Defeaturing: A posteriori error analysis via feature sensitivity

It is well known that small geometric features within a CAD model can significantly impact the computational cost, and often undermine the reliability, of finite element analysis. Engineers therefore resort to defeaturing or detail removal, wherein the offending features are suppressed prior to computational analysis. However, this results in a defeaturing‐induced analysis error. In this paper, we estimate this error in an a posteriori sense through the novel concept of feature sensitivity. The latter determines the first‐order change in quantities of interest when an arbitrary cluster of small geometric features is deleted from a model. A formal theory and a set of associated algorithms are provided to compute the feature sensitivity associated with a scalar elliptic partial differential equation. The theory is supported through numerical experiments in 2‐D, involving both internal and boundary features. Copyright © 2008 John Wiley & Sons, Ltd.     

Derivatives of complex eigenvectors with distinct and repeated eigenvalues

In this paper, we investigate the computation of the first‐order derivatives of complex eigenvectors for general non‐defective eigensystems. A new normalization condition is proposed, with which we can compute unique first‐order derivatives of arbitrary differentiable eigenvectors of systems with distinct and repeated eigenvalues. We also present an efficient algorithm to compute the particular solutions to the governing equations of the first‐order derivatives of eigenvectors. Finally, numerical examples are included to demonstrate the validity of the proposed method. Copyright © 2007 John Wiley & Sons, Ltd.     

A geometrically exact isogeometric Kirchhoff plate: Feature‐preserving automatic meshing and C1 rational triangular Bézier spline discretizations

The analysis of the Kirchhoff plate is performed using rational Bézier triangles in isogeometric analysis coupled with a feature‐preserving automatic meshing algorithm. Isogeometric analysis employs the same basis function for geometric design as well as for numerical analysis. The proposed approach also features an automatic meshing algorithm that admits localized geometric features (eg, small geometric details and sharp corners) with high resolution. Moreover, the use of rational triangular Bézier splines for domain triangulation significantly increases the flexibility in discretizing spaces bounded by complicated nonuniform rational B‐spline curves. To raise the global continuity to C¹ for the solution of the plate bending problem, Lagrange multipliers are leveraged to impose continuity constraints. The proposed approach also manipulates the control points at domain boundaries in such a way that the geometry is exactly described. A number of numerical examples consisting of static bending and free vibration analysis of thin plates bounded by complicated nonuniform rational B‐spline curves are used to demonstrate the advantage of the proposed approach.     

Fatigue growth prediction of center slant crack in rectangular plate

This paper presents a numerical prediction model of mixed‐mode crack fatigue growth in a plane elastic plate. It involves a formulations of fatigue growth of multiple crack tips under mixed‐mode loading and a displacement discontinuity method with crack‐tip elements (a boundary element method) proposed recently by Yan is extended to analyse the fatigue growth process of multiple crack tips. Due to an intrinsic feature of the boundary element method, a general growth problem of multiple cracks can be solved in a single‐region formulation. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not necessary. Crack extension is conveniently modelled by adding new boundary elements on the incremental crack extension to the previous crack boundaries. At the same time, the element characters of some related elements are adjusted according to the manner in which the boundary element method is implemented. As an example, the present numerical approach is used to analyse the fatigue growth of a centre slant crack in a rectangular plate. The numerical results illustrate the validation of the numerical prediction model and can reveal the effect of the geometry of the cracked plate on the fatigue growth.     

