Hybrid-Trefftz displacement element for spectral analysis of bounded and unbounded media

The hybrid-Trefftz displacement element is applied to the elastodynamic analysis of bounded and unbounded media in the frequency domain. The displacements are approximated in the domain of the element using local solutions of the wave equation, the Neumann conditions are enforced directly and the surface forces are approximated on the Dirichlet and inter-element boundaries of the finite element mesh. Two alternative elements are developed to model unbounded media, namely a finite element with absorbing boundaries and an unbounded element that satisfies explicitly the Sommerfeld condition. The finite element equations are derived from the fundamental relations of elastodynamics written in the frequency domain. The numerical implementation of these equations is discussed and numerical tests are presented to assess the performance of the formulation.     

Hybrid finite element formulations for elastodynamic analysis in the frequency domain

Three alternative sets of hybrid formulations to solve linear elastodynamic problems by the finite element method are presented. They are termed hybrid–mixed, hybrid and hybrid–Trefftz and differ essentially on the field conditions that the approximation functions are constrained to satisfy locally. Two models, namely the displacement and the stress models, are obtained for each formulation depending on whether the tractions or the boundary displacements are the field chosen to implement interelement continuity. A Fourier time discretization is used to uncouple the solving system in the frequency domain. The basic space discretization criterion is implemented directly on the fundamental relations of elastodynamics and used to derive the stress and displacement models of the hybrid–mixed formulation. The hybrid and hybrid–Trefftz formulations are presented in sequence as the variants of the hybrid–mixed formulation obtained by progressively increasing the constraints on the approximation bases. Numerical implementation aspects are briefly discussed and the performance of the finite element models is illustrated with numerical applications.     

US-FE-LSPIM四边形单元及其在几何非线性问题中的应用

为了提高在网格畸变时的数值计算精度,基于非对称有限单元的概念,提出US-FE-LSPIM四边形单元。该单元是利用传统的四节点等参元形函数集和FE—LSPIM四边形单元形函数集分别作为检验函数和试函数而构成。前者用于满足单元间和单元内的位移连续性要求,后者用于满足位移完备性要求。该单元结合了有限单元法和无网格法的优点,能...     

A variational formulation for plate buckling problems by the hybrid finite element method

A functional is derived for development of stress hybrid finite elements for plate buckling problems. The equilibrium equations inside the element are identically satisfied in terms of Southwell stress functionsand the transverse displacement. Along the boundary of the element further displacement and normal slope functions are employed. These functions are so chosen as to satisfy the interelement compatibility requirements when the elements are connected. The boundary and internal displacements are selected entirely independently and comments are made on the choice of interpolation functions for the internal displacement.The stationarity of the functional is shown to lead to satisfaction of the equilibrium conditions along interelement boundaries, and the compatibility conditions inside the elements. The paper includes the details of a simple rectangular element and the results of a number of plate buckling problems analysed by the developed element.     

A model reduction technique for laminated solids of revolution with a curved cross-section

This paper presents an extension of the numerical reduction method, which has been proposed in Lejeunes et al. (Arch Appl Mech, 76:311–326, 2006), for modeling curved laminated structures of revolution such as for instance rubber bearings. This method based on high-order finite elements is developed in the context of nearly incompressible hyperelastic behavior. The displacement is approximated with a sum of independent functions, leading to a separation of variables. Therefore, a one-dimensional finite element can be formulated, which represents a 3-dimensional solid in a general loading case. Comparisons with classical finite element models are provided and show the reliability of this model reduction. An important decrease in the model size and a greatly reduced computing time, compared to standard models, is observed.     

Boundary eigensolutions in elasticity II. Application to computational mechanics

The theory of fundamental boundary eigensolutions for elastostatic problems, developed in Part I, is applied to formulate methods for computational mechanics. This theory shows that every elastic solution can be written as a linear combination of some fundamental boundary orthogonal deformations, thus providing a generalized Fourier expansion. One finds that traditional boundary element and finite element methods are largely consistent with this theory, but do not harness its full power. This theory shows that these computational methods are indirectly a generalized discrete Fourier analysis. Furthermore, by utilizing suitable boundary weight functions, boundary element and finite element formulations may be written exclusively in terms of bounded quantities, even for non-smooth problems involving notches, cracks, mixed boundary conditions and bi-material interfaces. The close relationship between the resulting boundary element and finite element methods also becomes evident. Both use displacement and surface traction as primary variables. A new degree-of-freedom concept is introduced, along with a stiffness tensor that enables one to visualize a finite element method via a boundary discretization process, just as in a boundary element approach. Global convergence characteristics of the traction-oriented finite element method are also developed. Comparisons with closed-form fundamental boundary eigensolutions for a circular elastic disc are presented in order to provide a means for assessing the numerical methods. Several other numerical examples are solved efficiently by using the concept of boundary eigensolutions in an indirect fashion. The results indicate that the algorithms follow the underlying theory and that solutions to non-smooth problems can be obtained in a systematic manner. Beyond this, the concept of boundary eigensolutions provides an alternative view of computational continuum mechanics that may lead to the development of other non-traditional approaches.     

