Upper and lower bounds for natural frequencies: A property of the smoothed finite element methods

Node‐based smoothed finite element method (NS‐FEM) using triangular type of elements has been found capable to produce upper bound solutions (to the exact solutions) for force driving static solid mechanics problems due to its monotonic ‘soft’ behavior. This paper aims to formulate an NS‐FEM for lower bounds of the natural frequencies for free vibration problems. To make the NS‐FEM temporally stable, an α‐FEM is devised by combining the compatible and smoothed strain fields in a partition of unity fashion controlled by α∈[0, 1], so that both the properties of stiff FEM and the monotonically soft NS‐FEM models can be properly combined for a desired purpose. For temporally stabilizing NS‐FEM, α is chosen small so that it acts like a ‘regularization parameter’ making the NS‐FEM stable, but still with sufficient softness ensuring lower bounds for natural frequency solution. Our numerical studies demonstrate that (1) using a proper α, the spurious non‐zero energy modes can be removed and the NS‐FEM becomes temporally stable; (2) the stabilized NS‐FEM becomes a general approach for solids to obtain lower bounds to the exact natural frequencies over the whole spectrum; (3) α‐FEM can even be tuned for obtaining nearly exact natural frequencies. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

An edge‐based smoothed finite element method for 3D analysis of solid mechanics problems

The edge‐based smoothed finite element method (ES‐FEM) was proposed recently in Liu, Nguyen‐Thoi, and Lam to improve the accuracy of the FEM for 2D problems. This method belongs to the wider family of the smoothed FEM for which smoothing cells are defined to perform the numerical integration over the domain. Later, the face‐based smoothed FEM (FS‐FEM) was proposed to generalize the ES‐FEM to 3D problems. According to this method, the smoothing cells are centered along the faces of the tetrahedrons of the mesh. In the present paper, an alternative method for the extension of the ES‐FEM to 3D is investigated. This method is based on an underlying mesh composed of tetrahedrons, and the approximation of the field variables is associated with the tetrahedral elements; however, in contrast to the FS‐FEM, the smoothing cells of the proposed ES‐FEM are centered along the edges of the tetrahedrons of the mesh. From selected numerical benchmark problems, it is observed that the ES‐FEM is characterized by a higher accuracy and improved computational efficiency as compared with linear tetrahedral elements and to the FS‐FEM for a given number of degrees of freedom. Copyright © 2013 John Wiley & Sons, Ltd.     

Application of virtual crack closure integral method for interface cracks in low-k integrated circuit devices under thermal load

The virtual crack closure integral (VCCI) method is used to evaluate the stress intensity factor (SIF) and energy release rate (ERR) of an interface crack under thermal load. The VCCIs used in this work include the traditionally known “Mode I” and “Mode II” VCCIs and an additional coupling VCCI. The singularity element is used in the finite element method (FEM) implementation. The SIF and ERR calculated by the FEM are compared to the exact solution in the case of a joint dissimilar semi-infinite plates with double edge crack under thermal loading. The FEM result agrees well with the exact solution for relatively coarse meshes. The contribution of the mesh density and material mismatch to the FEM error is also explored. The VCCI method is used with the multi-scale FEM in a delamination risk assessment of a low-k integrated circuits device in flip-chip plastic ball grid array packages. The ERR is calculated for different package configurations and the prediction of the delamination risk is confirmed by reliability tests.     

Overcoming element quality dependence of finite elements with adaptive extended stencil FEM (AES‐FEM)

The finite element methods (FEMs) are important techniques in engineering for solving partial differential equations, but they depend heavily on element shape quality for stability and good performance. In this paper, we introduce the Adaptive Extended Stencil Finite Element Method (AES‐FEM) as a means for overcoming this dependence on element shape quality. Our method replaces the traditional basis functions with a set of generalized Lagrange polynomial basis functions, which we construct using local weighted least‐squares approximations. The method preserves the theoretical framework of FEM and allows imposing essential boundary conditions and integrating the stiffness matrix in the same way as the classical FEM. In addition, AES‐FEM can use higher‐degree polynomial basis functions than the classical FEM, while virtually preserving the sparsity pattern of the stiffness matrix. We describe the formulation and implementation of AES‐FEM and analyze its consistency and stability. We present numerical experiments in both 2D and 3D for the Poisson equation and a time‐independent convection–diffusion equation. The numerical results demonstrate that AES‐FEM is more accurate than linear FEM, is also more efficient than linear FEM in terms of error versus runtime, and enables much better stability and faster convergence of iterative solvers than linear FEM over poor‐quality meshes. Copyright © 2016 John Wiley & Sons, Ltd.     

