Finite cover method for progressive failure with cohesive zone fracture in heterogeneous solids and structures

The finite cover method (FCM), which is a cover-based generalized finite element method, is extended for analyses of progressive failure processes involving cohesive zone fracture, starting from an interface debonding and evolving toward one of the constituents of heterogeneous solids and structures. Assuming that the constituents fail according to the maximum principal stress, we are able to represent the evolution of the resulting failure surfaces of discontinuity independent of mesh alignment owing to the distinctive features of the FCM. Also, interface elements with Lagrange multipliers are introduced to impose compatibility conditions on the material interface so that debonding is judged by the multipliers. Representative numerical examples demonstrate the capability of the proposed method in tracing the smooth transition of crack paths from interfacial to internal failure, and vice versa.     

A face‐based smoothed finite element method (FS‐FEM) for 3D linear and geometrically non‐linear solid mechanics problems using 4‐node tetrahedral elements

This paper presents a novel face‐based smoothed finite element method (FS‐FEM) to improve the accuracy of the finite element method (FEM) for three‐dimensional (3D) problems. The FS‐FEM uses 4‐node tetrahedral elements that can be generated automatically for complicated domains. In the FS‐FEM, the system stiffness matrix is computed using strains smoothed over the smoothing domains associated with the faces of the tetrahedral elements. The results demonstrated that the FS‐FEM is significantly more accurate than the FEM using tetrahedral elements for both linear and geometrically non‐linear solid mechanics problems. In addition, a novel domain‐based selective scheme is proposed leading to a combined FS/NS‐FEM model that is immune from volumetric locking and hence works well for nearly incompressible materials. The implementation of the FS‐FEM is straightforward and no penalty parameters or additional degrees of freedom are used. The computational efficiency of the FS‐FEM is found better than that of the FEM. Copyright © 2008 John Wiley & Sons, Ltd.     

Implicit boundary method for finite element analysis using non‐conforming mesh or grid

In the finite element method (FEM), a mesh is used for representing the geometry of the analysis and for representing the test and trial functions by piece‐wise interpolation. Recently, analysis techniques that use structured grids have been developed to avoid the need for a conforming mesh. The boundaries of the analysis domain are represented using implicit equations while a structured grid is used to interpolate functions. Such a method for analysis using structured grids is presented here in which the analysis domain is constructed by Boolean combination of step functions. Implicit equations of the boundary are used in the construction of trial and test functions such that essential boundary conditions are guaranteed to be satisfied. Furthermore, these functions are constructed such that internal elements, through which no boundary passes, have the same stiffness matrix. This approach has been applied to solve linear elastostatic problems and the results are compared with analytical and finite element analysis solutions to show that the method gives solutions that are similar to the FEM in quality but is less computationally expensive for dense mesh/grid and avoids the need for a conforming mesh. Copyright © 2007 John Wiley & Sons, Ltd.     

Umbrella spherical integration: a stable meshless method for non‐linear solids

A stable meshless method for studying the finite deformation of non‐linear three‐dimensional (3D) solids is presented. The method is based on a variational framework with the necessary integrals evaluated through nodal integration. The method is truly meshless, requiring no 3D meshing or tessellation of any form. A local least‐squares approximation about each node is used to obtain necessary deformation gradients. The use of a local field approximation makes automatic grid refinement and the application of boundary conditions straightforward. Stabilization is achieved through the use of special ‘umbrella’ shape functions that have discontinuous derivatives at the nodes. Novel efficient algorithms for constructing the nodal stars and computing the nodal volumes are presented. The method is applied to four test problems: uniaxial tension, simple shear and bending of a bar, and cylindrical indentation. Convergence studies at infinitesimal strain show that the method is well‐behaved and converges with the number of nodes for both uniform and non‐uniform grids. Typical of meshless methods employing nodal integration, the total energy can be underestimated due to the approximate integration. At finite deformation the method reproduces known exact solutions. The bending example demonstrates an interesting example of torsional buckling resulting from the bending. Copyright © 2006 John Wiley & Sons, Ltd.     

Fully coupled non‐linear analysis of piezoelectric solids involving domain switching

Domain switching is the cause of significant non‐linearity in the response of piezoelectric materials to mechanical and electrical effects. In this paper, the response of piezoelectric solids is formulated by coupling thermal, electrical, and mechanical effects. The constitutive equations are non‐linear. Moreover, due to the domain switching phenomenon, the resulting governing equations become highly non‐linear. The corresponding non‐linear finite element equations are derived and solved by using an incremental technique. The developed formulation is first verified against a number of benchmark problems for which a closed‐form solution exists. Next, a cantilever beam made of PZT‐4 is studied to evaluate the effect of domain switching on the overall force–displacement response of the beam. A number of interesting observations are made with respect to the extent of non‐linearity and its progressive spread as the load on the beam increases. Copyright © 2002 John Wiley & Sons, Ltd.     