压电材料平面裂纹尖端场的杂交应力有限元分析

基于复势理论和杂交变分原理建立了一种适用于力电耦合分析的杂交应力有限元模 型. 给出了建立刚度矩阵的主要公式和推导过程，单元内的位移场和应力场采用满足平 衡方程的复变函数级数解，假设的复变函数级数解事先精确满足裂纹的无应力和电位移法向 分量为零的条件，单元外边界的位移场假设按抛物线变化, 单元的刚度矩阵采用Gauss积分的方法得出. 通过对力电耦合裂尖场的数值计算验证了程序 的正确性和单元的有效性，同时也用所得结果校验了理论解.     

Evaluation of stress intensity factors for bi-material interface cracks using displacement jump methods

The aim of the present work is to investigate the numerical modeling of interfacial cracks that may appear at the interface between two isotropic elastic materials. The extended finite element method is employed to analyze brittle and bi-material interfacial fatigue crack growth by computing the mixed mode stress intensity factors (SIF). Three different approaches are introduced to compute the SIFs. In the first one, mixed mode SIF is deduced from the computation of the contour integral as per the classical J-integral method, whereas a displacement method is used to evaluate the SIF by using either one or two displacement jumps located along the crack path in the second and third approaches. The displacement jump method is rather classical for mono-materials, but has to our knowledge not been used up to now for a bi-material. Hence, use of displacement jump for characterizing bi-material cracks constitutes the main contribution of the present study. Several benchmark tests including parametric studies are performed to show the effectiveness of these computational methodologies for SIF considering static and fatigue problems of bi-material structures. It is found that results based on the displacement jump methods are in a very good agreement with those of exact solutions, such as for the J-integral method, but with a larger domain of applicability and a better numerical efficiency (less time consuming and less spurious boundary effect).     

Non-linear viscoelastodynamic equations of three-dimensional rotating structures in finite displacement and finite element discretization

This paper deals with the non-linear viscoelastodynamics of three-dimensional rotating structure undergoing finite displacement. In addition, the non-linear dynamics is studied with respect to geometrical and mechanical perturbations. On part of the boundary of the structure, a rigid body displacement field is applied which moves the structure in a rotation motion. A time-dependent Dirichlet condition is applied to another part of the boundary. For instance, this corresponds to the cycle step of a helicopter rotor blade. A surface force field is applied to the third part of the boundary and depends on the time history of the structural displacement field. For example, this might corresponds to general unsteady aerodynamics forces applied to the structure. The objective of this paper is to model the non-linear dynamic behavior of such a rotating viscoelastic structure undergoing finite displacements, and to allow small geometrical and mechanical (mass, constitutive equations) perturbations analysis to be performed. The model is constructed by the introduction of a reference configuration which is deduced from the non-linear steady boundary value problem. A constitutive equation deduced from the Coleman and Noll theory concerning the viscoelasticity in finite displacement is used. Thereafter, the weak formulation of the boundary value problem is constructed and discretized using the finite element method. In order to simplify the mathematical study of the equations, multilinear forms are introduced in the algebraic calculation and their mathematical properties are presented.     