A level set‐based topology optimization method targeting metallic waveguide design problems

In this paper, we propose a level set‐based topology optimization method targeting metallic waveguide design problems, where the skin effect must be taken into account since the metallic waveguides are generally used in the high‐frequency range where this effect critically affects performance. One of the most reasonable approaches to represent the skin effect is to impose an electric field constraint condition on the surface of the metal. To implement this approach, we develop a boundary‐tracking scheme for the arbitrary Lagrangian Eulerian (ALE) mesh pertaining to the zero iso‐contour of the level set function that is given in an Eulerian mesh, and impose Dirichlet boundary conditions at the nodes on the zero iso‐contour in the ALE mesh to compute the electric field. Since the ALE mesh accurately tracks the zero iso‐contour at every optimization iteration, the electric field is always appropriately computed during optimization. For the sensitivity analysis, we compute the nodal coordinate sensitivities in the ALE mesh and smooth them by solving a Helmholtz‐type partial differential equation. The obtained smoothed sensitivities are used to compute the normal velocity in the level set equation that is solved using the Eulerian mesh, and the level set function is updated based on the computed normal velocity. Finally, the utility of the proposed method is discussed through several numerical examples. Copyright © 2011 John Wiley & Sons, Ltd.     

A spline‐based enrichment function for arbitrary inclusions in extended finite element method with applications to finite deformations

A novel enrichment function, which can model arbitrarily shaped inclusions within the framework of the extended finite element method, is proposed. The internal boundary of an arbitrary‐shaped inclusion is first discretized, and a numerical enrichment function is constructed ‘on the fly’ using spline interpolation. We consider a piecewise cubic spline which is constructed from seven localized discrete boundary points. The enrichment function is then determined by solving numerically a nonlinear equation which determines the distance from any point to the spline curve. Parametric convergence studies are carried out to show the accuracy of this approach compared with pointwise and linear segmentation of points for the construction of the enrichment function in the case of simple inclusions and arbitrarily shaped inclusions in linear elasticity. Moreover, the viability of this approach is illustrated on a neo‐Hookean hyperelastic material with a hole undergoing large deformation. In this case, the enrichment is able to adapt to the deformation and effectively capture the correct response without remeshing. Copyright © 2013 John Wiley & Sons, Ltd.     

Integral representations in elastostatics and their application to an alternative boundary element method

We obtain a new representation for derivatives and anti‐derivatives of any order of the displacement and stress fields for elastostatics problems when the boundary data is given in terms of polynomials of arbitrary degree. The result includes, as a special case, Somigliana's theorem. Based on this identity, we propose an alternative algorithm for the boundary element method that uses polynomial approximations of arbitrary order. The method provides accurate results for two‐dimensional elastostatics boundary value problems and has the advantage that it is easy to implement. The formula can also be used to accurately compute stresses and strains. For domains bounded by polygons, we provide closed‐form analytical expressions for the terms that appear in the stiffness matrix and the load vector for polynomials of arbitrary degree, thus avoiding numerical integration. We analyse the accuracy of the numerical solution as a function of the degree of the polynomial approximation by solving a representative boundary value problem. Copyright © 2004 John Wiley & Sons, Ltd.     

A boundary spectral method for elastostatic problems with multiple spherical cavities and inclusions

The problem of an infinite solid containing an arbitrary number of non-overlapping spherical cavities and inclusions with arbitrary sizes and locations is considered. The infinite solid and the spherical inclusions are made of different isotropic, linearly elastic materials. The spherical cavities are assumed to carry arbitrary tractions, and the spherical inclusions are assumed to be perfectly bonded to the infinite solid. The boundary and interfacial displacements and tractions are represented by truncated series of surface spherical harmonics. The problem involving multiple spherical features is replaced by a sequence of problems involving a single spherical feature via Schwarz's alternating method which accounts for the interactions in the course of an iterative process. Problems involving a single spherical feature are solved by employing the Papkovich–Neuber functions, and the interactions are evaluated by applying a least squares method. A robust scheme is introduced to control the total errors on the spherical boundaries and interfaces and to choose the number of terms in the series expansions. Several numerical examples are given to address the efficiency and the accuracy of the proposed method.     

