T-elements: a finite element approach with advantages of boundary solution methods

Since their introduction in 1977, the so-called T-elements have considerably evolved and have now become a highly efficient and well established tool for the solution of complex boundary value problems. This class of finite elements, associated with the Trefftz method, is based on enforcement of interelement continuity and boundary conditions on assumed displacement fields chosen so as to a priori satisfy the governing differential equations of the problem. Several alternative T-element formulations are available which yield for a particular subdomain the customary force-displacement relationship with a symmetric positive definite stiffness matrix which makes it possible for such elements to be implemented into the standard finite element (FE) codes.

Owing to their nature, the T-elements may either be considered as a new FE model or as a non-conventional symmetric substructure-oriented form of the boundary element method (BEM). From the point of view of the latter, the outstanding features of the T-element approaches are the use of T-complete sets of non-singular solutions (rather than the singular Kelvin's type fundamental solutions) and the replacement of the customary integral equations form by a symmetric variational formulation.

This paper reviews and critically assesses the most important T-element formulations developed over the past years. It shows that such elements not only cumulate the advantages and discard the drawbacks of the conventional finite element and boundary element methods, but also offer additional advantages not available in the standard form of these methods.     

Finite element modal analysis of an HDD considering the flexibility of spinning disk–spindle, head–suspension–actuator and supporting structure

This paper presents a finite element method to analyze the free vibration of a flexible HDD (hard disk drive) composed of the spinning disk–spindle system with fluid dynamic bearings (FDBs), the head–suspension–actuator with pivot bearings, and the base plate with complicated geometry. Finite element equations of each component of an HDD are consistently derived with the satisfaction of the geometric compatibility in the internal boundary between each component. The spinning disk, hub and FDBs are modeled by annular sector elements, beam elements and stiffness and damping elements, respectively. It develops a 2-D quadrilateral 4-node shell element with rotational degrees of freedom to model the thin suspension efficiently as well as to satisfy the geometric compatibility between the 3-D tetrahedral element and the 2-D shell element. Base plate, arm, E-block and fantail are modeled by tetrahedral elements. Pivot bearing of an actuator and air bearing between spinning disk and head are modeled by stiffness elements. The restarted Arnoldi iteration method is applied to solve the large asymmetric eigenvalue problem to determine the natural frequencies and mode shapes of the finite element model. Experimental modal testing shows that the proposed method well predicts the vibration characteristics of an HDD. This research also shows that even the vibration motion of the spinning disk corresponding to half-speed whirl and the pure disk mode are transferred to a head–suspension–actuator and base plate through the air bearing and the pivot bearing consecutively. The proposed method can be effectively extended to investigate the forced vibration of an HDD and to design a robust HDD against shock.     

A contact stiffness matrix for finite element problems involving external elastic restraint

A simple and fundamental definition is given for a contact stiffness matrix for finite element analysis. This matrix is applicable to all types of finite elements and can easily model elastic foundations supporting beam, plate or solid elements. Another application envisaged is to membrane elements, where the contact stiffness matrix can be used to simulate adhesion to rigid surfaces. In general the concept is an extension of the formulation of consistent loads and is as widely applicable as are these.     

Duabechies小波基梁单元的构造及应用研究

以Daubechies小波尺度函数作为单元插值函数，构造了一类小波基梁单元，由于采用节点挠度和转角作为单元自由度，单元连接和约束条件处理能够像传统单元一样方便地进行，从而使该小波基梁单元可有效地用于变截面、局部承载等复杂梁弯曲问题分析，数值算例表明，本文构造的小波基梁单元对不同约束条件、不同结构形式梁弯曲问题均有较高分析精度，为有限元方法提出了一种新的研究途径．     

Some results of finite element applications in finite elasticity

This paper examines a number of problems connected with the finite element analysis of finite elastic deformations. A brief review of formulation of equations governing finite deformations of highly elastic elements is given. The convergence of finite element approximations for static problems in elasticity is studied. Incremental stiffness equations are derived in general form and various types of incremental loading techniques are examined. A number of representative solved problems in finite elasticity are given.     

