Aspects of the formulation and finite element implementation of large strain isotropic elasticity

The paper presents a formulation of isotropic large strain elasticity and addresses some computational aspects of its finite element implementation. On the theoretical side, an Eulerian setting of isotropic elasticity is discussed exclusively in terms of the Finger tensor as a strain measure. Noval aspects are a direct representation of the Eulerian elastic moduli in terms of the Finger tensor and their rigorous decomposition into decoupled volumetric and isochoric contributions based on a multiplicative split of the Finger tensor into spherical and unimodular parts. The isochoric stress response is formulated in terms of the eigenvalues of the unimodular part of the Finger tensor. A constitutive algorithm for the computation of the stresses and tangent moduli for plane problems is developed and applied to a model problem of rubber elasticity. On the computational side, the implementation of the constitutive model in three possible finite element formulations is discussed. After pointing out algorithmic techniques for the treatment of incompressible elasticity, several numerical simulations are presented which show the performance of the proposed constitutive algorithm and the convergence behaviour of the different finite element fomulations for compressible and incompressible elasticity.     

Finite element analysis of two- and three-dimensional static problems in the asymmetric theory of elasticity as a basis for the design of experiments

In this paper, the constitutive relations of the finite element method are constructed and used for solving two- and three-dimensional problems of the asymmetric theory of elasticity. Different variants of finite elements are considered. The numerical experiments are carried out to evaluate the reliability and computational efficiency of the finite element algorithm based on the comparison between the numerical and analytical solutions, numerical estimation of the convergence and checking of the degree of accuracy, to which the natural boundary conditions are satisfied. The obtained solutions to the two- and three-dimensional problems are interpreted from the viewpoint of their applicability to a design of experiments capable of revealing the facts of couple-stress effects in material under elastic deformation and identification of material constants for the asymmetric theory of elasticity. The capabilities of the finite element algorithm to interpret experimental data and estimate the errors occurring in real experiments have been tested by solving several example problems.     

计算复合材料有效弹性模量的重心有限元方法

采用几何法构造出任意边数多边形单元的重心插值形函数, 应用Galerkin法提出了求解弹性力学问题的重心有限元方法。用重心有限元方法对SiC/Ti和B/Al 2种纤维复合材料横向截面的有效弹性模量进行了预报。计算模型取纤维呈六边形排列且为各向同性的代表性单胞, 对其杨氏模量、 剪切模量和体积模量在较大的体积分数范围内进行了数值模拟。通过与解析公式和传统有限元的计算结果对比, 重心有限元方法的计算结果符合解析公式解的趋势, 与传统有限元的计算结果吻合较好。与传统有限元方法相比, 重心有限元方法的单元划分不受三角形或四边形的形状限制, 能够再现材料的真实结构。由于单元较大且数目较少, 本文方法具有很高的计算效率。      

A generalized finite element method with global-local enrichment functions for confined plasticity problems

The main feature of partition of unity methods such as the generalized or extended finite element method is their ability of utilizing a priori knowledge about the solution of a problem in the form of enrichment functions. However, analytical derivation of enrichment functions with good approximation properties is mostly limited to two-dimensional linear problems. This paper presents a procedure to numerically generate proper enrichment functions for three-dimensional problems with confined plasticity where plastic evolution is gradual. This procedure involves the solution of boundary value problems around local regions exhibiting nonlinear behavior and the enrichment of the global solution space with the local solutions through the partition of unity method framework. This approach can produce accurate nonlinear solutions with a reduced computational cost compared to standard finite element methods since computationally intensive nonlinear iterations can be performed on coarse global meshes after the creation of enrichment functions properly describing localized nonlinear behavior. Several three-dimensional nonlinear problems based on the rate-independent J ₂ plasticity theory with isotropic hardening are solved using the proposed procedure to demonstrate its robustness, accuracy and computational efficiency.     

