The solution of elastostatic and elastodynamic problems with Chebyshev spectral finite elements

The Chebyshev spectral element approximation with exact quadratures

The computational efficiency of finite elements using Chebyshev basis functions is compared with that of standard equispaced lower order (bilinear and biquadratic) and higher order (bicubic and greater) p-elements in the solution of the two-dimensional scalar wave equation when using explicit time integration. All solutions use a central-difference-in-time temporal discretization and both consistent and lumped mass matrices are considered. The Chebyshev finite elements are shown to possess significant advantages in accuracy and in computational efficiency with respect to p-elements for an equivalent number of nodes. Copyright © 1999 John Wiley & Sons, Ltd.

The spectral element approach is applied for the first time to the control of chaos with arbitrary control delays. Three important cases are explored using the spectral approach: half-period delayed feedback control (DFC), control loop latency in DFC and extended delay feedback control (EDFC), and steady state control using multiple delay feedback control (MDFC). These cases are explored in self-excited, chaotic systems, and the Floquet exponents calculated using the spectral approach compare well with published results and numerical simulations.

In this paper, a frequency-domain spectral finite element model (SFEM) is developed for dynamic analysis of piezoelectric laminated composite beams. The displacement field of the beam is represented by the first-order shear deformation theory (FSDT). The electric potential for the piezoelectric layer is expressed in two ways: (i) distributed linearly through the thickness and (ii) layerwise through thickness distribution consistent with the FSDT. The governing differential equation of motion for piezoelectric laminated composite beam is obtained using Hamilton’s principle. These time-domain equations are transformed to frequency domain using the Fourier transformation. The spectral element is derived from the exact solution of the frequency domain governing equations of motion. The formulation is validated by comparing the results of the natural frequencies with the published finite element method (FEM) results. The developed element is used to perform dispersion, free vibration analysis, and elastic wave propagation in laminated composite beam fully or partially covered with surface-bonded piezoelectric layer.

Petrov-Galerkin formulations with weighting functions dependent upon spatial and temporal discretization: Applications to transient convection-diffusion problems

Conflation of modal strain energy, spectral finite element, and machine learning techniques for damage detection in multi-span steel girder bridges with variable sections

The simplest elasticity model of the foundation underlying a slender beam under flexure was conceived by Winkler, requiring local proportionality between soil reactions and beam deflection. Such an approach leads to well-posed elastostatic and elastodynamic problems, but as highlighted by Wieghardt, it provides elastic responses that are not technically significant for a wide variety of engineering applications. Thus, Winkler’s model was replaced by Wieghardt himself by assuming that the beam deflection is the convolution integral between soil reaction field and an averaging kernel. Due to conflict between constitutive and kinematic compatibility requirements, the corresponding elastic problem of an inflected beam resting on a Wieghardt foundation is ill-posed. Modifications of the original Wieghardt model were proposed by introducing fictitious boundary concentrated forces of constitutive type, which are physically questionable, being significantly influenced on prescribed kinematic boundary conditions. Inherent difficulties and issues are overcome in the present research using a displacement-driven nonlocal integral strategy obtained by swapping the input and output fields involved in Wieghardt’s original formulation. That is, nonlocal soil reaction fields are the output of integral convolutions of beam deflection fields with an averaging kernel. Equipping the displacement-driven nonlocal integral law with the bi-exponential averaging kernel, an equivalent nonlocal differential problem, supplemented with non-standard constitutive boundary conditions involving nonlocal soil reactions, is established. As a key implication, the integrodifferential equations governing the elastostatic problem of an inflected elastic slender beam resting on a displacement-driven nonlocal integral foundation are replaced with much simpler differential equations supplemented with kinematic, static, and new constitutive boundary conditions. The proposed nonlocal approach is illustrated by examining and analytically solving exemplar problems of structural engineering. Benchmark solutions for numerical analyses are also detected.

