A mixed variational principle for finite element analysis

A mixed variational principle is developed and utilized in a finite element formulation. The procedure is mixed in the sense that it is based upon a combination of modified potential and complementary energy principles. Compatibility and equilibrium are satisfied throughout the domain a priori, leaving only the boundary conditions to be satisfied by the variational principle. This leads to a finite element model capable of relaxing troublesome interelement continuity requirements. The nodal concept is also abandoned and, instead, generalized parameters serve as the degrees-of-freedom. This allows for easier construction of higher order elements with the displacements and stresses treated in the same manner. To illustrate these concepts, plane stress and plate bending analyses are presented.     

Linear and Geometrically nonlinear analysis of plates and shells by a new refined non-conforming triangular plate/shell element

A refined non-conforming triangular plate/shell element for linear and geometrically nonlinear analysis of plates and shells is developed in this paper based on the refined non-conforming element method (RNEM). A conforming triangle membrane element with drilling degrees of freedom in Cartesian coordinates and the refined non-conforming triangular plate-bending element RT9, in which Kirchhoff kinematic assumption was adopted, are used to construct the present element. The displacement continuity condition along the interelement boundary is satisfied in an average sense for plate analysis, and the coupled displacement continuity requirement at the interelement is satisfied in an average sense, thereby improving the performance of the element for shell analysis. Selectively reduced integration with stabilization scheme is employed in this paper to avoid membrane locking. Numerical examples demonstrate that the present element behaves quite satisfactorily either for the linear analysis of plate bending problems and plane problems or for the geometrically nonlinear analysis of thin plates and shells with large displacement, moderate rotation but small strain.     

A PARTIAL HYBRID DEGENERATED PLATE/SHELL ELEMENT FOR THE ANALYSIS OF LAMINATED COMPOSITES

A degenerated plate/shell element based on the combined energy variational principle and the equivalent single-layer model is proposed. It is derived from the 3-D continuum equation by imposing one constraint on the 3-D isoparametric solid element: a straight line normal to the mid-surface before deformation remains straight but not normal after deformation. The continuities of interlaminar stresses are satisfied at the interlaminar surface and the number of degrees of freedom per node is independent of the number of layers. In this work, the combined energy variational principle is used to overcome transverse stress continuity limitations of single-layer models. The traction-free conditions are satisfied on the upper and lower surfaces of a laminate by assuming the transverse stress components independently. The transverse normal strain is taken into account in order to consider the full 3-D effect in a laminated composite. The iso-function method and the classification method of the stress modes are used to construct the assumed stress field which contains a minimum number of stress modes and guarantees no zero energy mode in the element stiffness matrix. Three examples are presented to illustrate the efficiency and accuracy of the element.     

Refined non-conforming quadrilateral thin plate bending element

A refined non-conforming quadrilateral thin plate bending element RPQ4 which can satisfy the requirement of convergence is established such that the non-conforming displacement function can be derived directly. A simple explicit expression of a refined constant strain matrix can be introduced into the formulation of the standard displacement element which results in the constraint condition of interelement continuity being satisfied in an average sense. Numerical examples are presented to show that the present model can pass the patch test and possesses high accuracy. © 1997 John Wiley & Sons, Ltd.     

An improved theory and finite-element model for laminated composite and sandwich beams using first-order zig-zag sublaminate approximations

A new beam finite element based on a new discrete-layer laminated beam theory with sublaminate first-order zig-zag kinematic assumptions is presented and assessed for thick and thin laminated beams. The model allows a laminate to be represented as an assemblage of sublaminates in order to increase the model refinement through the thickness, when needed. Within each sublaminate, discrete-layer effects are accounted for via a modified form of DiSciuva's linear zig-zag laminate kinematics, in which continuity of interfacial transverse shear stresses is satisfied identically. In the computational model, each finite element represents one sublaminate. The finite element is developed with the topology of a fournoded rectangle, allowing the thickness of the beam to be discretized into several elements, or sublaminates, if necessary, to improve accuracy. Each node has three engineering degrees of freedom, two translations and one rotation. Thus, this element can be conveniently implemented into general purpose finite-element codes. The element stiffness coefficients are integrated exactly, yet the element exhibits no shear locking due to the use of a consistent interdependent interpolation scheme. Numerical performance of the current element is investigated for an arbitrarily layered beam, a symmetrically layered beam and a sandwich beam with low and high aspect ratios. The comparisons of numerical results with elasticity solutions show that the element is very accurate and robust.     

A variational formulation for finite elasticity with independent rotation and Biot-axial fields

The fundamentals of the geometrically nonlinear mechanics of the three-dimensional elastic continuum are derived, starting from a general variational framework established for the polar model and passing through a constitutive definition of the non-polar medium itself. A constrained variational setting follows, having as unknown vector fields the displacement, the rotation vector and the axial of the Biot stress. It embraces both the rotational equilibrium and the characterization of the rotation as Euler-Lagrange equations. These conditions can then be satisfied in a weak sense within discrete approximations. It is also shown that the classical approach of the non-polar continuum can be accomodated as a particular case of the present formulation. A consistent linearization is then proposed and a simple solid finite element developed to test the computational viability of the formulation. A few examples assess the capability of the element to represent large three-dimensional rotations. Communicated by S. N. Atluri, 2 August 1996     

A finite element collocation method for the exact control of a parabolic problem

A finite element method is given for the problem of exact control of a linear parabolic equation. The basis functions consist of piecewise bicubic polynomials and the differential equation is satisfied at Gaussian collocation points within each element. The overdetermined system of equations obtained is solved by the method of least squares, and a convergence argument is given for the complete procedure. Numerical results are given for two problems of boundary control.     

