力学方法。采用圆柱-圆柱的线接触力学模型和切片法,计算圆柱-圆柱的接触变形量和接触力,建立三维圆柱铰的轴销与轴套之间的间隙碰撞作用和三维动态接触关系。以等效刚体单元和Timoshenko梁单元描述柔性杆件的刚性有限元模型,建立柔性连杆和三维圆柱铰的具有多刚体系统的统一形式的动力学方程。计算分析了含三维圆柱铰的空间机构的运动精度和动力学特性,获得空间机构的位移响应,三维圆柱铰的相对运动轨迹和动态作用力等动力学响应。采用三维圆柱-圆柱动态线接触方法,能有效地预测空间机构系统的三维圆柱铰持续接触、间隙碰撞状态和摩擦作用下的动力学特性。计算结果表明三维圆柱铰的间隙碰撞作用对空间机构的运动精度和动态特性的影响较为显著。动力学模型和计算结果对复杂工况下高精度多间隙的空间机构的动态设计和动力学研究具有重要的理论意义。     

粗网格划分下的箱梁三维实体有限元分析方法

针对现有箱梁分析方法普遍存在的计算精度与计算效率之间矛盾的问题,提出了粗网格划分下的箱梁三维实体有限元分析方法。在充分考虑箱梁受力变形特点的基础上,以修正的Hellinger-Reissner变分原理为基础,通过合理引入非协调位移插值项,构造出直角坐标系下的六面体八结点杂交应力单元8N21β和柱坐标系下的六面体八结点杂交应力单元8N21βc,分别用于粗网格划分下的直箱梁和曲线箱梁的三维实体有限元分析。数值算例表明:8N21β单元和8N21βc单元在粗网格划分下具有较高的计算精度,能有效提高箱梁三维实体有限元分析的计算效率。     

3-D eddy-current computation with a self-adaptive refinementtechnique

A method for the adaptive computation of 3-D eddy-current problems is presented. The 3-D eddy-current problem is formulated in terms of the electric intensity (E formulation). Edge elements that impose tangential continuity of the approximation function are employed for the discretization of the problem with the finite element method. An a-posteriori error estimation technique is proposed with the introduction of two error criteria: a) the tangential discontinuity of the magnetic intensity H, and b) the normal discontinuity of the eddy current density J_e. The proposed error estimation technique is employed in a 3-D self-adaptive refinement procedure. Sufficient approximation of the skin effect and calculation of the eddy current distribution is obtained with the proposed method. The implementation of the proposed technique in a problem of 3-D eddy-current computation in a multiply connected conducting body is discussed     

Hybrid and mixed-penalty finite elements for 3-D analysis of soft hydrated tissue

Solution of biomechanics problems involving three-dimensional (3-D) behaviour of soft tissue on geometries representative of such tissue in vivo will require the use of numerical methods. Toward this end, a pair of tetrahedral finite elements has been developed. The equations which are used to model the tissue behaviour for both elements are those commonly known as the linear biphasic equations. This model assumes that hydrated soft tissue is a mixture of two incompressible, immiscible phases, and employs mixture theory to derive governing equations for its mechanical behaviour. The finite element techniques applied to these equations for the two elements are the mixed-penalty method and the hybrid method. Both elements are described here, and the special requirements for 3-D analysis are discussed. Results obtained by solving canonical problems in two and three dimensions using both elements are presented and compared. Both elements are found to produce excellent results. The hybrid element is also noted to have advantages for non-linear analyses involving finite deformation which will require solution in the future.     

A CO-ROTATIONAL FORMULATION FOR 2-D CONTINUA INCLUDING INCOMPATIBLE MODES

The co-rotational finite element method is well known for decomposing the motion of an element into a rigid body motion and a strain-producing deformation. A key feature is the definition of the rotating frame and, partially for this reason, the method has almost exclusively been applied to beams and shells. A novel feature of the current paper is that the method is applied to continua—in the current case, two-dimensional. The main motivation is to allow the analyst to quickly introduce the latest and best linear elements into a non-linear context. To this end, in the current work, a set of ‘incompatible modes’ or ‘enhanced strains’ is added to the conventional four-noded elements. While the main body of the paper considers small strains, as a further novel aspect, it later applies the co-rotational method to problems with large strains (here via hyperelasticity) and, to this end, establishes a link between the co-rotational technique and a Biot stress formulation.     

