Parallel programming of a peridynamics code coupled with finite element method

Using OpenMP (the Open Multi- Processing application programming interface), dynamic peridynamics code coupled with a finite element method is parallelized. The parallel implementation improves run-time efficiency and makes the realistic simulation of crack coalescence possible. To assess the accuracy and efficiency of the parallel code, we investigate its speedup and scalability. In addition, to validate the parallel code, experimental results for crack coalescence development sequences are compared. It is noted that this parallelized code markedly reduces computation time along with the coupling scheme. Moreover, the coupling approach used in this parallel code enables a more realistic and feasible numerical prediction of coalescing fractures. With the parallel implementation, two main types of crack coalescences between two flaws, formed by two short shear cracks and by a short central tensile segment and subsequent shear cracks are in detail discussed in terms of their development sequences. Consequently, this proposed coupled peridynamics code can be used to efficiently solve actual coalescence development sequences, thereby providing a numerical solution for fracture mechanics.     

A hybrid finite element/method of moment formulation for singlefrequency eddy current inversion

Eddy-current inverse techniques using single-frequency currents have been applied with limited success to the reconstruction of crack width and thickness profiles primarily for one-dimensional and axisymmetric geometries. Because of the diffusive nature of the induced low-frequency eddy currents, the reconstruction process differs from high-frequency wave propagation methods. On the physical basis that both diffusive and wave phenomena can be described by the same Green's function with either a complex or real wave number, an integral formulation for the low-frequency magnetic vector potential is presented. By employing an iterative Born approximation algorithm and the method of moments, a reconstruction method for the conductivity profile in a metallic specimen is developed. To make this formulation amenable to complex geometries, finite-element analysis techniques are utilized to compute the integral kernel. The inversion process is tested with synthetic data generated by the numerical solution of a generic embedded flaw in a full-space and a surface-breaking defect     

An embedded atom hyperelastic constitutive model and multiscale cohesive finite element method

Based on embedded atom method (EAM), an embedded atom hyperelastic (EAH) constitutive model is developed. The proposed EAH constitutive model provides a multiscale formalism to determine mesoscale or macroscale material behavior by atomistic information. By combining the EAH with cohesive zone model (CZM), a multiscale embedded atom cohesive finite element model (EA-cohesive FEM) is developed for simulating failure of materials at mesoscale and macroscale, e.g. fracture and crack propagation etc. Based on EAH, the EA-cohesive FEM applies the Cauchy-Born rule to calculate mesoscale or macroscale material response for bulk elements. Within the cohesive zone, a generalized Cauchy-Born rule is applied to find the effective normal and tangential traction-separation cohesive laws of EAH material. Since the EAM is a realistic semi-empirical interatomic potential formalism, the EAH constitutive model and the EA-cohesive FEM are physically meaningful when it is compared with experimental data. The proposed EA-cohesive FEM is validated by comparing the simulation results with the results of large scale molecular dynamics simulation. Simulation result of dynamic crack propagation is presented to demonstrate the capacity of EA-cohesive FEM in capturing the dynamic fracture.     

A restricted hybrid stress formulation based on a direct finite element method

A direct method, which uses stress and displacement modes obtained from the governing equations of a problem, is adopted for finite element formulation. It is shown that this method actually leads to a restricted hybrid stress formulation if the displacement modes are changed to ensure symmetry of the stiffness matrix. Through this direct method, however, the problem of selecting the appropriate number of stress modes in the regular hybrid stress model is bypassed. Only the minimum number of modes that are compatible with the number of nodal degrees-of-freedom of an element is needed in the formulation. Using more modes only leads to a combination of stress modes, and will not improve the order of performance of the element. It is shown through numerical examples that the restricted hybrid stress formulation leads to well-balanced elements.     

A hybrid smoothed extended finite element/level set method for modeling equilibrium shapes of nano-inhomogeneities

Interfacial energy plays an important role in equilibrium morphologies of nanosized microstructures of solid materials due to the high interface-to-volume ratio, and can no longer be neglected as it does in conventional mechanics analysis. When designing nanodevices and to understand the behavior of materials at the nano-scale, this interfacial energy must therefore be taken into account. The present work develops an effective numerical approach by means of a hybrid smoothed extended finite element/level set method to model nanoscale inhomogeneities with interfacial energy effect, in which the finite element mesh can be completely independent of the interface geometry. The Gurtin–Murdoch surface elasticity model is used to account for the interface stress effect and the Wachspress interpolants are used for the first time to construct the shape functions in the smoothed extended finite element method. Selected numerical results are presented to study the accuracy and efficiency of the proposed method as well as the equilibrium shapes of misfit particles in elastic solids. The presented results compare very well with those obtained from theoretical solutions and experimental observations, and the computational efficiency of the method is shown to be superior to that of its most advanced competitor.     