Coupling of Nonconforming Finite Elements and Boundary Elements I: A Priori Estimates

Nonconforming finite element methods are sometimes considered as a variational crime and so we may regard its coupling with boundary element methods. In this paper, the symmetric coupling of nonconforming finite elements and boundary elements is established and a priori error estimates are shown. The coupling involves a further continuous layer on the interface in order to separate the nonconformity in the domain from its boundary data which are required to be continuous. Numerical examples prove the new scheme useful in practice. A posteriori error control and adaptive algorithms will be studied in the forthcoming Part II. Received: November 26, 1997; revised February 10, 1999     

An improved formulation of space stiffeners

This paper presents a displacement based finite element model for predicting the constraint torsion effect of stiffeners. In structural modelling, the plate/shell and the stiffeners are treated as separate elements where the displacement compatibility transformation between these two types of elements takes into account the constraint torsional warping effect in the stiffeners. The development is based on a general beam theory which includes flexural-torsion coupling, constrained torsion warping, and shear-centre location. The virtual work principle includes the second order terms of finite beam rotations. For finite element analysis, cubic Hermitian polynomials are used as shape functions of the straight space frame element with two nodes. Elastic stiffness and geometric stiffness matrices for an arbitrary cross-section are evaluated in a closed form, and load correction stiffness for eccentric stiffener loads are considered. To demonstrate the importance of torsion warping constraints and to illustrate the accuracy of this formulation, finite element solutions are presented and compared with available solutions.     

A geometrical nonlinear eccentric 3D-beam element with arbitrary cross-sections

In this paper a finite element formulation of eccentric space curved beams with arbitrary cross-sections is derived. Based on a Timoshenko beam kinematic, the strain measures are derived by exploitation of the Green-Lagrangean strain tensor. Thus, the formulation is conformed with existing nonlinear shell theories. Finite rotations are described by orthogonal transformations of the basis systems from the initial to the current configuration. Since for arbitrary cross-sections the centroid and shear center do not coincide, torsion bending coupling occurs in the linear as well as in the finite deformation case. The linearization of the boundary value formulation leads to a symmetric bilinear form for conservative loads. The resulting finite element model is characterized by 6 degrees of freedom at the nodes and therefore is fully compatible with existing shell elements. Since the reference curve lies arbitrarily to the line of centroids, the element can be used to model eccentric stiffener of shells with arbitrary cross-sections.     

A combined finite and infinite element approach for modeling spherically symmetric transient subsurface flow

This paper presents a finite element-infinite element coupling approach for modeling a spherically symmetric transient flow problem in a porous medium of infinite extent. A finite element model is used to examine the flow potential distribution in a truncated bounded region close to the spherical cavity. In order to give an appropriate artificial boundary condition at the truncated boundary, a transient infinite element, that is developed to describe transient flow in the exterior unbounded domain, is coupled with the finite element model. The coupling procedure of the finite and infinite elements at their interface is described by means of the boundary integro-differential equation rather than through a matrix approach. Consequently, a Neumann boundary condition can be applied at the truncated boundary to ensure the C¹-continuity of the solution at the truncated boundary. Numerical analyses indicate that the proposed finite element-infinite element coupling approach can generate a correct artificial truncated boundary condition to the finite element model for the unbounded flow transport problem.     

谱元法在求解刚架结构动力学问题中的应用

把谱元法应用于刚架结构的动力学响应计算和分析中.建立了杆和梁的谱单元动力学刚度阵,针对刚架结构组装了整体动力学刚度阵,建立了整体结构的运动方程,计算了结构的固有频率和时域响应,并与采用有限元方法得到的结果进行了对比.从结果中可以看出谱元法在数值模拟中的独特优势.     

Torsion of perforated cores: Planar formulation

A planar finite element approximate solution for the torsional analysis of symmetric perforated cores is given. The solution involves neglecting the out-of-plane stiffness of the walls of the core. Only one-quarter of the core is analyzed, using super elements assembled from high order plane stress finite elements. The results of the analysis are compared with the results from three-dimensional finite element analysis. It is observed that for almost all realistic designs the planar approximation is justified. The solution is very close with significant reduction in computation time and data preparation effort.     