Hybrid Equilibrium Finite Elements for Wave Diffraction Problems

Hybrid equilibrium finite elements based on the direct approximation of the domain stress and boundary displacement fields are presented. The structure is divided into a far field, which is considered as an infinite super element, and a near field, which is in turn discretized into finite elements. The displacements in the domains of typical finite elements are obtained from the assumed domain stress field by using the dynamic equilibrium equations. The Helmholtz equation is satisfied in the domain of the infinite super element, and the domain stress fields are associated with elastic and compatible displacements. The resulting governing system is symmetric, sparse, and, if well done, positive. Numerical applications are presented to illustrate the performance of the formulation     

Mixed-mode thermal stress intensity factors from the finite element discretized symplectic method

A finite element discretized symplectic method is introduced to find the thermal stress intensity factors (TSIFs) under steady-state thermal loading by symplectic expansion. The cracked body is modeled by the conventional finite elements and divided into two regions: near and far fields. In the near field, Hamiltonian systems are established for the heat conduction and thermoelasticity problems respectively. Closed form temperature and displacement functions are expressed by symplectic eigen-solutions in polar coordinates. Combined with the analytic symplectic series and the classical finite elements for arbitrary boundary conditions, the main unknowns are no longer the nodal temperature and displacements but are the coefficients of the symplectic series after matrix transformation. The TSIFs, temperatures, displacements and stresses at the singular region are obtained simultaneously without any post-processing. A number of numerical examples as well as convergence studies are given and are found to be in good agreement with the existing solutions.     

Non-local finite element analysis of damped beams

Non-local viscoelastic beam models are used to analyse the dynamics of beams with different boundary conditions using the finite element method. Unlike local damping models the internal force of the non-local model is obtained as weighted average of state variables over a spatial domain via convolution integrals with spatial kernel functions that depend on a distance measure. In the finite element analysis, the interpolating shape functions of the element displacement field are identical to those of standard two-node beam elements. However, for non-local damping, nodes remote from the element do have an effect on the energy expressions, and hence on the damping matrix. The expressions of these direct and cross damping matrices may be obtained explicitly for some common spatial kernel functions and Euler–Bernoulli beam theory. Alternatively numerical integration may be applied to obtain solutions. Examples are given where the eigenvalues are compared to the exact solution for a pinned–pinned beam to demonstrate the convergence of the finite element method. The results for beams with other boundary conditions are used to demonstrate the versatility of the finite element technique.     

Calibrating a J2 plasticity material model using a 2D inverse finite element procedure

Material models are the key ingredients to accurately capture the global mechanical response of structural systems. The use of finite element analysis has proven to be effective in simulating nonlinear engineering applications. However, the choice of the appropriate material model plays a big role in the value of the numerical predictions. Such models are not expected to exactly reproduce global experimental response in all cases. Alternatively, the measured global response at specific domain or surface points can be used to guide the nonlinear analysis to successively extract a representative material model. By selecting an initial set of stress–strain data points, the load–displacement response at the monitoring points is computed in a forward incremental analysis without iterations. This analysis retains the stresses at the integration points. The corresponding strains are not accurate since the computed displacements are not anticipated to match the measured displacements at the monitoring points. Therefore, a corrective incremental displacement analysis is performed at the same load steps to adjust for displacements and strains everywhere by matching the measured displacements at the monitoring points. The stress–strain vectors at the most highly stressed integration point are found to establish an improved material model. This model is used within a multi-pass incremental nonlinear finite element analysis until the discrepancy between the measured and the predicted structural response at the monitoring points vanishes. The J2 flow theory of plasticity is used as a constitutive framework to build the tangent elastic–plastic matrices. The applicability of the proposed approach is demonstrated by solving 2D inverse continuum problems. The comparisons presented support the effectiveness of the proposed approach in accurately calibrating the J2 plasticity material model for such problems.     

Flexure analysis of circular elastic layers bonded between rigid plates

An elastic layer of circular cross-section which is bonded between rigid plates and subjected to pure bending moment is analyzed through a theoretical approach. Based on two kinematic assumptions, the governing equations for the two horizontal displacement functions are established from the equilibrium equations. The horizontal displacements are then solved by satisfying the stress boundary conditions in the elastic layer. Through these solved displacements, the vertical stress in the elastic layer, the shear stress on the bonding surfaces, and the tilting stiffness of the bonded layer are derived in closed-forms and are also compared with the results of finite element analysis.     

A mixed lumped stress–displacement approach to the elastic problem of masonry walls

We present a novel approach to the elastic problem of masonry walls, which generalizes the lumped stress method presented in [Fraternali, 2001], [Fraternali, 2007] and [Fraternali, 2010] and Fraternali et al. (2002). The generalization consists of a mixed lumped stress-displacement approach to the elastic problem of a wall that incorporates no-tension elements. Such an approach depends on the nodal values of the Airy stress function and the displacements of selected (“pivot”) nodes. The latter coincide with inter-element and boundary nodes. The mixed lumped stress-displacement method can be conveniently coupled with standard finite element and boundary element approaches. Numerical applications dealing with recurrent structural elements are given, showing that such a method is able to capture some essential features of the actual response of masonry constructions.     