Identification of Constitutive Material Model Parameters for High-Strain Rate Metal Cutting Conditions Using Evolutionary Computational Algorithms

Advances in plasticity-based analytical modeling and finite element methods (FEM) based numerical modeling of metal cutting have resulted in capabilities of predicting the physical phenomena in metal cutting such as forces, temperatures, and stresses generated. However, accuracy and reliability of these predictions rely on a work material constitutive model describing the flow stress, at which work material starts to plastically deform. This paper presents a methodology to determine deformation behavior of work materials in high-strain rate metal cutting conditions and utilizes evolutionary computational methods in identifying constitutive model parameters. The Johnson-Cook (JC) constitutive model and cooperative particle swarm optimization (CPSO) method are combined to investigate the effects of high-strain rate dependency, thermal softening and strain rate-temperature coupling on the material flow stress. The methodology is applied in predicting JC constitutive model parameters, and the results are compared with the other solutions. Evolutionary computational algorithms have outperformed the classical data fitting solutions. This methodology can also be extended to other constitutive material models.     

A Computational Analysis to Burgers Huxley Equation

The efficiency of solving computationally partial differential equations can be profoundly highlighted by the creation of precise, higher-order compact numerical scheme that results in truly outstanding accuracy at a given cost. The objective of this article is to develop a highly accurate novel algorithm for two dimensional non-linear Burgers Huxley (BH) equations. The proposed compact numerical scheme is found to be free of superiors approximate oscillations across discontinuities, and in a smooth flow region, it efficiently obtained a high-order accuracy. In particular, two classes of higher-order compact finite difference schemes are taken into account and compared based on their computational economy. The stability and accuracy show that the schemes are unconditionally stable and accurate up to a two-order in time and to six-order in space. Moreover, algorithms and data tables illustrate the scheme efficiency and decisiveness for solving such non-linear coupled system. Efficiency is scaled in terms of L₂ and L_∞ norms, which validate the approximated results with the corresponding analytical solution. The investigation of the stability requirements of the implicit method applied in the algorithm was carried out. Reasonable agreement was constructed under indistinguishable computational conditions. The proposed methods can be implemented for real-world problems, originating in engineering and science.     

One dimensional nonlocal integro-differential model & gradient elasticity model : Approximate solutions and size effects

In the present work, the close similarity that exists between Mindlin’s strain gradient elasticity and Eringens nonlocal integro-differential model is explored. A relation between length scales of nonlocal-differential model and gradient elasticity model has been arrived. Further, a relation has also been arrived between the standard and nonstandard boundary conditions in both the cases. C⁰-based finite element methods (FEMs) are extensively used for the implementation of integro-differential equations. This results in standard diagonally dominant global stiffness matrix with off diagonal elements occupied largely by the kernel values evaluated at various locations. The global stiffness matrix is enriched in this process by nonzero off diagonal terms and helps in incorporation of the nonlocal effect, there by accounting the long-range interactions. In this case, the diagonally dominant stiffness matrix has a band width equal to influence domain of basis function. In such cases, a very fine discretization with larger number of degrees of freedom is required to predict nonlocal effect, thereby making it computationally expensive. In the numerical examples, both nonlocal-differential and gradient elasticity model are considered to predict the size effect of tensile bar example. The solutions to integro-differential equations obtained by using various higher-order approximations are compared. Lagrangian, Bèzier and B-Spline approximations are considered for the analysis. It has been shown that such higher-order approximations have higher inter-element continuity there by increasing the band width and the nonlocal character of the stiffness matrix. The effect of considering the higher-order and higher-continuous approximation on computational effort is made. In conclusion, both the models predict size effect for one-dimensional example. Further, the higher-continuous approximation results in less computational effort for nonlocal-differential model.     

A finite difference-Galerkin finite element solution of compressible laminar boundary-layer flows

A finite difference-Galerkinfinite element method is presented for the solution of the two-dimensional compressible laminar boundary-layer flow problem. The streamwise derivatives in the momentum and energy equations are approximated by finite differences. An iterative scheme, due to the non-linearity of the problem, in conjunction with the Galerkin finite element method is then proposed for the solution of the problem through the boundary-layer thickness. Numerical results are presented and these are compared with other numerical and analytical solutions in order to show the applicability and the effectiveness of the proposed formulation. In all the cases here examined, the results obtained attained the same accuracy of other numerical methods for a much smaller number of points in the boundary-layer.     