A two-dimensional distributed feedback used for synchronization of a multibeam planar free-electron maser system

We suggest to use a two-dimensional distributed feedback for synchronizing the radiation of a multibeam generator representing a system of planar free-electron maser (FEM) units, each FEM being power supplied with a ribbon-shaped relativistic electron beam. It is demonstrated that various FEM units can be coupled by transverse electromagnetic energy fluxes arising in two-dimensional Bragg structures.     

Adaptive modeling of damage growth using a coupled FEM/BEM approach

We propose a coupled boundary element method (BEM) and a finite element method (FEM) for modelling localized damage growth in structures. BEM offers the flexibility of modelling large domains efficiently, while the non‐linear damage growth is accurately accounted by a local FEM mesh. An integral‐type nonlocal continuum damage mechanics with adapting FEM mesh is used to model multiple damage zones and follow their propagation in the structure. Strong form coupling, BEM hosted, is achieved using Lagrange multipliers. Because the non‐linearity is isolated in the FEM part of the system of equations, the system size is reduced using Schur complement approach, then the solution is obtained by a monolithic Newton method that is used to solve both domains simultaneously. The coupled BEM/FEM approach is verified by a set of convergence studies, where the reference solution is obtained by a fine FEM. In addition, the method is applied to multiple fractures growth benchmark problems and shows good agreement with the literature. Copyright © 2015 John Wiley & Sons, Ltd.     

High-order Finite-element Method to Solve the Nonlinear Coupled-wave Equations for Degenerate Two-wave and Four-wave Mixing

Abstract

We use the one-dimensional finite-element method (FEM) to solve the nonlinear coupled-wave equations for degenerate two-wave mixing (DTWM) and four-wave mixing (DFWM) in a reflection geometry. High-order (quadratic) elements have been used. A comparison is made between the results obtained using the high-order FEM, and those using the first-order FEM, which have been reported previously. High accuracy is achieved using the high-order FEM which employs substantially fewer elements.     

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.     

A locking‐free selective smoothed finite element method using tetrahedral and triangular elements with adaptive mesh rezoning for large deformation problems

A novel finite element (FE) formulation with adaptive mesh rezoning for large deformation problems is proposed. The proposed method takes the advantage of the selective smoothed FE method (S‐FEM), which has been recently developed as a locking‐free FE formulation with strain smoothing technique. We adopt the selective face‐based smoothed/node‐based smoothed FEM (FS/NS‐FEM‐T4) and edge‐based smoothed/node‐based smoothed FEM (ES/NS‐FEM‐T3) basically but modify them partly so that our method can handle any kind of material constitutive models other than elastic models. We also present an adaptive mesh rezoning method specialized for our S‐FEM formulation with material constitutive models in total form. Because of the modification of the selective S‐FEMs and specialization of adaptive mesh rezoning, our method is locking‐free for severely large deformation problems even with the use of tetrahedral and triangular meshes. The formulation details for static implicit analysis and several examples of analysis of the proposed method are presented in this paper to demonstrate its efficiency. Copyright © 2014 John Wiley & Sons, Ltd.     

Selective smoothed finite element methods for extremely large deformation of anisotropic incompressible bio‐tissues

Two dynamic selective smoothed FEM (selective S‐FEM) are proposed for analysis of extremely large deformation of anisotropic incompressible bio‐tissues using the simplest four‐node tetrahedron elements. In the present two Selective S‐FEMs, the method that consists of the face‐based smoothed FEM (FS‐FEM) used for the deviatoric part of deformation and the node‐based smoothed FEM (NS‐FEM) used for the volumetric part is called FS/NS‐FEM; another method that replaces the deviatoric part of deformation in the first one by the edge‐based smoothed FEM (3D‐ES‐FEM) is call 3D‐ES/NS‐FEM. Both selective S‐FEMs can achieve outstanding accuracy, and stability of volumetric locking free. This is because the NS‐FEM offers an ‘overly‐soft’ feature (in contrast to the standard FEM ‘overly‐stiff’ model), which can be used to effectively mitigate the volumetric locking, and on the other hand, the 3D‐ES‐FEM and FS‐FEM produce close to exact stiffness for the discretized model leading to accurate solution. Numerical examples are presented to examine the performance of the selective S‐FEM methods, including soft bio‐tissues that may be isotropic, transversely isotropic, and anisotropic arterial layered materials. The present methods are found having good accuracy and performance. The examples also demonstrate that the proposed methods are very robust and possess remarkable capabilities of handling element distortion, which is very useful for simulating soft materials including bio‐tissues. Copyright © 2014 John Wiley & Sons, Ltd.     