Analysis of plates stiffened by parallel beams

In this paper a general solution for the analysis of plates stiffened by parallel beams subjected to an arbitrary loading is presented. According to the proposed model, the stiffening beams are isolated from the plate by sections in the lower outer surface of the plate, taking into account the arising tractions in all directions at the fictitious interfaces. The aforementioned integrated tractions result in the loading of the beams as well as the additional loading of the plate. Their distribution is established by applying continuity conditions in all directions at the interfaces. The analysis of both the plate and the beams is accomplished on their deformed shape taking into account second‐order effects. Six boundary value problems with respect to the plate transverse deflection, to the plate inplane displacement components, to the beam transverse deflections, to the beam axial deformation and to the beam non‐uniform angle of twist are formulated and solved using the analog equation method (AEM), a boundary element method (BEM)‐based method employing a boundary integral equation approach. The solution of the aforementioned plate and beam problems, which are non‐linearly coupled, is achieved using iterative numerical methods. The adopted model describes better the actual response of the plate beams system and permits the evaluation of the shear forces at the interfaces in both directions, the knowledge of which is very important in the design of prefabricated ribbed plates. The evaluated lateral deflections of the plate–beams system are found to exhibit considerable discrepancy from those of other models, which neglect inplane and axial forces and deformations. Copyright © 2006 John Wiley & Sons, Ltd.     

Efficient DG‐like formulation equipped with curved boundary edges for solving elasto‐acoustic scattering problems

A Discontinuous Galerkin (DG)‐based approach is proposed for computing the scattered field from an elastic bounded object immersed in an infinite homogeneous fluid medium. The proposed method possesses two distinctive features. First, it employs higher‐order polynomial‐shape functions needed to address the high‐frequency propagation regime. Second, it is equipped with curved boundary edges to provide an accurate representation of the fluid–structure interface. The most salient benefits resulting from the latter feature, as demonstrated by the numerical investigation, are the following: (i) an improvement by—at least—two orders of magnitude on the relative error and (ii) the disappearance of spurious resonance frequencies in the surrounding fluid medium. In addition, the reported numerical results reveal that when using cubic polynomials with less than three elements per wavelength, the proposed DG method computes the scattered field with a relative error below 1% for an elastic scatterer of about 30 wavelengths. This observation highlights the potential of the proposed solution methodology for efficiently solving mid‐frequency to high‐frequency elasto‐acoustic scattering problems. Copyright © 2014 John Wiley & Sons, Ltd.     

Complete eigen-solutions for plane notches with multi-materials by the imbedding method

In this paper, a robust algorithm is proposed to compute accurate and complete eigen-solutions for plane cracks/notches with multi-materials, arbitrary opening angles and different surface conditions. Based on the imbedding method, the general-purpose Ordinary Differential Equation (ODE) solver COLSYS is adopted to compute the complex line integral accurately and efficiently after transforming it into an equivalent ODE form. The searching domain is carefully chosen to be a narrow annular band so that the eigen-values can be computed in an ascending sequence measured by modulus without missing any in between. A number of numerical examples are given to show the excellent accuracy, efficiency, reliability and versatility of the proposed approach.     

基于SPH方法对不同药型罩线性聚能射流形成及后效侵彻过程的模拟

为了解决传统基于网格的数值方法在模拟线性聚能射流问题时因大变形而导致网格畸变使计算难以进行的问题,本文通过自编程实现的光滑粒子法（SPH）对不同药型罩线性聚能装药射流形成及其侵彻金属靶板的过程开展了数值模拟研究,所实现的算法可以为线性聚能射流数值模拟研究提供新途径。本文所开展的研究首先基于已有的线性聚能射流试验模型进行模拟分析,采用SPH方法有效实现了线性聚能射流的形成过程,数值模拟获得的射流头部速度与试验比对误差在10%以内。然后建立了装药质量、药型罩质量和装药横截面宽度相同的前提下不同药型罩线性聚能射流模型,数值模拟获得不同药型罩形成的射流特征以及侵彻金属靶板的开口宽度和侵彻深度随时间的变化规律。研究得到的不同药型罩线性聚能射流形成及后效侵彻规律可为线性聚能射流的设计提供参考。     