A hybrid finite element method for heterogeneous materials with randomly dispersed rigid inclusions

A hybrid finite element approach is proposed for the mechanical response of two-dimensional heterogeneous materials with linearly elastic matrix and randomly dispersed rigid circular inclusions of arbitrary sizes. In conventional finite element methods, many elements must be used to represent one inclusion. In this work, each inclusion is embedded inside a polygonal element and only one element is required to represent one inclusion. In numerically approximating stress and displacement distributions around the inclusion, classical elasticity solutions for a multiply-connected region are employed. A modified hybrid functional is used as the basis of the element formulation where the displacement boundary conditions of the element are automatically considered in a variational sense. The accuracy and efficiency of the proposed method are demonstrated by two boundary value problems. In one example, the results based on the proposed method with only 64 hybrid elements (450 degrees of freedom) are shown to be almost identical to those based on the traditional method with 2928 conventional elements (5526 degrees of freedom).     

A hybrid finite element model for multilayered plates

We present a finite element model for multilayered plates, based on a primal-hybrid variational formulation. Namely, each layer is analyzed as it were a lonely structure, and the displacement continuity is imposed from one layer to the other by means of Lagrange multipliers. Then, a Mindlin-like displacement field is assumed for any layer; the resulting continuous problem is proven to be well-posed under rather general hypotheses. Finally, a finite element model is deduced, using a very simple scheme (piecewise linear approximation for the displacement components and piecewise constant Lagrange multipliers). The numerical results assess the good performance of the proposed model.     

A hybrid finite element analysis of interface cracks

A versatile hybrid finite element scheme consisting of special crack-tip elements and crack face contact elements is developed to analyse a partially closed interface crack between two dissimilar anisotropic elastic materials. The crack-tip element incorporates higher-order asymptotic solutions for an interfacial crack tip. These solutions are obtained from complex variable methods in Stroh formalism. For a closed interfacial crack tip, a generalized contact model in which the crack-tip oscillation is eliminated is adopted in the calculation. The hybrid finite element modelling allows the stress singularity at an open and closed crack tip to be accurately treated. The accuracy and convergence of the developed scheme are tested with respect to the known interface crack solutions. Utilizing this numerical scheme, the stress intensity factors and contact zone are calculated for a finite interface crack between a laminated composite material.     

New hybrid Laplace transform/finite element method for three-dimensional transient heat conduction problem

The paper presents results obtained by the implementation of a new hybrid Laplace transform/finite element method developed by the authors. The present method removes the time derivatives from the governing differential equation using the Laplace transform and then solves the associated equation with the finite element method. Previously reported hybrid Laplace transform/finite element methods¹ have been confined to one nodal solution at a time. When applied to many nodes it takes an excessive amount of computer time. By using a similarity transform method on the matrix of the complex number coefficients this restriction is removed and the reported new method provides a more useful tool for the solution of linear transient problems. Test examples are used to show that the basic accuracy is comparable to that obtainable by analytical, finite difference and finite element methods.     

An arbitrary multi-node extended multiscale finite element method for thermoelastic problems with polygonal microstructures

International Journal of Mechanics and Materials in Design - A coupling extended multiscale finite element method (P-CEMsFEM) is developed for the numerical analysis of thermoelastic problems with...     

Analysis of cracked anisotropic plates using the hybrid finite element method

This paper presents a convenient and efficient method to obtain accurate stress intensity factors for cracked anisotropic plates. In this method, a complex variable formulation in conjunction with a hybrid displacement finite element scheme is used to carry out the stiffness and stress calculations of finite cracked plates subjected to general boundary and loading conditions. Unlike other numerical methods used for local analysis such as the boundary element method, the present method results in a symmetric stiffness matrix, which can be directly incorporated into the stiffness matrix representing other structural parts modeled by conventional finite elements. Therefore, the present method is ideally suited for modeling cracked plates in a large complex structure.     