Spectral element model for axially loaded bending–shear–torsion coupled composite Timoshenko beams

The use of frequency-dependent spectral element matrix (or exact dynamic stiffness matrix) in structural dynamics is known to provide extremely accurate solutions, while reducing the total number of degrees-of-freedom to resolve the computational and cost problems. Thus, in this paper, the spectral element model is developed for an axially loaded bending–shear–torsion coupled composite laminated beam which is represented by the Timoshenko beam model based on the first-order shear deformation theory. The high accuracy of the spectral element model is then numerically verified by comparing with exact theoretical solutions or the solutions obtained by conventional finite element method. For the numerical verification, the finite element model is also provided for the composite laminated beam.     

A simple yet accurate finite element procedure for computing stress intensity factors

A new finite element procedure for calculating stress intensity factors in elastic crack problems is developed. In common with a number of other approaches in the literature, the procedure combines the analytical singular fields present in a problem with a finite element treatment of the residual regular problem. What distinguishes the procedure is the use of path-independent integrals to balance the analytical and numerical contributions. A set of test problems with exact solutions is analysed and demonstrates that the procedure is readily implemented and can accurately evaluate stress intensity factors with a modest amount of computational effort. The application of two competing methods to the test problems further demonstrates that the new procedure is markedly superior in both its initial accuracy and its rate of convergence. The paper concludes with two additional illustrations of the procedure as applied to the single-edge crack and the centre crack; these also yield accurate results for little computational effort.     

An analysis of the linear advection–diffusion equation using mesh-free and mesh-dependent methods

The numerical solution of advection–diffusion equations has been a long standing problem and many numerical methods that attempt to find stable and accurate solutions have to resort to artificial methods to stabilize the solution. In this paper, we present a meshless method based on thin plate radial basis functions (RBF). The efficiency of the method in terms of computational processing time, accuracy and stability is discussed. The results are compared with the findings from the dual reciprocity/boundary element and finite difference methods as well as the analytical solution. Our analysis shows that the RBFs method, with its simple implementation, generates excellent results and speeds up the computational processing time, independent of the shape of the domain and irrespective of the dimension of the problem.     

A continuum finite element for single-set jointed media

In response to the need for an advanced computational model for wave propagation in jointed-rock media a new finite element for jointed media with a single set of regularly spaced joints is developed. The element is a numerical implementation of the higher-order homogenization model recently proposed by Murakami and Hegemier. Due to the dispersive effects induced by regularly spaced joints, wave phenomena in jointed media are altered significantly. Therefore, in order to improve the interpretation of seismograms for accurate source identification, it is necessary to develop a higher-order continuum element. The accuracy and efficiency of the new element is investigated by applying it to wave-guide and wave-normal problems of a jointed half-space and by comparing the wave response with that of DYNA2D. The analyses by DYNA2D discretize explicitly the details of the joint microstructure, and are adopted as numerically exact measures for the assessment of the proposed finite element; good correlations were obtained. The validation study also confirmed the importance of wave dispersion for non-linear as well as linear joint responses. Finally, as a more practical application of the proposed element, the problem of a jointed full-space with a cylindrical cavity pressurized by step and pulse loadings was solved. Velocities at several observation points were compared with the numerically exact results of DYNA2D. Similar analyses carried out for elastic isotropic media predicted totally different velocity responses and confirmed the need for the proposed element.     

Finite element methods for an optimal steady-state control problem

An optimal steady-state control problem governed by an elliptic state equation is solved by several finite element methods. Finite element discretizations are applied to different variational formulations of the problem yielding accurate numerical results as compared with the given analytical solution. It is sated that, for minimum computational effort and high accuracy, ‘mixed’ finite elements requiring only C° continuity, and approximating the control and state functions simultaneously are better suited to similar ‘fourth order’ problems.     

Adaptive vibration control of a cylindrical shell with laminated PVDF actuator

A generalized higher-order theory describing the mechanical behavior of multi-layered composite plates with arbitrary lamination scheme is proposed. Ritz’s method is employed to determine the kinematic unknowns expressed in a complete polynomial power series of the thickness-wise coordinate whereas the dependence on the in-plane coordinates is such that the functions satisfy all boundary conditions. The correct constitutive laws of a three-dimensional orthotropic elastic continuum are employed for each individual layer. The convergence and accuracy of the computational scheme are investigated by comparing elastic static and buckling results with analytical or finite element solutions for complex cross- and angle-ply laminates. For further validation of the theory, laminated plates under a transverse pressure are investigated for technically relevant lamination schemes and the associated deformation and stress results are compared with those obtained through FE calculations.     