Numerical plate testing for linear two‐scale analyses of composite plates with in‐plane periodicity

A method of numerical plate testing (NPT) for composite plates with in‐plane periodic heterogeneity is proposed. In the two‐scale boundary value problem, a thick plate model is employed at macroscale, while three‐dimensional solids are assumed at microscale. The NPT, which is nothing more or less than the homogenization analysis, is in fact a series of microscopic analyses on a unit cell that evaluates the macroscopic plate stiffnesses. The specific functional forms of microscopic displacements are originally presented so that the relationship between the macroscopic resultant stresses/moments and strains/curvatures to be consistent with the microscopic equilibrated state. In order to perform NPT by using general‐purpose FEM programs, we introduce control nodes to facilitate the multiple‐point constraints for in‐plane periodicity. Numerical examples are presented to verify that the proposed method of NPT reproduces the plate stiffnesses in classical plate and laminate theories. We also perform a series of homogenization, macroscopic, and localization analyses for an in‐plane heterogeneous composite plate to demonstrate the performance of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.     

Error modeling for surrogates of dynamical systems using machine learning

A machine learning–based framework for modeling the error introduced by surrogate models of parameterized dynamical systems is proposed. The framework entails the use of high‐dimensional regression techniques (eg, random forests, and LASSO) to map a large set of inexpensively computed “error indicators” (ie, features) produced by the surrogate model at a given time instance to a prediction of the surrogate‐model error in a quantity of interest (QoI). This eliminates the need for the user to hand‐select a small number of informative features. The methodology requires a training set of parameter instances at which the time‐dependent surrogate‐model error is computed by simulating both the high‐fidelity and surrogate models. Using these training data, the method first determines regression‐model locality (via classification or clustering) and subsequently constructs a “local” regression model to predict the time‐instantaneous error within each identified region of feature space. We consider 2 uses for the resulting error model: (1) as a correction to the surrogate‐model QoI prediction at each time instance and (2) as a way to statistically model arbitrary functions of the time‐dependent surrogate‐model error (eg, time‐integrated errors). We apply the proposed framework to model errors in reduced‐order models of nonlinear oil‐water subsurface flow simulations, with time‐varying well‐control (bottom‐hole pressure) parameters. The reduced‐order models used in this work entail application of trajectory piecewise linearization in conjunction with proper orthogonal decomposition. When the first use of the method is considered, numerical experiments demonstrate consistent improvement in accuracy in the time‐instantaneous QoI prediction relative to the original surrogate model, across a large number of test cases. When the second use is considered, results show that the proposed method provides accurate statistical predictions of the time‐ and well‐averaged errors.     

A novel hybrid FS‐FEM/SEA for the analysis of vibro‐acoustic problems

Predicting the frequency response of a complex vibro‐acoustic system becomes extremely difficult in the mid‐frequency regime. In this work, a novel hybrid face‐based smoothed finite element method/statistical energy analysis (FS‐FEM/SEA) method is proposed, aiming to further improve the accuracy of ‘mid‐frequency’ predictions. According to this approach, the whole vibro‐acoustic system is divided into a combination of a plate subsystem with statistical behaviour and an acoustic cavity subsystem with deterministic behaviour. The plate subsystem is treated using the recently developed FS‐FEM, and the cavity subsystem is dealt with using the SEA. These two different types of subsystems can be coupled and interacted through the so‐called diffuse field reciprocity relation. The ensemble average response of the system is calculated, and the uncertainty is confined and treated in the SEA subsystems. The use of FS‐FEM ‘softens’ the well‐known ‘overly stiff’ behaviour in the standard FEM and reduces the inherent numerical dispersion error. The proposed FS‐FEM/SEA approach is verified and its features are examined by various numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.     

