418 results on '"Petrov-Galerkin method"'
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2. A Conservative Finite Element Scheme for the Kirchhoff Equation
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R. Z. Dautov and M. V. Ivanova
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kirchhoff equation ,finite element method ,petrov–galerkin method ,implicit scheme ,newton method ,Mathematics ,QA1-939 - Abstract
This article presents an implicit two-layer finite element scheme for solving the Kirchhoff equation, a nonlinear nonlocal equation of hyperbolic type with the Dirichlet integral. The discrete scheme was designed considering the solution of the problem and its derivative for the time variable. It ensures total energy conservation at a discrete level. The use of the Newton method was proven to be effective for solving the scheme on the time layer despite the nonlocality of the equation. The test problems with smooth solutions showed that the scheme can define both the solution of the problem and its time derivative with an error of O(h2+τ2) in the root-mean-square norm, where τ and h are the grid steps in time and space, respectively. more...
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- 2024
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3. طريقة بيتروف جاليركين الخطية المنفصلة لمعادلات الانتشار الكسرية للزمن.
- Author
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بشير صالح عبد الل
- Abstract
Copyright of Arab Journal of Science & Research Publishing is the property of Mussasat Al-Majallah Al-Arabiyah lil-Ulum Wa-Nashr Al-Abhath and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) more...
- Published
- 2023
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4. Simulation of linear elastic structural elements using the Petrov–Galerkin finite element method.
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Zähringer, Felix and Betsch, Peter
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- *
FINITE element method - Abstract
In this contribution, it is demonstrated that the mesh sensitivity of linear elastic Reissner–Mindlin finite‐element plate formulations can be significantly reduced by using a Petrov–Galerkin‐based approach. In contrast to the usual Bubnov–Galerkin method, Petrov–Galerkin methods are generally characterized by the fact that the test function and the trial function are approximated using different shape functions. To provide an overview, established Petrov–Galerkin methods for 2D solid elements, which have been shown to reduce mesh sensitivity, are reviewed first. It is then investigated whether a suitable Petrov– Galerkin plate formulation can be developed. In this context, it is demonstrated that a full Petrov–Galerkin method leads to problems in the treatment of transverse shear locking. However, the proposed partial Petrov–Galerkin method shows the desired mesh‐insensitive behavior. [ABSTRACT FROM AUTHOR] more...
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- 2023
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5. A Petrov-Galerkin finite element method using polyfractonomials to solve stochastic fractional differential equations.
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Abedini, Nazanin, Foroush Bastani, Ali, and Zohouri Zangeneh, Bijan
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STOCHASTIC differential equations , *FINITE element method , *EXISTENCE theorems , *TELECOMMUNICATION systems , *BROWNIAN motion , *FRACTIONAL differential equations , *INITIAL value problems , *WIENER processes - Abstract
• A theorem on existence and uniqueness for SFDEs of Riemann-Liouville type is proved. • A PG finite element method based on fractional Jacobi polyfractonomials is proposed. • Error estimates are derived and numerical experiments are provided. • Paths of the RL fBm widely used in flow in telecommunication networks is generated. In this paper, we are concerned with existence, uniqueness and numerical approximation of the solution process to an initial value problem for stochastic fractional differential equation of Riemann-Liouville type. We propose and analyze a Petrov-Galerkin finite element method based on fractional (non-polynomial) Jacobi polyfractonomials as basis and test functions. Error estimates in L 2 norm are derived and numerical experiments are provided to validate the theoretical results. As an illustrative application, we generate sample paths of the Riemann-Liouville fractional Brownian motion which is of importance in many applications ranging from geophysics to traffic flow in telecommunication networks. [ABSTRACT FROM AUTHOR] more...
- Published
- 2021
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6. A machine-learning minimal-residual (ML-MRes) framework for goal-oriented finite element discretizations.
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Brevis, Ignacio, Muga, Ignacio, and van der Zee, Kristoffer G.
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ARTIFICIAL neural networks , *PARTIAL differential equations , *FINITE element method , *SPACE frame structures , *NEWSVENDOR model - Abstract
We introduce the concept of machine-learning minimal-residual (ML-MRes) finite element discretizations of partial differential equations (PDEs), which resolve quantities of interest with striking accuracy, regardless of the underlying mesh size. The methods are obtained within a machine-learning framework during which the parameters defining the method are tuned against available training data. In particular, we use a provably stable parametric Petrov–Galerkin method that is equivalent to a minimal-residual formulation using a weighted norm. While the trial space is a standard finite element space, the test space has parameters that are tuned in an off-line stage. Finding the optimal test space therefore amounts to obtaining a goal-oriented discretization that is completely tailored towards the quantity of interest. We use an artificial neural network to define the parametric family of test spaces. Using numerical examples for the Laplacian and advection equation in one and two dimensions, we demonstrate that the ML-MRes finite element method has superior approximation of quantities of interest even on very coarse meshes. [ABSTRACT FROM AUTHOR] more...
- Published
- 2021
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7. A Petrov–Galerkin finite element method for simulating chemotaxis models on stationary surfaces.