Non-conforming spectral element-finite element approximations for partial differential equations

Computational modelling of physical and biochemical processes has emerged as a means of evaluating medical devices, offering new insights that explain current performance, inform future designs and even enable personalized use. Yet resource limitations force one to compromise with reduced order computational models and idealized assumptions that yield either qualitative descriptions or approximate, quantitative solutions to problems of interest. Considering endovascular drug delivery as an exemplary scenario, we used a supervised machine learning framework to process data generated from low fidelity coarse meshes and predict high fidelity solutions on refined mesh configurations. We considered two models simulating drug delivery to the arterial wall: (i) two-dimensional drug-coated balloons and (ii) three-dimensional drug-eluting stents. Simulations were performed on computational mesh configurations of increasing density. Supervised learners based on Gaussian process modelling were constructed from combinations of coarse mesh setting solutions of drug concentrations and nearest neighbourhood distance information as inputs, and higher fidelity mesh solutions as outputs. These learners were then used as computationally inexpensive surrogates to extend predictions using low fidelity information to higher levels of mesh refinement. The cross-validated, supervised learner-based predictions improved fidelity as compared with computational simulations performed at coarse level meshes—a result consistent across all outputs and computational models considered. Supervised learning on coarse mesh solutions can augment traditional physics-based modelling of complex physiologic phenomena. By obtaining efficient solutions at a fraction of the computational cost, this framework has the potential to transform how modelling approaches can be applied in the evaluation of medical technologies and their real-time administration in an increasingly personalized fashion.

SummaryIn this paper, we propose a fully smoothed extended finite element method for axisymmetric problems with weak discontinuities. The salient feature of the proposed approach is that all the terms in the stiffness and mass matrices can be computed by smoothing technique. This is accomplished by combining the Gaussian divergence theorem with the evaluation of indefinite integral based on smoothing technique, which is used to transform the domain integral into boundary integral. The proposed technique completely eliminates the need for isoparametric mapping and the computing of Jacobian matrix even for the mass matrix. When employed over the enriched elements, the proposed technique does not require sub‐triangulation for the purpose of numerical integration. The accuracy and convergence properties of the proposed technique are demonstrated with a few problems in elastostatics and elastodynamics with weak discontinuities. It can be seen that the proposed technique yields stable and accurate solutions and is less sensitive to mesh distortion. Copyright © 2016 John Wiley & Sons, Ltd.

The development of any atmosphere or ocean model warrants a suite of test cases (TCs) to verify its spatial and temporal discretizations, order of accuracy, stability, reproducibility, portability, scalability, etc. In this paper, we present a suite of shallow water TCs designed to verify the barotropic solver of atmosphere and ocean models. These include the non‐dispersive coastal Kelvin wave; the dispersive inertia‐gravity wave; the dispersive planetary and topographic Rossby waves; the barotropic tide; and a non‐linear manufactured solution. These TCs check the implementation of the linear pressure gradient term; the linear constant or variable‐coefficient Coriolis and bathymetry terms; and the non‐linear advection terms. Simulation results are presented for a variety of time‐stepping methods as well as two spatial discretizations: a mimetic finite volume method based on the TRiSK scheme, and a high‐order discontinuous Galerkin spectral element method. The experimental procedure for conducting these numerical experiments is detailed. It underscores several key considerations that vary depending on the chosen spatial discretization method. Finally, convergence studies of every TC are conducted with refinement in both space and time, only in space, and only in time. The convergence slopes match the expected theoretical predictions.

A coupled computational method for multi-solver, multi-domain transient problems in elastodynamics

Acoustic waves scattered by elastic waveguides using a spectral approach with a 2.5D coupled boundary-finite element method

PurposeThe purpose of this paper is to present the solution of a highly nonlinear fluid dynamics in a low Prandtl number regime, typical for metal‐like materials, as defined in the call for contributions to a numerical benchmark problem for 2D columnar solidification of binary alloys. The solution of such a numerical situation represents the first step towards understanding the instabilities in a more complex case of macrosegregation.Design/methodology/approachThe involved temperature, velocity and pressure fields are represented through the local approximation functions which are used to evaluate the partial differential operators. The temporal discretization is performed through explicit time stepping.FindingsThe performance of the method is assessed on the natural convection in a closed rectangular cavity filled with a low Prandtl fluid. Two cases are considered, one with steady state and another with oscillatory solution. It is shown that the proposed solution procedure, despite its simplicity, provides stable and convergent results with excellent computational performance. The results show good agreement with the results of the classical finite volume method and spectral finite element method.Originality/valueThe solution procedure is formulated completely through local computational operations. Besides local numerical method, the pressure velocity is performed locally also, retaining the correct temporal transient.

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