Axisymmetric bending of functionally graded circular magneto-electro-elastic plates

Axisymmetric bending of functionally graded circular magneto-electro-elastic plates of transversely isotropic materials is analyzed based on linear three-dimensional theory of elasticity coupled with magnetic and electric fields. The transverse loads are expanded in Fourier-Bessel series and therefore can be arbitrarily distributed along the radial direction. The radial distributions of the displacements are assumed in combination of Fourier-Bessel series and polynomials as well as the electric potential and magnetic potential. If the material properties obey the exponential law along the thickness of the plate, two three-dimensional exact solutions for two unusual boundary conditions can be derived since they satisfy the governing equations and specified boundary conditions point by point. For simply supported or clamped boundary, the obtained solutions satisfy the governing equations exactly and the boundary conditions approximately. A layer wise model is also introduced to treat with the plates whose material property components vary independently and arbitrarily along the thickness of the plates. The numerical results are finally tabulated and plotted to demonstrate the presented method and agree well with those from finite element methods.     

Refined beam elements with only displacement variables and plate/shell capabilities

This paper proposes a refined beam formulation with displacement variables only. Lagrange-type polynomials, in fact, are used to interpolate the displacement field over the beam cross-section. Three- (L3), four- (L4), and nine-point (L9) polynomials are considered which lead to linear, quasi-linear (bilinear), and quadratic displacement field approximations over the beam cross-section. Finite elements are obtained by employing the principle of virtual displacements in conjunction with the Unified Formulation (UF). With UF application the finite element matrices and vectors are expressed in terms of fundamental nuclei whose forms do not depend on the assumptions made (L3, L4, or L9). Additional refined beam models are implemented by introducing further discretizations over the beam cross-section in terms of the implemented L3, L4, and L9 elements. A number of numerical problems have been solved and compared with results given by classical beam theories (Euler-Bernoulli and Timoshenko), refined beam theories based on the use of Taylor-type expansions in the neighborhood of the beam axis, and solid element models from commercial codes. Poisson locking correction is analyzed. Applications to compact, thin-walled open/closed sections are discussed. The investigation conducted shows that: (1) the proposed formulation is very suitable to increase accuracy when localized effects have to be detected; (2) it leads to shell-like results in case of thin-walled closed cross-section analysis as well as in open cross-section analysis; (3) it allows us to modify the boundary conditions over the cross-section easily by introducing localized constraints; (4) it allows us to introduce geometrical boundary conditions along the beam axis which lead to plate/shell-like cases.     

Evaluating orthodontic force system in clinical condition with a numerical method

The aim of this paper is to quantitatively evaluating orthodontic force system during orthodontic treatment with a numerical method. Dental cast models were made to obtain digital models with impressions of a patient’s dental arch at regular intervals. Then the displacement of each bracket for a period of time was obtained by computer aided inspection. The finite element model of archwire in free status and brackets engaged in archwire was built. With the derived displacements applied as the boundary condition, the orthodontic force system at a time point was derived by a finite element analysis. The error of this method, especially the influences of friction and material of archwire on the orthodontic force system were discussed at last. This method can quantitatively evaluate the orthodontic force system generated from elastic material in clinic condition, the deformation history of archwire need to be traced for the orthodontic force generated from shape memory alloy.     

Modeling of voids/cracks and their interactions

Numerical modeling of a two-dimensional elastic body containing multiple voids/cracks to study the interaction between these defects can be significantly simplified by developing special finite elements, each containing an internal circular/elliptic hole or a slit crack. These finite elements are developed using complex potentials and the conformal mapping technique. The elements developed can be divided into two categories, namely, the semi-analytic-type and hybrid-type elements. The latter element type is an improved version of the former due to the implementation of displacement continuity along the inter-element boundary. All the proposed elements can be easily combined with the conventional displacement elements, such as isoparametric elements, to analyze the above-mentioned problems without using complicated finite element meshes. Numerical examples have been employed to illustrate the modeling of voids/cracks and their interactions. The results obtained using the semi-analytic-type elements are in good agreement with the theoretical results, and the corresponding results obtained using the hybrid-type elements show an improvement of the agreement with the theoretical results. However, the former element type is much easier to construct.