Toward a multifrequency quasi‐static Ritz vector method for frequency‐dependent acoustic system application

Computational issues concerning the calculation of acoustic responses of a complex finite element (FE) model for various noise and vibration inputs have become prevalent. Such a model requires a significant amount of computation time because of repeated inversions of dynamic stiffness matrices. Thus, even state‐of‐the‐art computer hardware and software often face limitations where a model order reduction (MOR) scheme can help. The established MOR schemes such as Ritz vector or quasi‐static Ritz vector methods are efficient for general engineering systems, but these MOR methods become inaccurate for frequency response analyses in some acoustic systems with frequency‐dependent mass and stiffness matrices and force vectors (hereinafter frequency‐dependent acoustic systems). To cope with the inaccurate prediction by these methods for frequency‐dependent acoustic systems, this research presents and applies the multifrequency quasi‐static Ritz vector method. Unlike the Ritz vector or quasi‐static Ritz vector methods, the present multifrequency quasi‐static Ritz vector method employs direct Krylov subspace bases without an orthonormal procedure at multiple center frequencies. In comparison with the existing MOR scheme, a significant gain in computational efficiency is achieved, as well as enhanced accuracy. A comparison of these methods based on criteria such as efficiency, accuracy, and reliability was also conducted. Copyright © 2011 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.     

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.     

A quasi‐linear reproducing kernel particle method

Reproducing kernel particle method (RKPM) has been applied to many large deformation problems. RKPM relies on polynomial reproducing conditions to yield desired accuracy and convergence properties but requires appropriate kernel support coverage of neighboring nodes to ensure kernel stability. This kernel stability condition is difficult to achieve for problems with large particle motion such as the fragment‐impact processes that commonly exist in extreme events. A new reproducing kernel formulation with ‘quasi‐linear’ reproducing conditions is introduced. In this approach, the first‐order polynomial reproducing conditions are approximately enforced to yield a nonsingular moment matrix. With proper error control of the first‐order completeness, nearly second‐order convergence rate in L2 norm can be achieved while maintaining kernel stability. The effectiveness of this quasi‐linear RKPM formulation is demonstrated by modeling several extremely large deformation and fragment‐impact problems. Copyright © 2016 John Wiley & Sons, Ltd.     

A spline strip kernel particle method and its application to two‐dimensional elasticity problems

In this paper we present a novel spline strip kernel particle method (SSKPM) that has been developed for solving a class of two‐dimensional (2D) elasticity problems. This new approach combines the concepts of the mesh‐free methods and the spline strip method. For the interpolation of the assumed displacement field, we employed the kernel particle shape functions in the transverse direction, and the B₃‐spline function in the longitudinal direction. The formulation is validated on several beam and semi‐infinite plate problems. The numerical results of these test problems are then compared with the existing solutions obtained by the exact or numerical methods. From this study we conclude that the SSKPM is a potential alternative to the classical finite strip method (FSM). Copyright © 2003 John Wiley & Sons, Ltd.     

Finite element analysis of quasi‐static indentation of woven fabric textile composites using different nose shape indenters

The finite element FE analysis of quasi‐static indentation event of various nose shape rigid indenters into woven fabric composite with carbon fiber as reinforcement has been performed and discussed in detail. It was found that indenter nose shape has large influence in terms of absorbed energy, indentation at failure and damage area. The FE software, ABAQUS^® was employed to simulate quasi‐static response of woven composite unit cell. Exhaustive parametric studies have been conducted with an aim to analyze the effect of change in indenter geometry on the indentation response of the woven composite unit cell. The developed FE model for the purpose of validation was compared with available experimental results and was found to be in reasonably good agreement. The failure morphologies, damage shape and damage size were evaluated, compared and deeply discussed for different nose shape indenters. Largest damaged areas were observed for flat and truncated indenters while the smallest for the conical one.     

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.     

A continuum damage mechanics method for fracture simulation of quasi‐brittle materials

This paper addresses a novel continuum damage‐based method for simulating failure process of quasi‐brittle materials starting from local damage initiation to final fracture. In the developed method, the preset characteristic length field is used to evaluate damage instead of element, which is used to reduce the spurious sensitivity. In addition, damage is only updated in the most dangerous location at a time for considering stress redistribution due to damage evolution, which is used to simulate competitive fracture process. As cases study, representative numerical simulations of two benchmark tests are given to verify the performance of the developed continuum damage‐based method together with a used damage model. The simulation results of the crack paths for two concrete specimens obtained from the developed method matched well with the corresponding experimental results. The results show that the developed continuum damage‐based method is effective and can be used to simulate damage and fracture process of brittle or quasi‐brittle materials. And the simulation results based on the developed method depend only the preset characteristic length field and not grid mesh.     