A quasi‐static crack growth simulation based on the singular ES‐FEM

In the edge‐based smoothed finite element method (ES‐FEM), one needs only the assumed displacement values (not the derivatives) on the boundary of the edge‐based smoothing domains to compute the stiffness matrix of the system. Adopting this important feature, a five‐node crack‐tip element is employed in this paper to produce a proper stress singularity near the crack tip based on a basic mesh of linear triangular elements that can be generated automatically for problems with complicated geometries. The singular ES‐FEM is then formulated and used to simulate the crack propagation in various settings, using a largely coarse mesh with a few layers of fine mesh near the crack tip. The results demonstrate that the singular ES‐FEM is much more accurate than X‐FEM and the existing FEM. Moreover, the excellent agreement between numerical results and the reference observations shows that the singular ES‐FEM offers an efficient and high‐quality solution for crack propagation problems. Copyright © 2011 John Wiley & Sons, Ltd.     

3D non-linear time domain FEM–BEM approach to soil–structure interaction problems

Dynamic soil–structure interaction is concerned with the study of structures supported on flexible soils and subjected to dynamic actions. Methods combining the finite element method (FEM) and the boundary element method (BEM) are well suited to address dynamic soil–structure interaction problems. Hence, FEM–BEM models have been widely used. However, non-linear contact conditions and non-linear behavior of the structures have not usually been considered in the analyses. This paper presents a 3D non-linear time domain FEM–BEM numerical model designed to address soil–structure interaction problems. The BEM formulation, based on element subdivision and the constant velocity approach, was improved by using interpolation matrices. The FEM approach was based on implicit Green's functions and non-linear contact was considered at the FEM–BEM interface. Two engineering problems were studied with the proposed methodology: the propagation of waves in an elastic foundation and the dynamic response of a structure to an incident wave field.     

Numerical solutions for unsteady flow in unconfined aquifers

Two numerical methods for solving the problem of unsteady flow in unconfined aquifers are studied. They are an explicit finite difference method (FDM), and the finite element method (FEM). The FEM is further subdivided into three schemes: vertical displacement approach, explicit scheme (FEM1), normal velocity approach, explicit scheme (FEM2), and vertical displacement approach, implicit scheme (FEM3). Results from the above methods are compared with experimental results from a sand box model. Various factors affecting the accuracy and numerical stability are investigated. Results indicate that, for a similar degree of accuracy, the FEM requires less computational effort than the explicit FDM. Amongst the three FEM schemes, FEM3 appears to be most attractive as it is the most stable and economical of the three schemes compared.     

GPGPU-based parallel computing applied in the FEM using the conjugate gradient algorithm: a review

Parallelization of the finite-element method (FEM) has been contemplated by the scientific and high-performance computing community for over a decade. Most of the computations in the FEM are related to linear algebra that includes matrix and vector computations. These operations have the single-instruction multiple-data (SIMD) computation pattern, which is beneficial for shared-memory parallel architectures. General-purpose graphics processing units (GPGPUs) have been effectively utilized for the parallelization of FEM computations ever since 2007. The solver step of the FEM is often carried out using conjugate gradient (CG)-type iterative methods because of their larger convergence rates and greater opportunities for parallelization. Although the SIMD computation patterns in the FEM are intrinsic for GPU computing, there are some pitfalls, such as the underutilization of threads, uncoalesced memory access, lower arithmetic intensity, limited faster memories on GPUs and synchronizations. Nevertheless, FEM applications have been successfully deployed on GPUs over the last 10 years to achieve a significant performance improvement. This paper presents a comprehensive review of the parallel optimization strategies applied in each step of the FEM. The pitfalls and trade-offs linked to each step in the FEM are also discussed in this paper. Furthermore, some extraordinary methods that exploit the tremendous amount of computing power of a GPU are also discussed. The proposed review is not limited to a single field of engineering. Rather, it is applicable to all fields of engineering and science in which FEM-based simulations are necessary.     