An invariant eight-node hybrid-stress element for thin and thick multilayer laminated plates

A hybrid-stress eight-node isoparametric element is developed for the analysis of thin or thick multilayer fiber-reinforced composite plates. Transverse shear deformation effects are included by allowing for individual layer cross-section warping for thick laminates, or alternatively, laminate non-normal cross-section rotations for thin to moderately thick laminates. All stress components are included and are interpolated independently within each layer. Interlayer surface traction continuity and appropriate upper/lower surface traction-free conditions are exactly satisfied. The layer stress field is selected on the basis of earlier single-layer element studies so that the resulting element is naturally invariant with respect to co-ordinate translation or rotations, is non-locking in the thin-plate limit, and the element stiffness is of correct rank. An example for which an elasticity solution is available is used to demonstrate the element performance. Schemes for reduction of element stiffness computation time are also presented.     

Refined 15‐DOF triangular discrete degenerated shell element with high performances

A refined discrete degenerated 15‐DOF triangular shell element RDTS15 with high performances is proposed. For constructing the element displacement function, the exact displacement function of the Timoshenko's beam is used as the displacement on the element boundary, and the re‐constitute method for shear strain matrix is adopted. The proposed element can be used in the analysis of both moderate thick and thin plates/shells. Numerical examples presented show that the new model indeed possesses higher accuracy in the analysis of thin and thick plates/shells, and that it can pass the patch test required for the Kirchhoff thin plate elements, and also passed the inf–sup test for free cylindrical shell problems and satisfied both the bending‐ and membrane‐dominated test. Copyright © 2004 John Wiley Sons, Ltd.     

雪崩光电二极管高精度三维微调架结构设计与分析

通过利用计算机软件实体建模及有限元分析和测试实验的方法,为某卫星激光测距系统的雪崩光电二极管接收器设计了新型结构的高精度三维微调架.首先,使用UG软件进行三维实体建模;然后,利用有限元分析软件MSC.Patran对微调架模型进行静力分析和模态分析;最后,对微调架实物进行振动和精度测试实验.实验结果和有限元分析结果较为吻合,验证了有限元分析的可靠性.软件设计及分析和测试实验的结果表明微调架的设计满足预期的设计要求,能够实现对雪崩二极管的精密调整,使其正常工作.     

BEM for crack-hole problems in thermopiezoelectric materials

The boundary element formulation for analysing interaction between a hole and multiple cracks in piezoelectric materials is presented. Using Green's function for hole problems and variational principle, a boundary element model (BEM) for a 2-D thermopiezoelectric solid with cracks and holes has been developed and used to calculate stress intensity factors of the crack-hole problem. In BEM, the boundary condition on the hole circumference is satisfied a priori by Green's function, and is not involved in the boundary element equations. The method is applicable to multiple-crack problems in both finite and infinite solids. Numerical results for stress and electric displacement intensity factors at a particular crack tip in a crack-hole system of piezoelectric materials are presented to illustrate the application of the proposed formulation.     

A hybrid finite element formulation of the linear biphasic equations for hydrated soft tissue

A hybrid finite element method has been developed for application to the linear biphasic model of soft tissues. The biphasic model assumes that hydrated soft tissue is a mixture of two incompressible, immiscible phases, one solid and one fluid, and employs mixture theory to derive governing equations for its mechanical behaviour. These equations are time dependent, involving both fluid and solid velocities and solid displacement, and will be solved by spatial finite element and temporal finite difference approximation. The first step in the derivation of this hybrid method is application of a finite difference rule to the solid phase, thus obtaining equations with only velocities at discrete times as primary variables. A weighted residual statement of the temporally discretized governing equations, employing C° continuous interpolations of the solid and fluid phase velocities and discontinuous interpolations of the pore pressure and elastic stress, is then derived. The stress and pressure functions are chosen so that the total momentum equation of the mixture is satisfied; they are jointly referred to as an equilibrated stress and pressure field. The corresponding weighting functions are chosen to satisfy a relationship analogous to this equilibrium relation. The resulting matrix equations are symmetric. As an illustration of the hybrid biphasic formulation, six-noded triangular elements with complete linear, several incomplete quadratic, and complete quadratic stress and pressure fields in element local co-ordinates are developed for two dimensional analysis and tested against analytical solutions and a mixed-penalty finite element formulation of the same equations. The hybrid method is found to be robust and produce excellent results; preferred elements are identified on the basis of these results.     

