Thermo‐elasto‐plastic finite element analysis of quasi‐state processes in Eulerian reference frames

An Eulerian finite element formulation for quasi‐state one way coupled thermo‐elasto‐plastic systems is presented. The formulation is suitable for modeling material processes such as welding and laser surfacing. In an Eulerian frame, the solution field of a quasi‐state process becomes steady state for the heat transfer problem and static for the stress problem. A mixed small deformation displacement elasto‐plastic formulation is proposed. The formulation accounts for temperature dependent material properties and exhibits a robust convergence. Streamline upwind Petrov–Galerkin (SUPG) is used to remove spurious oscillations. Smoothing functions are introduced to relax the non‐differentiable evolution equations and allow for the use of gradient (stiffness) solution scheme via the Newton–Raphson method. A 3‐dimensional simulation of a laser surfacing process is presented to exemplify the formulation. Results from the Eulerian formulation are in good agreement with results from the conventional Lagrangian formulation. However, the Eulerian formulation is approximately 15 times faster than the Lagrangian. Copyright © 2001 John Wiley & Sons, Ltd.     

Isogeometric shape optimization for quasi‐static processes

A framework to solve shape optimization problems for quasi‐static processes is developed and implemented numerically within the context of isogeometric analysis (IGA). Recent contributions in shape optimization within IGA have been limited to static or steady‐state loading conditions. In the present contribution, the formulation of shape optimization is extended to include time‐dependent loads and responses. A general objective functional is used to accommodate both structural shape optimization and passive control for mechanical problems. An adjoint sensitivity analysis is performed at the continuous level and subsequently discretized within the context of IGA. The methodology and its numerical implementation are tested using benchmark static problems of optimal shapes of orifices in plates under remote bi‐axial tension and pure shear. Under quasi‐static loading conditions, the method is validated using a passive control approach with an a priori known solution. Several applications of time‐dependent mechanical problems are shown to illustrate the capabilities of this approach. In particular, a problem is considered where an external load is allowed to move along the surface of a structure. The shape of the structure is modified in order to control the time‐dependent displacement of the point where the load is applied according to a pre‐specified target. Copyright © 2015 John Wiley & Sons, Ltd.     

Interface‐reduction for the Craig–Bampton and Rubin method applied to FE–BE coupling with a large fluid–structure interface

Component mode‐based model‐order reduction (MOR) methods like the Craig–Bampton method or the Rubin method are known to be limited to structures with small coupling interfaces. This paper investigates two interface‐reduction methods for application of MOR to systems with large coupling interfaces: for the Craig–Bampton method a direct reduction method based on strain energy considerations is investigated. Additionally, for the Rubin method an iterative reduction scheme is proposed, which incrementally constructs the reduction basis. Hereby, attachment modes are tested if they sufficiently enlarge the spanned subspace of the current reduction basis. If so, the m‐orthogonal part is used to augment the basis. The methods are applied to FE–BE coupled systems in order to predict the vibro‐acoustic behavior of structures, which are partly immersed in water. Hereby, a strong coupling scheme is employed, since for dense fluids the feedback of the acoustic pressure onto the structure is not negligible. For two example structures, the efficiency of the reduction methods with respect to numerical effort, memory consumption and computation time is compared with the exact full‐order solution. Copyright © 2008 John Wiley & Sons, Ltd.     

Quasi‐static and Impact Responses of Multi‐layered Corrugated Paperboard Cushion by Virtual Mass Method

Quasi‐static compressive and impact behaviours of multi‐layered corrugated paperboard (MLCP) cushioning structure were analysed by a recently proposed virtual mass method. First, virtual mass method was applied and verified analytically to solve quasi‐static compressive responses for representative two‐layer corrugated paperboard cushioning structure. The results show that the two layers in the cushioning structure reach the buckling state in chronological order because of the existence of the small perturbations triggered by inertial force related to virtual mass, which leads to the two typical stress peaks in stress–strain curves. Second, the quasi‐static compressive behaviours of MLCP cushioning structure were further studied numerically, showing that the buckling order of multi‐layer cushioning structure depends on virtual mass, but the stress–strain curves remain unchanged when the virtual mass is smaller than some certain value. Finally, quasi‐static and dynamic impact tests of MLCP cushioning structure composed of C‐flute corrugated paperboard were carried out to further validate the capacity of the virtual mass method to describe layer‐wise collapse mechanism given the constitutive relationship of the monolayer corrugated paperboard. Copyright © 2014 John Wiley & Sons, Ltd.     