Linear finite elements with orthogonal discontinuous basis functions for explicit earthquake ground motion modeling

The dynamic explicit finite element method is commonly used in earthquake ground motion modeling. In this method, the element mass matrix is approximately lumped, which may lead to numerical dispersion. On the other hand, the orthogonal finite element method, based on orthogonal polynomial basis functions, naturally derives a lumped diagonal mass matrix and can be applied to dynamic explicit finite element analysis. In this paper, we propose finite elements based on orthogonal discontinuous basis functions, the element mass matrices of which are lumped without approximation. Orthogonal discontinuous basis functions are used to improve the accuracy and reduce the numerical dispersion in earthquake ground motion modeling. We present a detailed formulation of the 4‐node tetrahedral and 8‐node hexahedral elements. The relationship between the proposed finite elements and conventional finite elements is investigated, and the solutions obtained from the conventional explicit finite element method are compared with analytical solutions to verify the numerical dispersion caused by the lumping approximation. Comparison of solutions obtained with the proposed finite elements to analytical solutions demonstrates the usefulness of the technique. Examples are also presented to illustrate the effectiveness of the proposed method in earthquake ground motion modeling in the actual three‐dimensional crust structure. Copyright © 2010 John Wiley & Sons, Ltd.     

Error estimates for mixed finite element approximations of nonlinear problems

The finite element methods have proved a very effective tool for the numerical solutions of nonlinear problems arising in elasticity and other related engineering sciences. Relative to linear elliptic theory, little is known about the accuracy and convergence properties of mixed finite element approximation of nonlinear elliptic boundary value problems. The nonlinear problems are much more complicated, since each problem has to be treated individually. This is one of the reasons that there is no unified and general theory for the nonlinear problems. In this paper, the application of the mixed finite element method to a highly nonlinear Dirichlet problem, which arises in the field of oceanography and elasticity is studied and new results involving the error estimates are derived. In fact, some of the results and methods to be described in this paper may be extended to more complicated problems or problems with other boundary conditions. As a special case, we obtain the well known error estimates for the corresponding linear and mildly nonlinear elliptic boundary value problems.     

Closed-form effective elastic constants of frame-like periodic cellular solids by a symbolic object-oriented finite element program

In this study, the effective elastic constants of several 2D and 3D frame-like periodic cellular solids with different unit-cell topologies are analytically derived using the homogenization method based on equivalent strain energy. The analytical expressions of strain energy of a unit cell under different strain modes are determined using a generic symbolic object-oriented finite element (FE) program written in MATLAB. The obtained analytical expressions of the strain energy are then used to symbolically compute the effective elastic constants that include Young’s moduli, Poisson’s ratios, and shear moduli. The obtained analytical effective elastic constants are numerically verified using results from an ordinary numerical FE program. The obtained closed-form effective elastic constants are also compared with some existing solutions from the literature. This study demonstrates that symbolic computation platforms can be properly used to provide efficient methodologies for finding useful analytical solutions of mechanical problems. Without the symbolic object-oriented FE program in this study, elaborate and tedious analytical analysis has to be manually performed for each different unit cell. The symbolic object-oriented FE program provides analytical analysis of unit cells that is accurate and fast. The object-oriented programming technique allows the symbolic FE program in this study to be efficiently implemented.     

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.     

Improved finite difference method for equilibrium problems based on differentiation of the partial differential equations and the boundary conditions

A numerical algorithm for producing high-order solutions for equilibrium problems is presented. The approximated solutions are improved by differentiating both the governing partial differential equations and their boundary conditions. The advantages of the proposed method over standard finite difference methods are: the possibility of using arbitrary meshes; the possibility of using simultaneously approximations with different (distinct) orders of accuracy at different locations in the problem domain; an improvement in approximating the boundary conditions; the elimination of the need for ‘fictitious’ or ‘external’ nodal points in treating the boundary conditions. Furthermore, the proposed method is capable of reaching approximate solutions which are more accurate than other finite difference methods, when the same number of nodal points participate in the local scheme. A computer program was written for solving two-dimensional problems in elasticity. The solutions of a few examples clearly illustrate these advantages.