Mesh‐free radial point interpolation method for the buckling analysis of Mindlin plates subjected to in‐plane point loads

In this paper, a mesh‐free approach is employed for buckling analysis of Mindlin plates that are subjected to in‐plane point loads. The radial point interpolation method (RPIM) is used to approximate displacements based on nodes. Variational forms of the system equations are established. Two‐step solution procedures are implemented. The non‐uniform pre‐stress distribution of plate is first obtained using the RPIM based on a two‐dimensional (2D) elastic plane stress problem. This predetermined non‐uniform pre‐stress distribution is then used to compute buckling loads of plate using the RPIM based on Mindlin's plate assumption. The RPIM can easily handle any number and location of nodes in the plate domain for a desired computational accuracy without major difficulties in solving the initial stresses and buckling loads. Numerical examples considered here include circular and rectangular Mindlin plates that are subjected to in‐plane uniform and point loads with different aspect ratios and boundary conditions. The present results are validated against the available analytical and numerical solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

Simulation of ductile crack propagation and determination of CTOAs in pipeline steels using cohesive zone modelling

This paper presents recent results of numerical studies on stable crack extension of high toughness steels typical of those in modern gas pipelines using cohesive zone modelling (CZM). The main focus of the work is on the determination of crack‐tip opening angles (CTOAs) of these steels from CZM. Two sets of materials are modelled. The first material set models a typical structural steel, with variable toughness described by four traction–separation (TS) laws. The second set models an X70 pipe steel, with three different TS laws. For each TS law, there are three defining parameters: the maximum cohesive strength, the final separation and the work of separation. The specimens analysed include a crack in an infinite plate (small‐scale yielding, SSY) and a standard drop‐weight tear test (DWTT). Fracture propagation characteristics and values of CTOA are obtained from these two types of specimens. It is shown that cohesive zone models can be successfully used to simulate ductile crack propagation and to numerically measure CTOAs. The ductile crack propagation characteristics and CTOAs obtained from SSY and DWTT specimens are compared for each set of steels. In addition, the CTOA results from numerical CZM of DWTT specimens of X70 steel are compared with those from laboratory tests.     

An Application of the Cell Method to Multiaxial Fatigue Assessment of a Test Component under Different Criteria

Abstract: A recent numerical method, the cell method, was applied for the dynamic analysis of an L‐shaped steel plate subjected to a sinusoidal load. The calculated stress time histories were post‐processed in order to assess fatigue life under four different high‐cycle fatigue criteria: two formulations of the equivalent Von Mises approach and two critical plane methods. The latter require the definition of amplitude and mean value of the shear stress acting on a material plane: when considering periodic stress histories, this is commonly achieved by the construction of the minimum circumscribed circle (MCC), encompassing the shear stress load path on the assumed fracture plane. In this study, a novel algorithm to determine the MCC has also been proposed and applied. The fatigue life assessment results were discussed and compared to point out the relevant characteristics of each method.     

Identification of Added Mass in the Composite Plate Structure Based on Wavelet Packet Transform

A new concept has been introduced that the combination of rotational mode shape with two‐dimensional wavelet packet transform to detect the added mass (damage) in a glass fibre reinforced polymer composite plate structure. Wavelet packet transform is an advanced signal processing tool that can magnify the abnormality features in the signal. Rotational mode shapes are sensitive to damage in beam and plate structures. The proposed method employs an added mass, which slides to different locations to alter the local and global dynamic characteristics of the structure. Finite element analysis is carried out to obtain the first three rotational bending mode shapes, from the damaged plate structure, then used as input to two‐dimensional wavelet packet transform. The numerical results of normalised diagonal detail wavelet packet coefficients show a peak at single or multiple added mass (damage) locations of a plate structure for two different boundary conditions. This method seems to be sensitive to relatively small amount of damage to the plate structure. A simple parametric study is carried out for the damage extent quantification. In addition, investigation with noise‐contaminated signals shows its feasibility in the real applications.