Dynamic stiffness matrix of non-symmetric thin-walled curved beam on Winkler and Pasternak type foundations

Using the technical computing program Mathematica, the dynamic stiffness matrix for the spatially coupled free vibration analysis of thin-walled curved beam with non-symmetric cross-section on two-types of elastic foundation is newly presented based on the power series method. For this, the elastic strain energy considering the axial/flexural/torsional coupled terms, the kinetic energy including the rotary inertia effect, and the energy due to the elastic foundation are introduced. Then, equations of motion are derived from the energy principle and explicit expressions for displacement parameters are derived based on power series expansions of displacement components. Finally, the exact dynamic stiffness matrix is determined using force–displacement relations. In order to demonstrate the validity and the accuracy of this study, the natural frequencies of thin-walled curved beams with mono-symmetric and non-symmetric cross-sections are evaluated and compared with the analytical solutions and finite element solutions using Hermitian curved beam elements and ABAQUS’s shell elements. In addition, some results by a parametric study are reported.     

A symmetric variational finite element-boundary integral equation coupling method

This paper is concerned with the development of a mixed variational principle for coupling finite element and boundary integral methods in interface problems, using the generalized Poisson's equation as a prototype situation. One of its primary objectives is to compare the performance of fully variational procedures with methods that use collocation for the treatment of boundary integral equations. A distinctive feature of the new variational principle is that the discretized algebraic equations for the coupled problem are automatically symmetric since they are all derived from a single functional. In addition, the condition that the flux remain continuous across interfaces is satisfied naturally. In discretizing the problem within inhomogeneous or loaded regions, domain finite elements are used to approximate the field variable. On the other hand, only boundary elements are used for regions where the medium is homogeneous and free of external agents. The corresponding integral equations are discretized both by fully variational and by collocation techniques. Results of numerical experiments indicate that the accuracy of the fully variational procedure is significantly greater than that of collocation for the complete interface problem, especially for complex disturbances, at little additional computational cost. This suggests that fully variational procedures may be preferable to collocation, not only in dealing with interface problems, but even for solving integral equations by themselves.     

A mesh-geometry based method for coupling 1D and 3D elements

When conducting a finite element analysis (FEA) one way to reduce the total number of degrees of freedom is to use a mixed-dimensional model. Using beam elements to model long and slender components can significantly reduce the total number of elements. Problems arise when trying to connect elements with different dimensions in part due to incompatible degrees of freedom between different types of finite elements. This paper focuses on problems that occur in coupling beams and solids, which means coupling 1D and 3D finite elements. This paper presents a mesh-based solution to these problems only using specific arrangements of classical 1D and 3D finite elements without requiring the use of additional constraint equations. Two alternative solutions are detailed, evaluated and compared in this paper through series of computational experiments. The implementation of both solutions is also presented and involves mesh and geometry processing operations along with an adaptation of boundary representation (BREP) classical data structures.     

Finite element analysis of stiffened plates

A finite element method is presented in which the constraint between stiffener and member is imposed by means of Lagrange multipliers. This is performed on the functional level, forming augmented variational principles. In order to simplify the initial development and implementation of the proposed method, two-dimensional stiffened beam finite elements are developed. Several such elements are formulated, each showing monotonic convergence in numerical tests. In the development of stiffened plate finite elements, the bending and membrane behaviors are treated seperately. For each, the stiffness matrix of a standard plate element is modified to account for an added beam element (representing the stiffener) and additional terms imposing the constraint between the two. The resulting stiffened plate element was implemented in the SAPIV finite element code. Exact solutions are not known for rib-reinforced plated structures, but results of numerical tests converge monotonically to a value in the vicinity of an approximate “smeared” series solution.     