An integrated procedure for three‐dimensional structural analysis with the finite cover method

In this paper an integrated procedure for three‐dimensional (3D) structural analyses with the finite cover method (FCM) is introduced. In the pre‐process of this procedure, the geometry of a structure is modelled by 3D‐CAD, followed by digitization to have the corresponding voxel model, and then the structure is covered by a union of mathematical covers, namely a mathematical mesh independently generated for approximation purposes. Since the mesh topology in the FCM does not need to conform to the physical boundaries of the structure, the mesh can be regular and structured. Thus, the numerical analysis procedure is free from the difficulties mesh generation typically poses and, in this sense, enables us to realize the mesh‐free analysis. After formulating the FCM with interface elements for the static equilibrium state of a structure, we detail the procedure of the finite cover modelling, including the geometry modelling with 3D‐CAD and the identification of the geometry covered by a regular mesh for numerical integration. Prior to full 3D modelling and analysis, we present a simple numerical example to confirm the equivalence of the performance of the FCM and that of the standard finite element method (FEM). Finally, representative numerical examples are presented to demonstrate the capabilities of the proposed analysis procedure. Copyright © 2005 John Wiley & Sons, Ltd.     

Integration of singular enrichment functions in the generalized/extended finite element method for three‐dimensional problems

A mapping method is developed to integrate weak singularities, which result from enrichment functions in the generalized/extended finite element method. The integration scheme is applicable to 2D and 3D problems including arbitrarily shaped triangles and tetrahedra. Implementation of the proposed scheme in existing codes is straightforward. Numerical examples for 2D and 3D problems demonstrate the accuracy and convergence properties of the technique. Copyright © 2008 John Wiley & Sons, Ltd.     

Multi‐time‐step explicit–implicit method for non‐linear structural dynamics

We present a method with domain decomposition to solve time‐dependent non‐linear problems. This method enables arbitrary numeric schemes of the Newmark family to be coupled with different time steps in each subdomain: this coupling is achieved by prescribing continuity of velocities at the interface. We are more specifically interested in the coupling of implicit/explicit numeric schemes taking into account material and geometric non‐linearities. The interfaces are modelled using a dual Schur formulation where the Lagrange multipliers represent the interfacial forces. Unlike the continuous formulation, the discretized formulation of the dynamic problem is unable to verify simultaneously the continuity of displacements, velocities and accelerations at the interfaces. We show that, within the framework of the Newmark family of numeric schemes, continuity of velocities at the interfaces enables the definition of an algorithm which is stable for all cases envisaged. To prove this stability, we use an energy method, i.e. a global method over the whole time interval, in order to verify the algorithms properties. Then, we propose to extend this to non‐linear situations in the following cases: implicit linear/explicit non‐linear, explicit non‐linear/explicit non‐linear and implicit non‐linear/explicit non‐linear. Finally, we present some examples showing the feasibility of the method. Copyright © 2001 John Wiley & Sons, Ltd.     

An enriched meshless method for non‐linear fracture mechanics

This paper presents an enriched meshless method for fracture analysis of cracks in homogeneous, isotropic, non‐linear‐elastic, two‐dimensional solids, subject to mode‐I loading conditions. The method involves an element‐free Galerkin formulation and two new enriched basis functions (Types I and II) to capture the Hutchinson–Rice–Rosengren singularity field in non‐linear fracture mechanics. The Type I enriched basis function can be viewed as a generalized enriched basis function, which degenerates to the linear‐elastic basis function when the material hardening exponent is unity. The Type II enriched basis function entails further improvements of the Type I basis function by adding trigonometric functions. Four numerical examples are presented to illustrate the proposed method. The boundary layer analysis indicates that the crack‐tip field predicted by using the proposed basis functions matches with the theoretical solution very well in the whole region considered, whether for the near‐tip asymptotic field or for the far‐tip elastic field. Numerical analyses of standard fracture specimens by the proposed meshless method also yield accurate estimates of the J‐integral for the applied load intensities and material properties considered. Also, the crack‐mouth opening displacement evaluated by the proposed meshless method is in good agreement with finite element results. Furthermore, the meshless results show excellent agreement with the experimental measurements, indicating that the new basis functions are also capable of capturing elastic–plastic deformations at a stress concentration effectively. Copyright © 2003 John Wiley & Sons, Ltd.     