An upscale theory of particle simulation for two‐dimensional quasi‐static problems

An upscale theory of the particle simulation, which is based on the distinct element method, is presented for two‐dimensional quasi‐static problems. Since the present upscale theory is comprised of four similarity criteria between different length‐scale particle‐simulation models, it reveals the intrinsic relationship between the particle‐simulation solution obtained from a small length‐scale (e.g. a laboratory length‐scale) model and that obtained from a large length‐scale (e.g. a geological length‐scale) one. The present upscale theory is of significant theoretical value in the particle simulation of two‐dimensional systems, at least from the following two points of view. (1) If the mechanical response of a particle model of a small length‐scale is used to indirectly investigate that of a large length‐scale, then the present upscale theory provides the necessary conditions, under which the particle model of the small length‐scale needs to be satisfied so that a similarity between the mechanical responses of two different length‐scale particle models can be maintained. (2) If a particle model of a large length‐scale is used to directly investigate the mechanical response of the model, then the present upscale theory can be used to determine the necessary particle‐scale mechanical properties from the macroscopic mechanical properties that are obtained from either a laboratory test or an in situ measurement. The related simulation results from two typical examples of significantly different length‐scales (i.e. a metre‐scale and a kilometre‐scale) have demonstrated the usefulness and correctness of the proposed upscale theory for simulating different length‐scale problems in quasi‐static geological systems. Copyright © 2007 John Wiley & Sons, Ltd.     

An improved continued‐fraction‐based high‐order transmitting boundary for time‐domain analyses in unbounded domains

A high‐order local transmitting boundary to model the propagation of acoustic or elastic, scalar or vector‐valued waves in unbounded domains of arbitrary geometry is proposed. It is based on an improved continued‐fraction solution of the dynamic stiffness matrix of an unbounded medium. The coefficient matrices of the continued‐fraction expansion are determined recursively from the scaled boundary finite element equation in dynamic stiffness. They are normalised using a matrix‐valued scaling factor, which is chosen such that the robustness of the numerical procedure is improved. The resulting continued‐fraction solution is suitable for systems with many DOFs. It converges over the whole frequency range with increasing order of expansion and leads to numerically more robust formulations in the frequency domain and time domain for arbitrarily high orders of approximation and large‐scale systems. Introducing auxiliary variables, the continued‐fraction solution is expressed as a system of linear equations in iω in the frequency domain. In the time domain, this corresponds to an equation of motion with symmetric, banded and frequency‐independent coefficient matrices. It can be coupled seamlessly with finite elements. Standard procedures in structural dynamics are directly applicable in the frequency and time domains. Analytical and numerical examples demonstrate the superiority of the proposed method to an existing approach and its suitability for time‐domain simulations of large‐scale systems. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical simulation of substation grounding grids buried in both horizontal and vertical multilayer earth model

In the simulation of quasi‐static electromagnetic fields produced by a point source located in both horizontal and vertical multilayer media, the conventional image method requires an infinite number of images to get an accurate solution, while the method of quasi‐static complex images needs only a few ones. Based on the method of quasi‐static complex images, the closed form of Green's function of a point source in both horizontal and vertical multilayer earth model is derived through matrix pencil (MP) approach. The fast convergent Galerkin's type of boundary element method (BEM) is taken to simulate and analyse a grounding system including floating electrode with any complicated structure, which can be located anywhere in horizontal or vertical multilayer earth model. Copyright © 2006 John Wiley & Sons, Ltd.     