A particle method for two‐phase flows with large density difference

A new numerical approach for solving incompressible two‐phase flows is presented in the framework of the recently developed Consistent Particle Method (CPM). In the context of the Lagrangian particle formulation, the CPM computes spatial derivatives based on the generalized finite difference scheme and produces good results for single‐phase flow problems. Nevertheless, for two‐phase flows, the method cannot be directly applied near the fluid interface because of the abrupt discontinuity of fluid density resulting in large change in pressure gradient. This problem is resolved by dealing with the pressure gradient normalized by density, leading to a two‐phase CPM of which the original singlephase CPM is a special case. In addition, a new adaptive particle selection scheme is proposed to overcome the problem of ill‐conditioned coefficient matrix of pressure Poisson equation when particles are sparse and non‐uniformly spaced. Numerical examples of Rayleigh–Taylor instability, gravity current flow, water‐air sloshing and dam break are presented to demonstrate the accuracy of the proposed method in wave profile and pressure solution. Copyright © 2015 John Wiley & Sons, Ltd.     

A new fast multipole boundary element method for solving large‐scale two‐dimensional elastostatic problems

A new fast multipole boundary element method (BEM) is presented in this paper for large‐scale analysis of two‐dimensional (2‐D) elastostatic problems based on the direct boundary integral equation (BIE) formulation. In this new formulation, the fundamental solution for 2‐D elasticity is written in a complex form using the two complex potential functions in 2‐D elasticity. In this way, the multipole and local expansions for 2‐D elasticity BIE are directly linked to those for 2‐D potential problems. Furthermore, their translations (moment to moment, moment to local, and local to local) turn out to be exactly the same as those in the 2‐D potential case. This formulation is thus very compact and more efficient than other fast multipole approaches for 2‐D elastostatic problems using Taylor series expansions of the fundamental solution in its original form. Several numerical examples are presented to study the accuracy and efficiency of the developed fast multipole BEM formulation and code. BEM models with more than one million equations have been solved successfully on a laptop computer. These results clearly demonstrate the potential of the developed fast multipole BEM for solving large‐scale 2‐D elastostatic problems. Copyright © 2005 John Wiley & Sons, Ltd.     

A rapid simulation of nano‐particle transport in a two‐dimensional human airway using POD/Galerkin reduced‐order models

An approach is proposed for the rapid prediction of nano‐particle transport and deposition in the human airway, which requires the solution of both the Navier–Stokes and advection–diffusion equations and for which computational efficiency is a challenge. The proposed method builds low‐order models that are representative of the fully coupled equations by means of the Galerkin projection and proper orthogonal decomposition technique. The obtained reduced‐order models (ROMs) are a set of ordinary differential equations for the temporal coefficients of the basis functions. The numerical results indicate that the ROMs are highly efficient for the computation (the speedup factor is approximately 3 × 10³) and have reasonable accuracy compared with the full model (relative error of ≈7 × 10^?3). Using ROMs, the deposition of particles is studied for 1≤d_n≤100 nm, where d_n is the diameter of a nano‐particle. The effectiveness of this approach is promising for applications of health risk assessment. Copyright © 2015 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.     

The boundary node method for three‐dimensional problems in potential theory

The boundary node method (BNM) is developed in this paper for solving potential problems in three dimensions. The BNM represents a coupling between boundary integral equations (BIE) and moving least‐squares (MLS) interpolants. The main idea here is to retain the dimensionality advantage of the former and the meshless attribute of the later. This results in decoupling of the ‘mesh’ and the interpolation procedure for the field variables. A general BNM computer code for 3‐D potential problems has been developed. Several parameters involved in the BNM need to be chosen carefully for a successful implementation of the method. An in‐depth and systematic study has been carried out in this paper in order to better understand the effects of various parameters on the performance of the method. Numerical results for spheres and cubes, subjected to different types of boundary conditions, are extremely encouraging. Copyright © 2000 John Wiley & Sons, Ltd.     

Coupled meshfree and fractal finite element method for mixed mode two‐dimensional crack problems

This paper presents a coupling technique for integrating the element‐free Galerkin method (EFGM) with the fractal finite element method (FFEM) for analyzing homogeneous, isotropic, and two‐dimensional linear‐elastic cracked structures subjected to mixed‐mode (modes I and II) loading conditions. FFEM is adopted for discretization of the domain close to the crack tip and EFGM is adopted in the rest of the domain. In the transition region interface elements are employed. The shape functions within interface elements which comprise both the EFG and the finite element (FE) shape functions, satisfies the consistency condition thus ensuring convergence of the proposed coupled EFGM–FFEM. The proposed method combines the best features of EFGM and FFEM, in the sense that no special enriched basis functions or no structured mesh with special FEs are necessary and no post‐processing (employing any path independent integrals) is needed to determine fracture parameters, such as stress‐intensity factors (SIFs) and T‐stress. The numerical results show that SIFs and T‐stress obtained using the proposed method are in excellent agreement with the reference solutions for the structural and crack geometries considered in the present study. Also, a parametric study is carried out to examine the effects of the integration order, the similarity ratio, the number of transformation terms, and the crack length to width ratio on the quality of the numerical solutions. A numerical example on mixed‐mode condition is presented to simulate crack propagation. As in the proposed coupled EFGM–FFEM at each increment during the crack propagation, the FFEM mesh (around the crack tip) is shifted as it is to the new updated position of the crack tip (such that FFEM mesh center coincides with the crack tip) and few meshless nodes are sprinkled in the location where the FFEM mesh was lying previously, crack‐propagation analysis can be dramatically simplified. Copyright © 2010 John Wiley & Sons, Ltd.     