Theoretical modeling of microfabricated beams with elasticallyrestrained supports

A mathematical model is developed to analyze the mechanical behavior of step-up supports which typically result from surface micromachining. This model accounts for the finite stiffness of the step-up support, which is more accurately represented by an elastically restrained boundary condition rather than a fixed or built-in boundary condition. Based on the model developed, the deflection of cantilever and doubly-supported beams under arbitrarily located concentrated and distributed loads is determined for a wide range of beam geometries. In addition, the buckling load of doubly-supported beams with elastically restrained boundary conditions is analyzed. Finite element analysis verifies the accuracy of the models developed. The models show that significant error in predicting the beam mechanical behavior may result if the finite stiffness of the step-up support is not correctly modeled     

Nonlinear finite element analysis of elastic frames

A very simple and effective formulation and numerical procedure to remove the restriction of small rotations between two successive increments for the geometrically nonlinear finite element analysis of in-plane frames is presented. A co-rotational formulation combined with small deflection beam theory with the inclusion of the effect of axial force is adopted. A body attached coordinate is used to distinguish between rigid body and deformational rotations. The deformational nodal rotational angles are assumed to be small, and the membrane strain along the deformed beam axis obtained from the elongation of the arc length of the deformed beam element is assumed to be constant. The element internal nodal forces are calculated using the total deformational nodal rotations in the body attached coordinate. The element stiffness matrix is obtained by superimposing the bending and the geometric stiffness matrices of the elementary beam element and the stiffness matrix of the linear bar element. An incremental iterative method based on the Newton-Raphson method combined with a constant arc length control method is employed for the solution of the nonlinear equilibrium equations. In order to improve convergence properties of the equilibrium iteration, a two-cycle iteration scheme is introduced. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.     

Nonlinear bending of beams using the finite element method

In this paper a finite element formulation for determining the finite deflection of thin bars is presented. The nonlinear stiffness equations are generated after simple approximate expressions involving the nodal parameters are used to replace the nonlinear terms in the energy functional. The procedure used results in a simplified set of nonlinear algebraic equations which are more amenable to solution than the equations usually presented. The applicability and accuracy of the method together with an evaluation of three incremental solution techniques, a step by step method, a one step Newton-Raphson procedure, and a variable interpolation technique is demonstrated by solving a cantilever beam with a point load acting on the end. Curves showing the sensitivity to increment size and to the number of elements are also presented. The results indicate that the formulation is accurate and inexpensive in terms of computational effort.     

A nonlinear two-node superelement for use in flexible multibody systems

A nonlinear two-node superelement is proposed for the modeling of flexible complex-shaped links for use in multibody simulations. Assuming that the elastic deformations with respect to a corotational reference frame remain small, substructuring methods may be used to obtain reduced mass and stiffness matrices from a linear finite element model. These matrices are used in the derivation of potential and kinetic energy expressions of the nonlinear two-node superelement. By evaluating Lagrange’s equations, expressions for the internal and external forces acting on the superelement can be obtained. The inertia forces of the superelement are derived in terms of absolute nodal velocities and accelerations, which greatly simplifies the dynamic formulation. Three examples are included. The first two examples are used to validate the method by comparing the results with those obtained from nonlinear beam element solutions. We consider a benchmark simulation of the spin-up motion of a flexible beam with uniform cross-section and a similar simulation in which the beam is simultaneously excited in the out-of-plane direction. Results from both examples show good agreement with simulation results obtained using nonlinear finite beam elements. In a third example, the method is applied to an unbalanced rotating shaft, illustrating the potential of the proposed methodology for a more complex geometry.     

A finite element formulation for beams with thin walled cross-sections

The paper presents a finite element model for calculation of stresses and deformations of beams with thin walled cross-sections. The beam model takes into account deformations due to shear. Warping is accounted for by a modified sector coordinate formulation. As interpolation functions between the seven degrees of freedom at each node are used the analytical solutions for the special case of a double symmetric cross-section. Therefore, depending on the external loading, each prismatic beam can in most cases be treated as a single element. The assembly of the beam elements to the global model is performed by use of transition matrices which assures compatibility between the elements in the sence of least squares.