A topological optimization approach for structural design of a high‐speed low‐load mechanism using the equivalent static loads method

In high‐speed low‐load mechanisms, the principal loads are the inertial forces caused by the high accelerations and velocities. Hence, mechanical design should consider lightweight structures to minimize such loads. In this paper, a topological optimization method is presented on the basis of the equivalent static loads method. Finite element (FE) models of the mechanism in different positions are constructed, and the equivalent loads are obtained using flexible multibody dynamics simulation. Kinetic DOFs are used to simulate the motion joints, and a quasi‐static analysis is performed to obtain the structural responses. The element sensitivity is calculated according to the static‐load‐equivalent equilibrium, in such a way that the influence on the inertial force is considered. A dimensionless component sensitivity factor (strain energy caused by unit load divided by kinetic energy from unit velocity) is used, which quantifies the significance of each element. Finally, the topological optimization approach is presented on the basis of the evolutionary structural optimization method, where the objective is to find the maximum ratio of strain energy to kinetic energy. In order to show the efficiency of the presented method, we presented two numerical cases. The results of these analyses show that the presented method is more efficient and can be easily implemented in commercial FE analysis software. Copyright © 2011 John Wiley & Sons, Ltd.     

Coupled three‐layered analysis of smart piezoelectric beams with different electric boundary conditions

This paper presents a theoretical and finite element (FE) formulation of a three‐layered smart beam with two piezoelectric layers acting as sensors or actuators. For the definition of the mechanical model a partial layerwise theory is considered for the approximation of the displacement field of the core and piezoelectric face layers. An electrical model for different electric boundary conditions (EBC), namely, electroded layers with either closed‐ or open‐circuit electrodes with electric potential prescribed or layers without electrodes, is considered. Using a variational formulation, the direct piezoelectric effect for the different EBC is physically incorporated into the mechanical model through appropriate approximations of the electric field in the axial and transverse directions. An FE model of a three‐layered smart beam with different EBC is proposed considering a fully coupled electro‐mechanical theory through the use of effective stiffness parameters and a modified static condensation. FE solutions of the quasi‐static electrical and mechanical actuations and natural frequencies are presented. Comparisons with numerical FE and analytical solutions available in the literature demonstrate the representativeness of the developed theory and the effectiveness of the proposed FE model for different EBC. Copyright © 2005 John Wiley & Sons, Ltd.     

Exact static element stiffness matrices of non‐symmetric thin‐walled curved beams

An evaluation procedure of exact static stiffness matrices for curved beams with non‐symmetric thin‐walled cross section are rigorously presented for the static analysis. Higher‐order differential equations for a uniform curved beam element are first transformed into a set of the first‐order simultaneous ordinary differential equations by introducing 14 displacement parameters where displacement modes corresponding to zero eigenvalues are suitably taken into account. This numerical technique is then accomplished via a generalized linear eigenvalue problem with non‐symmetric matrices. Next, the displacement functions of displacement parameters are exactly calculated by determining general solutions of simultaneous non‐homogeneous differential equations. Finally an exact stiffness matrix is evaluated using force–deformation relationships. In order to demonstrate the validity and effectiveness of this method, displacements and normal stresses of cantilever thin‐walled curved beams subjected to tip loads are evaluated and compared with those by thin‐walled curved beam elements as well as shell elements. Copyright © 2004 John Wiley & Sons, Ltd.     

Time‐domain soil–structure interaction analysis in two‐dimensional medium based on analytical frequency‐dependent infinite elements

A direct method for soil–structure interaction analysis in two‐dimensional medium is presented in time domain, which is based on the transformation of the analytical frequency‐dependent dynamic stiffness matrix. The present dynamic stiffness matrix for the far‐field region is constructed by assembling stiffness matrices of the analytical frequency‐dependent dynamic infinite elements, so that the equation of motion can be analytically transformed into the time‐domain equation. An efficient procedure is devised to evaluate the dynamic responses in time domain. Verification of the present formulation is carried out by comparing the compliances for a strip foundation on a homogeneous and layered half‐spaces with those obtained by other methods. Numerical analyses are also carried out for the transient responses of an elastic block and tunnel in a homogeneous and a layered half‐space. The comparisons with those by other approaches indicate that the proposed time‐domain method for soil–structure interaction analysis gives good solutions. Copyright © 2000 John Wiley & Sons, Ltd.     