Average and complete incompressibility in the finite element method

The constraint of incompressibility is incorporated into the finite element method for plane strain through the use of a Lagrange multiplier. Depending on the approximating function chosen for this multiplier, the constraint condition can be satisfied everywhere within the element or only in an average sense for the entire element. A study of these two approaches from the point of view of rate of convergence and computer time is presented.     

基于接触的水下夹桩器卡环连接结构设计

水下夹桩器是一种广泛应用于海洋石油平台建设的液压夹具。卡环式连接结构被应用于水下夹桩器。在水下夹桩器卡环连接结构设计中引入基于多体接触的有限元优化设计过程。阐述了基于扩张的拉格朗日算法有限元分析的接触优化模型建立,利用ANSYS软件参数化设计语言完成了优化计算。优化结果显示,采用非线性有限元法和优化方法对卡环连接结构进行设计能够取得满意的效果,设计结果满足工程要求。     

Modeling of crack propagation kinetics

Central crack development in a metal stip under tension is considered using a regular-finiteelement model. Crack development is modeled by removing the finite element in which the fracture conditions are satisfied. It has been shown that this approach allows one to qualitatively trace crack development and to represent some experimentally observed effects, such as stepwise crack development in metal.     

A traction‐based equilibrium finite element free from spurious kinematic modes for linear elasticity problems

This paper presents equilibrium elements for dual analysis. A traction‐based equilibrium element is proposed in which tractions of an element instead of stresses are chosen as DOFs, and therefore, the interelement continuity and the Neumann boundary balance are directly satisfied. To be solvable, equilibrated tractions with respect to the space of rigid body motion are required for each element. As a result, spurious kinematic modes that may inflict troubles on stress‐based equilibrium elements do not appear in the element because only equilibrium constraints on tractions are required. An admissible stress field is eventually constructed in terms of the equilibrated tractions for the element, and hence, equilibrium finite element procedures can proceed. The element is also generalized to accommodate non‐zero body forces, nonlinear boundary tractions and curved Neumann boundaries. Numerical tests including a single equilibrium element, error estimation of a cantilever beam and an infinite plate with a circular hole are conducted, displaying excellent convergence and effectiveness of the element for error estimation. Copyright © 2014 John Wiley & Sons, Ltd.     

Locking of thin plate/shell elements

Often, finite element solutions of thin plate/shell elements become very stiff and the displacement field solutions diverge from those predicted by Kirchhoff's theory. This phenomenon is known as the locking phenomenon. A theoretical fomulation demonstrating its existence is developed, and results of finite element analysis of a single element and mesh are discussed. This leads to a sufficient and necessary criterion which must be satisfied to avoid the locking phenomenon.     

HYBRID-TREFFTZ EQUILIBRIUM MODEL FOR CRACK PROBLEMS

A formulation based on the approximation of the stress field is used to compute directly the stress intensity factors in crack problems. The boundary displacements are independently approximated. In each finite element, the assumed stresses may model multipoint singularities of variable order. The differential equilibrium equations are locally satisfied as solutions of the governing differential system are used to build the stress approximation basis. The approximation on the boundary displacements is constrained to satisfy locally the kinematic boundary conditions. The remaining fundamental conditions, namely the differential compatibility equations, the constitutive relations and the static boundary conditions are enforced through weighted residual statements. The approximation criteria are so chosen as to ensure that the finite element model is described by a sparse, adaptive and symmetric governing system described by structural matrices with boundary integral expressions. Numerical applications are presented to show that accurate solutions can be obtained using structural discretizations based on coarse meshes of few but highly rich elements, each of which may have different geometries and alternative approximation laws.     

圆管网格结构的自适应有限元分析方法研究

为了判断网格结构有限元模型中梁单元的长度和插值函数是否合理并对此进行调整,首先推导了梁在受拉、受压、纯弯3种情况下的挠曲微分方程,以有限元试算得到的梁单元两端节点力为边界条件,求出了梁单元广义力分布场的解析解;然后,根据Zienkiewicz-Zhu后验误差估计理论,以该解析解为广义力相对精确解,推导了广义力有限元解和广义力相对精确解的能量范数以确定梁单元的相对误差。在试算过程中,如果网格结构中每个梁单元的相对误差都满足精度要求,则终止试算过程,否则调整梁单元的插值函数或长度后再进行试算。以单层球面网壳的自适应有限元静力分析为例验证了该方法的正确性和可行性。     

An assumed displacement hybrid finite element model for linear fracture mechanics

This paper deals with a procedure to calculate the elastic stress intensity factors for arbitrary-shaped cracks in plane stress and plane strain problems. An assumed displacement hybrid finite element model is employed wherein the unknowns in the final algebraic system of equations are the nodal displacements and the elastic stress intensity factors. Special elements, which contain proper singular displacement and stress fields, are used in a fixed region near the crack tip; and the interelement displacement compatibility is satisfied through the use of a Lagrangean multiplier technique. Numerical examples presented include: central as well as edge cracks in tension plates and a quarter-circular crack in a tension plate. Excellent correlations were obtained with available solutions in all the cases. A discussion on the convergence of the present solution is also included.