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.     

AN OBJECT-ORIENTED ARCHITECTURE FOR A FINITE ELEMENT METHOD KNOWLEDGE-BASED SYSTEM

A finite element method (FEM) system is complex in nature and still faces bottleneck problems in its maintenance, extension, etc., and it is yet necessary to be dealt with using some new methodologies. An object-orientation is a promising paradigm for treating complexities. In this paper, first an object-oriented FEM knowledge base system architecture is proposed. Then, an object model in the FEM domain is established through entity abstractions, action abstractions, category abstractions, agent abstractions, etc. Finally, a semantic network results from a semantic analysis of the FEM object model to represent the generative relationships among the FEM objects. Through an illustrative example, it is shown that a control task of finite element elasto-static analysis can be represented by a traveling path in the semantic network.     

A quasi‐implicit characteristic–based penalty finite‐element method for incompressible laminar viscous flows

In this paper, a novel characteristic–based penalty (CBP) scheme for the finite‐element method (FEM) is proposed to solve 2‐dimensional incompressible laminar flow. This new CBP scheme employs the characteristic‐Galerkin method to stabilize the convective oscillation. To mitigate the incompressible constraint, the selective reduced integration (SRI) and the recently proposed selective node–based smoothed FEM (SNS‐FEM) are used for the 4‐node quadrilateral element (CBP‐Q4SRI) and the 3‐node triangular element (CBP‐T3SNS), respectively. Meanwhile, the reduced integration (RI) for Q4 element (CBP‐Q4RI) and NS‐FEM for T3 element (CBP‐T3NS) with CBP scheme are also investigated. The quasi‐implicit CBP scheme is applied to allow a large time step for sufficient large penalty parameters. Due to the absences of pressure degree of freedoms, the quasi‐implicit CBP‐FEM has higher efficiency than quasi‐implicit CBS‐FEM. In this paper, the CBP‐Q4SRI has been verified and validated with high accuracy, stability, and fast convergence. Unexpectedly, CBP‐Q4RI is of no instability, high accuracy, and even slightly faster convergence than CBP‐Q4SRI. For unstructured T3 elements, CBP‐T3SNS also shows high accuracy and good convergence but with pressure oscillation using a large penalty parameter; CBP‐T3NS produces oscillated wrong velocity and pressure results. In addition, the applicable ranges of penalty parameter for different proposed methods have been investigated.     

高层建筑岩石洞室地基稳定性分析方法与应用

随着地面与地下空间的综合开发利用日愈增多,地面建筑与地下工程的冲突也日愈突出,地处山区的山城重庆,这类问题尤为突出,研究岩石洞室地基的稳定性已成为重庆市建设中急需解决的一项重要课题。论文结合重庆市岩石洞室地基工程,通过理论研究、数值模拟实验、现场测试和智能识别等技术,较为全面系统地分析了岩石洞室地基的稳定性问题,明确了开洞地基和地基开洞的安全系数确定方法,建立了岩石洞室地基的智能识别系统,提出了研究复杂洞室地基稳定性的综合分析方法推荐思路框架。     

Hybrid Finite Element Method Based on Novel General Solutions for Helmholtz-Type Problems

This paper presents a hybrid finite element model (FEM) with a new type of general solution as interior trial functions, named as HGS-FEM. A variational functional corresponding to the proposed general solution is then constructed for deriving the element stiffness matrix of the proposed element model and the corresponding existence of extremum is verified. Then the assumed intra-element potential field is constructed by a linear combination of novel general solutions at the points on the element boundary under consideration. Furthermore, the independent frame field is introduced to guarantee the intra-element continuity. The present scheme inherits the advantages of hybrid Trefftz FEM (HT-FEM) over the conventional FEM and BEM, and avoids the difficulty in choosing appropriate terms of Trefftz functions in HT-FEM and also removing the troublesome for determining fictitious boundary in hybrid fundamental solution-based FEM (HFS-FEM). The efficiency and accuracy of the proposed model are assessed through several numerical examples.