Ritz analysis of vibrating rectangular and skew multilayered plates based on advanced variable-kinematic models

A variable-kinematic Ritz formulation based on two-dimensional higher-order layerwise and equivalent single-layer theories is described in this paper to accurately predict free vibration of thick and thin, rectangular and skew multilayered plates with clamped, free and simply-supported boundary conditions. The main result is the derivation at a layer level of so-called Ritz fundamental nuclei for the stiffness and mass matrices which are invariant with respect to both the assumed kinematic model and the type of Ritz functions. In this work, products of Chebyshev polynomials and boundary-compliant functions are chosen as admissible trial set. After studying the convergence of the method, its accuracy is evaluated, in terms of frequency parameters and through-the-thickness distribution of modal displacements, by comparison with some reference results available in the literature. Results for sandwich plates with soft core are given for the first time, which may serve as benchmark values for future research.     

广义特征值问题求解的改进Ｒｉｔｚ向量法

从提高算法的稳定性和计算效率入手,采取迭代及防止漏根、多根的措施,对传统的Ｒｉｔｚ向量法进行改进,提出改进的Ｒｉｔｚ向量法。此算法仅需生成ｒ维的Ｋｒｙｌｏｖ空间,大大降低投影矩阵阶数,减少投影矩阵特征值计算时间。引入重正交方案和模态比较法,并给出Ｒｉｔｚ向量块宽ｑ与生成步数ｒ的建议取值。最后通过四参数的谱变换法,不     

Particle simulation of spontaneous crack generation problems in large‐scale quasi‐static systems

To extend the application range of the distinct element method from a laboratory scale into a large scale such as a geological scale, we need to deal with an upscale issue associated with simulating spontaneous crack generation problems in large‐scale quasi‐static systems. Toward this direction, three important simulation issues, which may affect the quality of the particle simulation results of a quasi‐static system, have been addressed in details in this paper. The first simulation issue is how to determine the particle‐scale mechanical properties of a particle from the measured macroscopic mechanical properties of rocks. The second simulation issue is that the fictitious time, rather than the physical time, is used in the particle simulation of a quasi‐static problem. The third simulation issue is that the conventional loading procedure used in the distinct element method is conceptually inaccurate, at least from the force propagation point of view. A new loading procedure is proposed to solve the conceptual problem resulting from the third simulation issue. The proposed loading procedure is comprised of two main types of periods, a loading period and a frozen period. Using the proposed loading procedure, the parameter selection problem stemming from the first issue can be somewhat solved. Since the second issue is an inherent one, it is strongly recommended that a particle‐size sensitivity analysis of at least two different models, which have the same geometry but different smallest particle sizes, be carried out to confirm the particle simulation result of a large‐scale quasi‐static system. The related simulation results have demonstrated the usefulness and correctness of the proposed loading procedure for dealing with spontaneous crack generation problems in large‐scale quasi‐static geological systems. Copyright © 2006 John Wiley & Sons, Ltd.     

Adaptive frequency windowing for multifrequency solutions in structural acoustics based on the matrix Padé‐via‐Lanczos algorithm

For many problems in structural acoustics, it is desired to obtain solutions at many frequencies over a large range in the frequency domain. A reduced‐order multifrequency algorithm based on matrix Padé approximation, using the matrix Padé‐via‐Lanczos (MPVL) connection, has been previously used to solve both exterior and interior acoustic problems. However, the method is not guaranteed to give the correct solution across the entire frequency region of interest, but only locally around a reference frequency. An adaptive frequency windowing scheme is introduced to address this shortcoming for practical application of this method. The application of this algorithm to tightly coupled problems in interior structural acoustics is presented. Copyright © 2007 John Wiley & Sons, Ltd.     

A quasi‐static discontinuous Galerkin configurational force crack propagation method for brittle materials

This paper presents a framework for r‐adaptive quasi‐static configurational force (CF) brittle crack propagation, cast within a discontinuous Galerkin (DG) symmetric interior penalty (SIPG) finite element scheme. Cracks are propagated in discrete steps, with a staggered algorithm, along element interfaces, which align themselves with the predicted crack propagation direction. The key novelty of the work is the exploitation of the DG face stiffness terms existing along element interfaces to propagate a crack in a mesh‐independent r‐adaptive quasi‐static fashion, driven by the CF at the crack tip. This adds no new degrees of freedom to the data structure. Additionally, as DG methods have element‐specific degrees of freedom, a geometry‐driven p‐adaptive algorithm is also easily included allowing for more accurate solutions of the CF on a moving crack front. Further, for nondeterminant systems, we introduce an average boundary condition that restrains rigid body motion leading to a determinant system. To the authors' knowledge, this is the first time that such a boundary condition has been described. The proposed formulation is validated against single and multiple crack problems with single‐ and mixed‐mode cracks, demonstrating the predictive capabilities of the method.     