‘Infinite domain’ elements

A parametric element is formulated which enables the economic modelling of ‘infinite domain’ type problmes. A typical problem is an opening in a stress field in an infinite medium, either in two or three dimensions. The strategy is to model around the opening with two or three layers of conventional isoparametric finite elements and surround these with a single layer of ‘infinite domain’ elements. Several sample problems has been analysed for circular, square and spherical openings in infinite media, and the results compared with either theoretical or boundary element solutions which include the ‘infinite’ boundary in their solution technique.     

The mathematical shell model underlying general shell elements

We show that although no actual mathematical shell model is explicitly used in ‘general shell element’ formulations, we can identify an implicit shell model underlying these finite element procedures. This ‘underlying model’ compares well with classical shell models since it displays the same asymptotic behaviours—when the thickness of the shell becomes very small—as, for example, the Naghdi model. Moreover, we substantiate the connection between general shell element procedures and this underlying model by mathematically proving a convergence result from the finite element solution to the solution of the model. Copyright © 2000 John Wiley & Sons, Ltd.     

Finite element simulation and modeling of 2-D arrays for 3-D ultrasonic imaging

Issues of modeling and design of 2-D arrays in three dimensions with finite element code are discussed. These ultrasonic arrays are used for real time dynamic imaging of the heart. Topics include optimization, sensitivity, and performance and methods to speed up the run times required for computer simulations of large three-dimensional models. Empirical results from 45×45 2-D arrays are also presented     

STRESS INTENSITY FACTORS FOR CORNER CRACKS AT A HOLE BY A 3-D WEIGHT FUNCTION METHOD WITH STRESSES FROM THE FINITE ELEMENT METHOD

Abstract— Stress intensity factors for quarter-elliptical corner cracks emanating from a circular hole are determined using a 3-D weight function method combined with a 3-D finite element method. The 3-D finite element method is used to analyze uncracked configurations and provide stress distributions in the region where a crack is likely to occur. Using this stress distribution as input, the 3-D weight function method is used to determine stress intensity factors. Three different loading conditions, i.e. remote tension, remote bending and wedge loading, are considered for a wide range of geometrical parameters. The significance of using 3-D uncracked stress distributions is studied. Comparisons are made with solutions available in the literature.     

A new 3-D finite element model to evaluate added mass for analysis of fluid-structure interaction problems

The paper presents the results of investigations conducted to evaluate the added mass to represent fluid-structure interaction effects in vibration/dynamic analysis of floating bodies such as ship hulls. While the structural plating is idealized by 9-noded plate/shell finite elements, the fluid domain is modelled by 20-noded/21-noded 3-D finite elements in the investigations conducted. A new 8-noded element has been developed to model the interface between the structure and the fluid. An efficient computational methodology has been used for computation of added mass. The finite element models are validated by comparing the results with those given by analytical solution for a submerged sphere. The efficacy of the finite element model is demonstrated through convergence of the results obtained for a floating barge problem. A better convergence rate and distribution of added mass in three orthogonal directions have been obtained.     

Computation of 3-D magnetostatic fields using a reduced scalarpotential

Some improvements to the finite element computation of static magnetic fields in three dimensions using a reduced magnetic scalar potential are presented. Methods are described for obtaining an edge element representation of the rotational part of the magnetic field from a given source current distribution. When the current distribution is not known in advance, a boundary value problem is set up in terms of a current vector potential. An edge element representation of the solution can be directly used in the subsequent magnetostatic calculation. The magnetic field in a DC arc furnace is calculated by first determining the current distribution in terms of a current vector potential. A 3-D problem involving a permanent magnet as well as a coil is solved, and the magnetic field in some points is compared with measurement results