F‐bar aided edge‐based smoothed finite element method using tetrahedral elements for finite deformation analysisof nearly incompressible solids

A new smoothed finite element method (S‐FEM) with tetrahedral elements for finite strain analysis of nearly incompressible solids is proposed. The proposed method is basically a combination of the F‐bar method and edge‐based S‐FEM with tetrahedral elements (ES‐FEM‐T4) and is named ‘F‐barES‐FEM‐T4’. F‐barES‐FEM‐T4 inherits the accuracy and shear locking‐free property of ES‐FEM‐T4. At the same time, it also inherits the volumetric locking‐free property of the F‐bar method. The isovolumetric part of the deformation gradient ( F ^iso) is derived from the F of ES‐FEM‐T4, whereas the volumetric part ( F ^vol) is derived from the cyclic smoothing of J(=det( F )) between elements and nodes. Some demonstration analyses confirm that F‐barES‐FEM‐T4 with a sufficient number of cyclic smoothings suppresses the pressure oscillation in nearly incompressible materials successfully with no increase in DOF. Moreover, they reveal that our method is capable of relaxing the corner locking issue arising at the corner in the cylinder barreling analysis. Copyright © 2016 John Wiley & Sons, Ltd.     

The probability density evolution method for dynamic response analysis of non‐linear stochastic structures

The probability density evolution method (PDEM) for dynamic responses analysis of non‐linear stochastic structures is proposed. In the method, the dynamic response of non‐linear stochastic structures is firstly expressed in a formal solution, which is a function of the random parameters. In this sense, the dynamic responses are mutually uncoupled. A state equation is then constructed in the augmented state space. Based on the principle of preservation of probability, a one‐dimensional partial differential equation in terms of the joint probability density function is set up. The numerical solving algorithm, where the Newmark‐Beta time‐integration algorithm and the finite difference method with Lax–Wendroff difference scheme are brought together, is studied. In the numerical examples, free vibration of a single‐degree‐of‐freedom non‐linear conservative system and dynamic responses of an 8‐storey shear structure with bilinear hysteretic restoring forces, subjected to harmonic excitation and seismic excitation, respectively, are investigated. The investigations indicate that the probability density functions of dynamic responses of non‐linear stochastic structures are usually irregular and far from the well‐known distribution types. They exhibit obvious evolution characteristics. The comparisons with the analytical solution and Monte Carlo simulation method demonstrate that the proposed PDEM is of fair accuracy and efficiency. Copyright © 2005 John Wiley & Sons, Ltd.     

A high‐order generalized FEM for through‐the‐thickness branched cracks

This paper presents high‐order implementations of a generalized finite element method for through‐the‐thickness three‐dimensional branched cracks. This approach can accurately represent discontinuities such as triple joints in polycrystalline materials and branched cracks, independently of the background finite element mesh. Representative problems are investigated to illustrate the accuracy of the method in combination with various discretizations and refinement strategies. The combination of local refinement at crack fronts and high‐order continuous and discontinuous enrichments proves to be an excellent combination which can deliver convergence rates close to that of problems with smooth solutions. Copyright © 2007 John Wiley & Sons, Ltd.     

A general non‐linear optimization algorithm for lower bound limit analysis

The non‐linear programming problem associated with the discrete lower bound limit analysis problem is treated by means of an algorithm where the need to linearize the yield criteria is avoided. The algorithm is an interior point method and is completely general in the sense that no particular finite element discretization or yield criterion is required. As with interior point methods for linear programming the number of iterations is affected only little by the problem size. Some practical implementation issues are discussed with reference to the special structure of the common lower bound load optimization problem, and finally the efficiency and accuracy of the method is demonstrated by means of examples of plate and slab structures obeying different non‐linear yield criteria. Copyright © 2002 John Wiley & Sons, Ltd.     

A discontinuous Galerkin finite element method for dynamic and wave propagation problems in non‐linear solids and saturated porous media

A time‐discontinuous Galerkin finite element method (DGFEM) for dynamics and wave propagation in non‐linear solids and saturated porous media is presented. The main distinct characteristic of the proposed DGFEM is that the specific P3–P1 interpolation approximation, which uses piecewise cubic (Hermite's polynomial) and linear interpolations for both displacements and velocities, in the time domain is particularly proposed. Consequently, continuity of the displacement vector at each discrete time instant is exactly ensured, whereas discontinuity of the velocity vector at the discrete time levels still remains. The computational cost is then obviously saved, particularly in the materially non‐linear problems, as compared with that required for the existing DGFEM. Both the implicit and explicit algorithms are developed to solve the derived formulations for linear and materially non‐linear problems. Numerical results illustrate good performance of the present method in eliminating spurious numerical oscillations and in providing much more accurate solutions over the traditional Galerkin finite element method using the Newmark algorithm in the time domain. Copyright © 2003 John Wiley & Sons, Ltd.     

