International Journal for Numerical Methods in Engineering

doi: 10.1002/nme.6474pmid: N/A

Kapuria, Santosh; Jain, Mayank

doi: 10.1002/nme.6612pmid: N/A

A C1‐continuous time‐domain spectral finite element (SFE) is developed for efficient and accurate analysis of flexural‐guided wave propagation in Euler–Bernoulli beam‐type structures. A new C1‐continuous spectral interpolation using the Lobatto basis is proposed, which is shown to eliminate the Runge phenomenon observed in the conventional higher order Hermite interpolation. It is also able to diagonalize the mass matrix, an attractive feature of existing C0‐continuous SFEs, which enhances computational efficiency. The developed element is validated by comparing the results for natural frequencies of first 20 modes with analytical solutions, and its performance for wave propagation problems is assessed in comparison with converged ABAQUS solutions obtained with a very fine mesh using the classical beam element. It is shown that the present element yields excellent accuracy with much faster convergence, higher computational efficiency, and many‐fold reduction in computational time than the conventional FE for narrowband high‐frequency flexural guided wave propagation problems in both undamaged and damaged beams. It also shows excellent performance for wave propagation under broadband impact excitations and initial displacements. The C1‐continuous interpolation proposed here will pave the way for developing several new SFEs for elastic‐ and piezoelectric‐laminated beams using advanced higher order laminated theories, which require C1‐continuity of displacements.

Gu, Yan; Sun, Linlin

doi: 10.1002/nme.6635pmid: N/A

The aim of the present study is to present an effect boundary element method (BEM) for electroelastic analysis of ultrathin piezoelectric films/coatings. The troublesome nearly singular integrals, which are crucial in applying the BEM for thin‐structural problems, are calculated accurately by using a nonlinear coordinate transformation method. The advanced BEM presented requires no remeshing procedure regardless of the thickness of the thin structure. Promising BEM results with only a small number of boundary elements can be achieved with the relative thickness of the thin piezoelectric film is as small as 10−8, which is sufficient for modeling many ultrathin piezoelectric films as used in smart materials and micro‐electro‐mechanical systems. The present BEM procedure with thin‐body capabilities is also extended to general multidomain problems and used to model ultrathin coating/substrate piezoelectric structures. The influence of relative layer‐to‐substrate thickness and the bimaterial mismatch parameters are carefully investigated. Excellent agreement between numerical and theoretical solutions has been demonstrated.

Wang, Gang; Wang, Ying; He, Yinnian

doi: 10.1002/nme.6636pmid: N/A

In this paper, we introduce a least‐squares virtual element method for the convection‐diffusion‐reaction problem in mixed form. We use the H(div) virtual element and continuous virtual element to approximate the flux and the primal variables, respectively. The method allows for the use of very general polygonal meshes. Optimal order a priori error estimates are established for the flux and the primal variables in suitable norms. The least‐squares method offers an efficient a posteriori error estimator without extra effort. Moreover, the hanging nodes are naturally treated in the virtual element method, which provides the high flexibility in mesh refinement because the local mesh postprocessing is never required. Both attractive features motivate us to develop the a posteriori error estimate of the method. Numerical experiments are shown to illustrate the accuracy of the theoretical analysis and demonstrate that the adaptive mesh refinement driven by the proposed estimator can efficiently capture the boundary and the interior layers.

Qin, Jincheng; Isakari, Hiroshi; Taji, Kouichi; Takahashi, Toru; Matsumoto, Toshiro

doi: 10.1002/nme.6637pmid: N/A

We propose a novel robust topology optimization for designing acoustic devices that are effective for broadband sound waves. Here, we define the objective function as the weighted sum of the acoustic response to an incident wave consisting of a single frequency and its standard deviation (SD) against the frequency perturbation. By approximating the SD, under the assumption that the incident frequency follows the normal distribution, with the high‐order Taylor expansion of the (conventional) objective function, we deal with significant frequency variations. To calculate such an approximation, we need to calculate the high‐order frequency derivatives of the state variable. Here, we define them by integral representations, which enables us to characterize them even when the state variable is defined in an unbounded domain as is often the case with wave scattering problems. We further show that, based on this definition, the high‐order derivatives can efficiently be computed by a combination of the boundary element method and automatic differentiation. We also present a derivation and calculation of the topological derivative for the newly defined objective function. We install all the proposed techniques into a topology optimization method based on the level‐set method to design wideband acoustic devices. The validity and effectiveness of the proposed topology optimization are confirmed through several numerical examples.

Sachse, Renate; Geiger, Florian; Bischoff, Manfred

doi: 10.1002/nme.6638pmid: N/A

The design of adaptive structures is one method to improve sustainability of buildings. Adaptive structures are able to adapt to different loading and environmental conditions or to changing requirements by either small or large shape changes. In the latter case, also the mechanics and properties of the deformation process play a role for the structure's energy efficiency. The method of variational motion design, previously developed in the group of the authors, allows to identify deformation paths between two given geometrical configurations that are optimal with respect to a defined quality function. In a preliminary, academic setting this method assumes that every single degree of freedom is accessible to arbitrary external actuation forces that realize the optimized motion. These (nodal) forces can be recovered a posteriori. The present contribution deals with an extension of the method of motion design by the constraint that the motion is to be realized by a predefined set of actuation forces. These can be either external forces or prescribed length chances of discrete, internal actuator elements. As an additional constraint, static stability of each intermediate configuration during the motion is taken into account. It can be accomplished by enforcing a positive determinant of the stiffness matrix.