DSC‐Ritz method for the free vibration analysis of Mindlin plates

This paper introduces a novel method for the free vibration analysis of Mindlin plates. The proposed method takes the advantage of both the local bases of the discrete singular convolution (DSC) algorithm and the pb‐2 Ritz boundary functions to arrive at a new approach, called DSC‐Ritz method. Two basis functions are constructed by using DSC delta sequence kernels of the positive type. The energy functional of the Mindlin plate is represented by the newly constructed basis functions and is minimized under the Ritz variational principle. Extensive numerical experiments are considered by different combinations of boundary conditions of Mindlin plates of rectangular and triangular shapes. The performance of the proposed method is carefully validated by convergence analysis. The frequency parameters agree very well with those in the literature. Numerical experiments indicate that the proposed DSC‐Ritz method is a very promising new method for vibration analysis of Mindlin plates. Copyright © 2004 John Wiley & Sons, Ltd.     

An efficient sparse matrix‐vector multiplication on CUDA‐enabled graphic processing units for finite element method simulations

Finite element method (FEM) is a well‐developed method to solve real‐world problems that can be modeled with differential equations. As the available computational power increases, complex and large‐size problems can be solved using FEM, which typically involves multiple degrees of freedom (DOF) per node, high order of elements, and an iterative solver requiring several sparse matrix‐vector multiplication operations. In this work, a new storage scheme is proposed for sparse matrices arising from FEM simulations with multiple DOF per node. A sparse matrix‐vector multiplication kernel and its variants using the proposed scheme are also given for CUDA‐enabled GPUs. The proposed scheme and the kernels rely on the mesh connectivity data from FEM discretization and the number of DOF per node. The proposed kernel performance was evaluated on seven test matrices for double‐precision floating point operations. The performance analysis showed that the proposed GPU kernel outperforms the ELLPACK (ELL) and CUSPARSE Hybrid (HYB) format GPU kernels by an average of 42% and 32%, respectively, on a Tesla K20c card. Copyright © 2016 John Wiley & Sons, Ltd.     

Non‐linear analysis of thin‐walled members of open cross‐section

The paper presents a means of determining the non‐linear stiffness matrices from expressions for the first and second variation of the Total Potential of a thin‐walled open section finite element that lead to non‐linear stiffness equations. These non‐linear equations can be solved for moderate to large displacements. The variations of the Total Potential have been developed elsewhere by the authors, and their contribution to the various non‐linear matrices is stated herein. It is shown that the method of solution of the non‐linear stiffness matrices is problem dependent. The finite element procedure is used to study non‐linear torsion that illustrates torsional hardening, and the Newton–Raphson method is deployed for this study. However, it is shown that this solution strategy is unsuitable for the second example, namely that of the post‐buckling response of a cantilever, and a direct iteration method is described. The good agreement for both of these problems with the work of independent researchers validates the non‐linear finite element method of analysis. Copyright © 1999 John Wiley & Sons, Ltd.     

大跨度屋盖结构等效静力风荷载中共振分量的确定方法研究

大跨度屋盖结构等效静力风荷载中的共振分量确定,包括对结构共振响应的准确计算以及如何将其等效为静力风荷载这两个主要问题。针对第一个问题,提出Ritz-POD法分析该类结构的风振响应,以解决在结构风振响应分析中存在的多振型参与结构振动且高阶振型对结构风振响应贡献可能较大这一问题;针对第二个问题,将等效静力风荷载的共振分量表示为各Ritz向量的等效惯性力的组合。对一鞍型索网结构的等效静力风荷载分布情况进行了分析;结果表明,能够有效地解决大跨度屋盖结构等效静力风荷载的确定问题。