The variational collocation method Gomez, Hector; De Lorenzis, Laura
Computer methods in applied mechanics and engineering,
09/2016, Volume:
309
Journal Article
Peer reviewed
We propose the variational collocation method for the numerical solution of partial differential equations. The conceptual basis is the establishment of a direct connection between the Galerkin ...method and the classical collocation methods, with the perspective of achieving the accuracy of the former with a computational cost of one point evaluation per degree of freedom as in the latter. Variational collocation requires a discrete space constructed by smooth and pointwise non-negative basis functions, which makes the approach immediately applicable to isogeometric analysis and some meshfree methods. In this paper, we concentrate on isogeometric analysis and demonstrate that there exists a set of points such that collocation of the strong form at these points produces the Galerkin solution exactly. We provide an estimate of these points and show that applying isogeometric collocation at the estimated points completely solves the well-known odd/even discrepancy in the order of spatial convergence. We demonstrate the potential of variational collocation with examples of linear and non-linear elasticity as well as Kirchhoff plates.
•We propose the variational collocation method for the numerical solution of PDEs.•We find a direct connection between the Galerkin method and classical collocation.•We show the existence of collocation sites that produce the Galerkin solution exactly.•Variational collocation is applicable to IGA and some meshfree methods.•In IGA collocation we remove the odd/even degree discrepancy in the rate of convergence.
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Fractional integro-differential equations have been recently solved by many methods, such as Adomian decomposition method, differential transform method, collocation method and Taylor expansion ...approach. In this paper a hybrid collocation method is used which combines a non-polynomial collocation used on the first subinterval and graded piecewise polynomial collocation used on the rest of the interval. A theoretical analysis for the convergence order of the method is presented. Some numerical examples are given which confirm the theoretical results.
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Optimization modeling tools are essential to determine optimal design specifications and operation conditions of polymerization processes, especially when quality indices based on molecular weight ...distributions (MWDs) must be enforced. This study proposes a generalized MWD‐based optimization strategy using orthogonal collocation in two dimensions, which can capture the dynamic features of MWDs accurately. To enable the strategy, this study considers generalized initialization methods for large‐scale simulation and optimization. Here, a homotopy method based on intermediate solutions is adopted to generate initial values for general steady‐state simulation models, starting from an arbitrary known solution for any steady‐state simulation model. For dynamic simulation models, the response of a first‐order linear system is adopted to initialize the state variables. Case studies show the effectiveness of this procedure to enable systematic, reliable, and efficient solution of the optimization problem.
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The degree of collocational knowledge influences learners' competence in the production of L2 speech and text. However, collocation learning is complex because learners might co-produce words ...incorrectly according to L1 inference. This study aims at the design of an online video-assisted collocation learning system, VACLS, that integrated video captioning and concordancing to foster collocational knowledge acquisition. English collocations with three patterns, namely verb-noun, verb-adverb, and verb-preposition, were selected as target collocations in VACLS. While learning with VACLS, the learners are able to watch videos with full captions and then looked for additional knowledge and instances of the target collocations via an online concordancer incorporated in VACLS. Besides the development of VACLS, effectiveness of the system and learners' perception were also evaluated. 20 university EFL students in Taiwan enrolled the experiment. After 3 weeks of learning with the aid of VACLS, the learning outcome for collocation retention was significant and the participants with less initial collocational knowledge exhibited greater improvement in collocation retention after learning VACLS. Besides, these learners highly agreed VACLS is helpful for English collocation acquisition. Thus the VACLS can be useful for English educators and EFL/ESL learners.
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Summary
In this work, an adaptive simplex stochastic collocation method is introduced in which sample refinement is informed by variability in the solution of the system. The proposed method is based ...on the concept of multi‐element stochastic collocation methods and is capable of dealing with very high‐dimensional models whose solutions are expressed as a vector, a matrix, or a tensor. The method leverages random samples to create a multi‐element polynomial chaos surrogate model that incorporates local anisotropy in the refinement, informed by the variance of the estimated solution. This feature makes it beneficial for strongly nonlinear and/or discontinuous problems with correlated non‐Gaussian uncertainties. To solve large systems, a reduced‐order model (ROM) of the high‐dimensional response is identified using singular value decomposition (higher‐order SVD for matrix/tensor solutions) and polynomial chaos is used to interpolate the ROM. The method is applied to several stochastic systems of varying type of response (scalar/vector/matrix) and it shows considerable improvement in performance compared to existing simplex stochastic collocation methods and adaptive sparse grid collocation methods.