A hybrid finite element method for near bodies of revolution (EM scattering)

A genetic formulation for a hybrid finite element solution for three-dimensional electromagnetic scattering is given using the equivalent current approach. The major computational tasks involved in monostatic scattering calculations are analyzed and compared as a function of the method of implementing the near-field radiation condition, i.e. method of moments, model expansion, and body of revolution (BOR). A method utilizing a BOR formulation that addresses these computational issues is given. This BOR implementation utilizes Hermite cubic basis functions and a variable number of modes per basis function in order to achieve the greatest efficiency. The combined field integral equation formulation is used to eliminate nonphysical resonance of the mesh boundary. Examples are given showing the efficiency and accuracy of this BOR code by itself, and as part of this hybrid finite-element method.<>     

Extended multiscale finite element method for elasto-plastic analysis of 2D periodic lattice truss materials

An extended multiscale finite element method is developed for small-deformation elasto-plastic analysis of periodic truss materials. The base functions constructed numerically are employed to establish the relationship between the macroscopic displacement and the microscopic stress and strain. The unbalanced nodal forces in the micro-scale of unit cells are treated as the combined effects of macroscopic equivalent forces and microscopic perturbed forces, in which macroscopic equivalent forces are used to solve the macroscopic displacement field and microscopic perturbed forces are used to obtain the stress and strain in the micro-scale to make sure the correctness of the results obtained by the downscale computation in the elastic-plastic problems. Numerical examples are carried out and the results verify the validity and efficiency of the developed method by comparing it with the conventional finite element method.     

Alternate hybrid stress finite element models

Alternate hybrid stress finite element models in which the internal equilibrium equations are satisfied on the average only, while the equilibrium equations along the interelement boundaries and the static boundary conditions are adhered to exactly a priori, are developed. The variational principle and the corresponding finite element formulation, which allows the standard direct stiffness method of structural analysis to be used, are discussed. Triangular elements for a moderately thick plate and a doubly-curved shallow thin shell are developed. Kinematic displacement modes, convergence criteria and bounds for the direct flexibility-influence coefficient are examined.     

A coupled meshfree/finite element method for automotive crashworthiness simulations

Mesh distortion induced numerical instability is a major roadblock in automotive crashworthiness finite element simulations. Remedies such as wrapping elements with null shells and deletion of distorted meshes have been adopted but none of them seems robust enough to survive various scenarios. Meshfree methods have been developed over the past almost twenty years in view of their capabilities in dealing with large material deformation and separation, but have remained in academic research due to their unaffordable high computational cost in solving large-scale industrial applications. This paper presents a coupled meshfree/finite-element method which allows engineers to model the severe deformation area with the meshfree method while keeping the remaining area modeled by the finite element methods. The method is implemented into LS-DYNA version 971 and its later versions so that it is available for automotive crashworthiness simulations. In the paper, one linear patch test and three crash examples are presented to demonstrate the accuracy of the meshfree formulation, its effectiveness in resolving mesh distortion difficulty, and the efficiency of the coupled meshfree/finite element solver in handling large-scale models.     

A finite element method for non-self-adjoint problems

This paper presents a general theory and application of the finite element method for some special class of non-self-adjoint problems. The formulation employed here is based on the Galerkin method for linear boundary value and eigenvalue problems described by the partial differential equations of elliptic type, and it can be regarded as an extension of the usual displacement method formulated by the use of the principle of minimum potential energy. In order to illustrate its validity and feasibility, the method is applied to the problems of the two-group neutron diffusion equations and of the stability of a non-conservative system.     

A new finite element method in micromagnetics

A finite-element method is presented in which the magnetization is linearly interpolated within each tetrahedral element and the magnetostatic interaction is accurately obtained by integration. The equilibrium magnetizations at the nodal points can be found by minimizing the total energy of the system. Two different minimization schemes are compared     

A refined global-local finite element analysis method

A refined global-local method was proposed to improve the efficiency of finite element analysis. The proposed method was based on the regular finite element method in conjunction with three basic step, i.e. the global analysis, the local analysis and the refined global analysis. In the first two steps, a coarse finite element mesh was used to analyse the entire structure to obtain the nodal displacements which were subsequently used as displacement boundary conditions for local regions of interest. These local regions with the prescribed boundary conditions were then analysed with refined meshes to obtain more accurate stresses. In the third step, a new global displacement distribution based on the results of the previous two steps was assumed for the analysis, from which much improved solutions for both stresses and displacements were produced. Numerical examples showed that the proposed method yielded accurate solutions with significant savings in computing time compared with the regular finite element method. Further, this method is suitable for parallel computation.