An adaptive non‐conforming finite‐element method for Reissner–Mindlin plates

Adaptive algorithms are important tools for efficient finite‐element mesh design. In this paper, an error controlled adaptive mesh‐refining algorithm is proposed for a non‐conforming low‐order finite‐element method for the Reissner–Mindlin plate model. The algorithm is controlled by a reliable and efficient residual‐based a posteriori error estimate, which is robust with respect to the plate's thickness. Numerical evidence for this and the efficiency of the new algorithm is provided in the sense that non‐optimal convergence rates are optimally improved in our numerical experiments. Copyright © 2003 John Wiley & Sons, Ltd.     

A rotation‐free corotational plane beam element for non‐linear analyses

This paper presents a plane beam element without rotational degrees of freedom that can be used for the analysis of non‐linear problems. The element is based on two main ideas. First, a corotational approach is adopted, which means that the kinematics of the element is decomposed into a rigid body motion part and a deformational part. Next, in the deformational part, the local nodal rotations are extrapolated as a function of the local displacements of the two nodes of the element and the first nodes to the left and right of the element. Six numerical applications are presented in order to assess the performance of the formulation. Copyright © 2007 John Wiley & Sons, Ltd.     

Finite heterogeneous element method using sliced microstructures for linear elastic analysis

A new finite heterogeneous element consisting of sliced microstructures (FHES) is applied in a multi?scale technique. The FHES represents a heterogeneous material with microscopic constituents without homogenization or microscopic finite element analysis. A representative volume element extracted from a heterogeneous structure is thinly sliced. Each slice is modeled as a combined spring to calculate properties of the FHES. Each FHES has the same number of nodes as an ordinary finite element, and the macroscopic analysis cost is the same as that for ordinary finite element analysis. However, the FHES retains information about the microscopic material layout (i.e., the distribution of a material's property) in itself that is lost during homogenization. In the proposed approach, materials are not homogenized. The FHES does not have a constant (homogenized) material property and can ‘change stiffness’ depending on its deformation behavior. This reduces error due to coarse?graining and allows us to calculate the macroscopic deformation behavior with sufficient accuracy even if a large gradient of strain is generated in the macroscopic field. The novelty of the research is the development of rational heterogeneous finite elements. The paper presents the theory behind the FHES and its practical application to a linear elastic problem. Copyright © 2014 John Wiley & Sons, Ltd.     

Geometrically non‐linear thermoelastic analysis of functionally graded shells using finite element method

A finite element formulation governing the geometrically non‐linear thermoelastic behaviour of plates and shells made of functionally graded materials is derived in this paper using the updated Lagrangian approach. Derivation of the formulation is based on rewriting the Green–Lagrange strain as well as the 2nd Piola–Kirchhoff stress as two second‐order functions in terms of a through‐the‐thickness parameter. Material properties are assumed to vary through the thickness according to the commonly used power law distribution of the volume fraction of the constituents. Within a non‐linear finite element analysis framework, the main focus of the paper is the proposal of a formulation to account for non‐linear stress distribution in FG plates and shells, particularly, near the inner and outer surfaces for small and large values of the grading index parameter. The non‐linear heat transfer equation is also solved for thermal distribution through the thickness by the Rayleigh–Ritz method. Advantages of the proposed approach are assessed and comparisons with available solutions are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Remarks on the interpretation of current non‐linear finite element analyses as differential–algebraic equations

For the numerical solution of materially non‐linear problems like in computational plasticity or viscoplasticity the finite element discretization in space is usually coupled with point‐wise defined evolution equations characterizing the material behaviour. The interpretation of such systems as differential–algebraic equations (DAE) allows modern‐day integration algorithms from Numerical Mathematics to be efficiently applied. Especially, the application of diagonally implicit Runge–Kutta methods (DIRK) together with a Multilevel‐Newton method preserves the algorithmic structure of current finite element implementations which are based on the principle of virtual displacements and on backward Euler schemes for the local time integration. Moreover, the notion of the consistent tangent operator becomes more obvious in this context. The quadratical order of convergence of the Multilevel‐Newton algorithm is usually validated by numerical studies. However, an analytical proof of this second order convergence has already been given by authors in the field of non‐linear electrical networks. We show that this proof can be applied in the current context based on the DAE interpretation mentioned above. We finally compare the proposed procedure to several well‐known stress algorithms and show that the inclusion of a step‐size control based on local error estimations merely requires a small extra time‐investment. Copyright © 2001 John Wiley & Sons, Ltd.