A particle method for two‐phase flows with compressible air pocket

When using particle methods to simulate water–air flows with compressible air pockets, a major challenge is to deal with the large differences in physical properties (e.g., density and viscosity) between water and air. In addition, the accurate modeling of air compressibility is essential. To this end, a new two‐phase strategy is proposed to simulate incompressible and compressible fluids simultaneously without iterations between the solvers for incompressible and compressible flows. Water is modeled by the recently developed 2‐phase Consistent Particle Method for incompressible flows. For air modeling, a new compressible solver is proposed based on the ideal gas law and thermodynamics. The formulation avoids the problem of determining the actual sound speed that is dependent on the temperature and is therefore not necessarily constant. In addition, the compressible air solver is seamlessly integrated with the incompressible solver 2‐phase Consistent Particle Method because they both use the same predictor–corrector scheme to solve the governing equations. The performance of the proposed method is demonstrated by three benchmark problems as well as an experimental study of sloshing impact with entrapped air pockets in an oscillating tank. Copyright © 2016 John Wiley & Sons, Ltd.     

Experimental Analysis of the folding force in thin wall extruded cells under quasi‐static in‐plane loading

In most of the engineering structures, specially moving ones and generally in structures under dynamic and static loadings, energy absorber systems are implemented for preventing or reducing damages. These systems are employed in all on‐road vehicles, train wagons, airplanes and ships. Energy absorbers are subjected to two forms of in‐plane and out‐of‐plane loadings. In this paper, the effects of geometrical parameters such as thickness and height of structure and mechanical parameters such as yield stress in the cell structure are investigated. The effect of changing boundary conditions on the folding force is also investigated. According to the results, the energy absorbed by the cell, directly relates to the increase in the number of rows; the main reason of this increase is the increase in shared walls. Moreover, doubling the wall thickness has resulted in a 4.7 times higher energy absorption. In addition, the accuracy of different analytical models is compared, and the model with most precise predictions is introduced.     

A contact detection algorithm for multi‐sphere particles by means of two‐level‐grid‐searching in DEM simulations

In discrete element method simulations, multi‐sphere particle is extensively employed for modeling the geometry shape of non‐spherical particle. A contact detection algorithm for multi‐sphere particles has been developed through two‐level‐grid‐searching. In the first‐level‐grid‐searching, each multi‐sphere particle is represented by a bounding sphere, and global space is partitioned into identical square or cubic cells of size D, the diameter of the greatest bounding sphere. The bounding spheres are mapped into the cells in global space. The candidate particles can be picked out by searching the bounding spheres in the neighbor cells of the bounding sphere for the target particle. In the second‐level‐grid‐searching, a square or cubic local space of size (D + d) is partitioned into identical cells of size d, the diameter of the greatest element sphere. If two bounding spheres of two multi‐sphere particles are overlapped, the contacts occurring between the element spheres in the target multi‐sphere particle and in the candidate multi‐sphere particle are checked. Theoretical analysis and numerical tests on the memory requirement and contact detection time of this algorithm have been performed to verify the efficiency of this algorithm. The results showed that this algorithm can effectively deal with the contact problem for multi‐sphere particles. Copyright © 2015 John Wiley & Sons, Ltd.     

Differential quadrature element method for static analysis of shear deformable cross‐ply laminates

In this paper, the two‐dimensional differential quadrature element method (DQEM) is developed for the static analysis of symmetric cross‐ply laminates using the first‐order shear deformation plate theory. In this study, the laminated plate, which may contain different discontinuities in loading, geometry, material, and boundary conditions, is first divided into several simple plate elements and then the differential quadrature method (DQM) is applied to each simple element. Compatibility conditions are derived to connect the plate elements so that the overall matrix equation system for the whole plate is obtained and solved. The reliability of the DQEM for solving the titled problems is examined carefully through convergence and accuracy studies and finally some numerical test examples are given to demonstrate the applicability and flexibility of this method for practical use. The methodology presented here has overcome some critical drawbacks of the global DQM but is different from the Quadrature Element Method (QEM) since only one grid point is employed to represent the interface point. Copyright © 1999 John Wiley & Sons, Ltd.     

Three‐dimensional theory of elasticity for free vibration analysis of composite laminates via layerwise differential quadrature modelling

In this paper, the free vibration analysis of simply‐supported and clamped composite laminates, especially thick laminates, is carried out. The three‐dimensional theory of elasticity is integrated into a layerwise model via differential quadrature discretization. All physical governing equations are satisfied, including the additional constraints of the characteristics of continuity and discontinuity of interfacial transverse and in‐plane strains and stresses along the interfaces of composite laminates. Effects of plate aspect and thickness ratios on the free vibration of these laminates are examined in detail. This study demonstrates the applicability, accuracy, and stability of the present methodology, for vibration analyses of composite structures of thick laminated constitution. Copyright © 2003 John Wiley & Sons, Ltd.