Bruno, Oscar P.; Xu, Liwei; Yin, Tao

doi: 10.1002/nme.6639pmid: N/A

We present new methodologies for the numerical solution of problems of elastic scattering by open arcs in two dimensions. The algorithms utilize weighted versions of the classical elastic integral operators associated with Dirichlet and Neumann boundary conditions, where the integral weight accounts for (and regularizes) the singularity of the integral‐equation solutions at the open‐arc endpoints. Crucially, the method also incorporates a certain “open‐arc elastic Calderón relation” introduced in this article, whose validity is demonstrated on the basis of numerical experiments, but whose rigorous mathematical proof is left for future work. (In fact, the aforementioned open‐arc elastic Calderón relation generalizes a corresponding elastic Calderón relation for closed surfaces, which is also introduced in this article, and for which a rigorous proof is included.) Using the open‐surface Calderón relation in conjunction with spectrally accurate quadrature rules and the Krylov‐subspace linear algebra solver GMRES, the proposed overall open‐arc elastic solver produces results of high accuracy in small number of iterations, for both low and high frequencies. A variety of numerical examples in this article demonstrate the accuracy and efficiency of the proposed methodology.

Lin, Chao‐Ping; Xie, Huiqing; Grimes, Roger; Bai, Zhaojun

doi: 10.1002/nme.6640pmid: N/A

We consider the problem of extracting a few desired eigenpairs of the buckling eigenvalue problem Kx=λKGx, where K is symmetric positive semi‐definite, KG is symmetric indefinite, and the pencil K−λKG is singular, namely, K and KG share a nontrivial common nullspace. Moreover, in practical buckling analysis of structures, bases for the nullspace of K and the common nullspace of K and KG are available. There are two open issues for developing an industrial strength shift‐invert Lanczos method: (1) the shift‐invert operator (K−σKG)−1 does not exist or is extremely ill‐conditioned, and (2) the use of the semi‐inner product induced by K drives the Lanczos vectors rapidly toward the nullspace of K, which leads to a rapid growth of the Lanczos vectors in norms and causes permanent loss of information and the failure of the method. In this paper, we address these two issues by proposing a generalized buckling spectral transformation of the singular pencil K−λKG and a regularization of the inner product via a low‐rank updating of the semi‐positive definiteness of K. The efficacy of our approach is demonstrated by numerical examples, including one from industrial buckling analysis.

Li, Kai; Cheng, Gengdong; Wang, Yu; Liang, Yuan

doi: 10.1002/nme.6641pmid: N/A

The shakedown load of elastoplastic structures under multiple variable loading is an important factor in structural design and integrity analysis. In classical plasticity shakedown analysis is an essential and challenging problem. Most existing methods are based on the solution of super large‐scale mathematical programming, basis reduction or mechanics insight method which have their own limitations in practical engineering problem. In the present study, the proposed method explores the Karush–Kuhn–Tucker (KKT) conditions and the physical reinterpretation of its primal‐dual variables of the Melan's lower bound theorem, and establishes the relation of the primal‐dual variables in finite element formulation. Based on the primal‐dual theory, the primal‐dual eigenstress‐driven method is proposed which is a two‐level nested algorithm: the inner level is the eigenstress‐driven incremental algorithm for constructing the beneficial residual stress field and calculating the safe multiplier and the outer level is the load multiplier descent algorithm for searching the shakedown multiplier. For the purpose of algorithm's universality, the proposed algorithm is fully integrated with the commercial finite element software ANSYS APDL without special optimization solvers, which is well suited for large‐scale practical engineering structures. Several numerical examples are provided to demonstrate the accuracy and efficiency of the proposed algorithm.

Li, Xiaopeng; Gao, Liang; Zhou, Ying; Li, Hao

doi: 10.1002/nme.6642pmid: N/A

A hybrid level set method is proposed for devising structures with embedded components, where the supporting structure as well as the positions and orientations of the components are optimized simultaneously. Two different types of level sets, namely the explicit and implicit level set are introduced to respectively represent the components and supporting structure. In this fashion, smooth geometries and clear interfaces for both the components and their underpinning structure can be obtained, which can facilitate the following analysis or manufacture of the optimized design. The positions and orientation of the components are described by a set of explicit level sets. To facilitate the solution of these multiconstraint optimization problems, a parametric mechanism is formulated by approximating the implicit level sets with the Gaussian radial basis function. Distinguished from the existing approaches, we use two different sets of design variables in a uniform optimization loop, that is, the geometric parameters for the positions and orientations of the components, and the expansion coefficients of the level set interpolation for the supporting structure. In this way, the overall design variables for the optimization problem can be greatly reduced. Several examples are provided to demonstrate the effectiveness and efficiency of the proposed method.