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In this article, a generalized log orthogonal functions (GLOFs)‐spectral collocation method to two dimensional weakly singular Volterra integral equations of the second kind is proposed. The mild ...singularities of the solution at the interval endpoint can be captured by Gauss‐GLOFs quadrature and the shortcoming of the traditional spectral method which cannot well deal with weakly singular Volterra integral equations with limited regular solutions is avoided. A detailed L∞$$ {L}^{\infty } $$ convergence analysis of the numerical solution is carried out. The efficiency of the proposed method is demonstrated by numerical examples.
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Spectral methods solve elliptic partial differential equations (PDEs) numerically with errors bounded by an exponentially decaying function of the number of modes when the solution is analytic. For ...time dependent problems, almost all focus has been on low-order finite difference schemes for the time derivative and spectral schemes for spatial derivatives. Spectral methods that converge spectrally in both space and time have appeared recently. This paper is a continuation of the authors' previous works on Legendre and Chebyshev space-time methods for the heat equation. Here space-time spectral collocation methods for the Schrodinger, wave, Airy and beam equations are proposed and analyzed. In particular, a condition number estimate of each global Chebyshev space-time operator is shown. The analysis requires new estimates of eigenvalues of some spectral derivative matrices. In particular, it is shown that the real part of every eigenvalue of the third-order Chebyshev derivative matrix is positive and bounded away from zero, settling a twenty-year-old conjecture. Similarly, the real part of every eigenvalue of the fourth-order Chebyshev derivative matrix with Dirichlet boundary conditions is shown to be also positive and bounded away from zero. Numerical results verify the theoretical results, and demonstrate that the space-time methods also work well for some common nonlinear PDEs.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
In view of the general inertia and damping features as well as the inevitable uncertainty factors in engineering structures, a novel dynamic reliability-based topology optimization (DRBTO) strategy ...is investigated for time-variant mechanical systems with overall consideration of material dispersion and loading deviation effects. The static interval-set model is first utilized to quantify multi-source uncertainty inputs and the transient interval-process model is then established to characterize unknown-but-bounded response results, which can be readily solved through the proposed interval-process collocation approach combined with a classical Newmark difference scheme. Different from the traditional deterministic design framework, the present DRBTO scheme will directly consider new reliability constraints, for which the non-probabilistic time-variant reliability (NTR) index is mathematically deduced using the first-passage principle. In addition, the issues related to uncertainty-oriented design sensitivity and filtering method are discussed. The usage and effectiveness of DRBTO are demonstrated with three numerical examples.
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This paper proposes a full grid interval collocation method (FGICM) and a sparse grid interval collocation method (SGICM) to solve the uncertain heat convection-diffusion problem with interval input ...parameters in material properties, applied loads and boundary conditions. The Legendre polynomial series is adopted to approximate the functional dependency of temperature response with respect to the interval parameters. In the process of calculating the expansion coefficients, FGICM evaluates the deterministic solutions directly on the full tensor product grids, while the Smolyak sparse grids are reconstructed in SGICM to avoid the curse of dimensionality. The eventual lower and upper bounds of temperature responses are easily predicted based on the continuously-differentiable property of the approximate function. Comparing results with traditional Monte Carlo simulations and perturbation method, the numerical example evidences the remarkable accuracy and effectiveness of the proposed methods for interval temperature field prediction in engineering.
•Collocation methods are developed to the interval uncertainty analysis.•High-order Legendre polynomial series is adopted for temperature approximation.•Continuously-differentiable property of polynomial function is used for extrema.•Smolyak-type sparse grids are constructed to avoid the curse of dimensionality.
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A collocation mixed finite element method (MFEM) for direct and converse flexoelectricity in piezoelectric materials is developed for 2D problems. The size-effect phenomenon in micro/nano structures ...is considered by the strain- and electric intensity vector-gradient effects. C0 continuous finite element method is inadequate to treat flexoelectricity problems involving the size-effect. To this end, the MFEM with Lagrangian multipliers to treat these solids has been reported recently. With existing MFEM, the computational efficiency is low due to the additional nodal degrees of freedom (DOFs) for the Lagrangian multipliers. In this study, a new collocation MFEM is proposed, in which the number of the DOFs, when compared to the traditional Lagrangian approach, can be reduced. At the same time, the kinematic constraints between the displacement and strain are guaranteed. These kinematic constraints are satisfied by the collocation method at some internal points in the finite elements. The present collocation MFEM can be used to solve flexoelectricity problems with higher efficiency. Its accuracy is verified by comparing the numerical results with available analytical solutions for the bending of a cantilever beam and the compression of a truncated pyramid, respectively. The results indicate that flexoelctricity is strongly related to the geometry of the physical problem. It is shown that flexoelectricity increases significantly with the decrease of the sample size. The same occurs when, for the beam problem, the ratio of the length to depth dimensions increases; similarly, for the truncated pyramid problem, when the ratio of the width of the bottom and top surfaces increases.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
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