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Zhao, Shubo, Xiao, Xufeng, Zhao, Jianping, and Feng, Xinlong
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CONSERVATION of mass , *GALERKIN methods , *FINITE element method , *CHEMICAL reactions - Abstract
In this paper, we present a Petrov–Galerkin finite element method for a class of chemotaxis models defined on surfaces, which describe the movement by one community in reaction to one chemical or biological signal on manifolds. It is desired for numerical methods to satisfy discrete maximum principle and discrete mass conservation property, which is a challenge due to the singular behavior of numerical solution. Thus a Petrov–Galerkin method is combined with an effective mass conservation factor to overcome the challenge. Furthermore, we prove two facts, this method maintains positivity and discrete mass conservation property. In addition, decoupled approach is applied based on the gradient and Laplacian recoveries to solve the coupling system. The relevant stability analyses is provided. Finally, numerical simulations of blowing-up problems and pattern formulations demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR] more...
- Published
- 2020
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8. An h-p version of the continuous Petrov-Galerkin finite element method for Riemann-Liouville fractional differential equation with novel test basis functions.
- Author
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Bu, Weiping and Xiao, Aiguo
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FRACTIONAL differential equations , *FINITE element method , *GALERKIN methods - Abstract
In this paper, we develop an h-p version of finite element method for one-dimensional fractional differential equation − 0 D x α u + Au = f (x) with Dirichlet boundary condition. First, we introduce the existence and uniqueness of the considered problem and show the wellposedness of the corresponding weak form. To solve the mentioned equation, the classical hierarchical polynomials are employed as the trial basis functions. Then, we develop a kind of novel test basis functions for the Petrov-Galerkin finite element method such that the stiffness matrix becomes an identity matrix and the coefficient matrix often has a small condition number. Moreover, we give some properties of the developed test basis functions, and discuss the implementation of the developed finite element method in detail. It is shown that the implementation of our method is easier than that of other finite (and spectral) element methods. Finally, we give a numerical example, and the numerical results show the effectiveness of the develped method. [ABSTRACT FROM AUTHOR] more...
- Published
- 2019
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9. The Petrov–Galerkin finite element method for the numerical solution of time-fractional Sharma–Tasso–Olver equation.
- Author
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Gupta, A. K. and Ray, S. Saha
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FRACTIONAL calculus ,GALERKIN methods ,FINITE element method ,QUINTIC equations ,ITERATIVE methods (Mathematics) - Abstract
In this paper, time-fractional Sharma–Tasso–Olver (STO) equation has been solved numerically through the Petrov–Galerkin approach utilizing a quintic B-spline function as the test function and a linear hat function as the trial function. The Petrov–Galerkin technique is effectively implemented to the fractional STO equation for acquiring the approximate solution numerically. The numerical outcomes are observed in adequate compatibility with those obtained from variational iteration method (VIM) and exact solutions. For fractional order, the numerical outcomes of fractional Sharma–Tasso–Olver equations are also compared with those obtained by variational iteration method (VIM) in Song et al. [Song L., Wang Q., Zhang H., Rational approximation solution of the fractional Sharma–Tasso–Olver equation, J. Comput. Appl. Math. 224:210–218, 2009]. Numerical experiments exhibit the accuracy and efficiency of the approach in order to solve nonlinear fractional STO equation. [ABSTRACT FROM AUTHOR] more...
- Published
- 2019
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10. A finite element method for extended KdV equations
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Karczewska Anna, Rozmej Piotr, Szczeciński Maciej, and Boguniewicz Bartosz
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shallow water wave problem ,nonlinear equations ,second order kdv equations ,finite element method ,petrov–galerkin method ,Mathematics ,QA1-939 ,Electronic computers. Computer science ,QA75.5-76.95 - Abstract
The finite element method (FEM) is applied to obtain numerical solutions to a recently derived nonlinear equation for the shallow water wave problem. A weak formulation and the Petrov–Galerkin method are used. It is shown that the FEM gives a reasonable description of the wave dynamics of soliton waves governed by extended KdV equations. Some new results for several cases of bottom shapes are presented. The numerical scheme presented here is suitable for taking into account stochastic effects, which will be discussed in a subsequent paper. more...
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- 2016
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11. A Petrov-Galerkin finite element method using polyfractonomials to solve stochastic fractional differential equations
- Author
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Ali Foroush Bastani, Nazanin Abedini, and Bijan Z. Zangeneh
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Computational Mathematics ,Numerical Analysis ,Fractional Brownian motion ,Basis (linear algebra) ,Applied Mathematics ,Petrov–Galerkin method ,Initial value problem ,Applied mathematics ,Ranging ,Uniqueness ,Type (model theory) ,Finite element method ,Mathematics - Abstract
In this paper, we are concerned with existence, uniqueness and numerical approximation of the solution process to an initial value problem for stochastic fractional differential equation of Riemann-Liouville type. We propose and analyze a Petrov-Galerkin finite element method based on fractional (non-polynomial) Jacobi polyfractonomials as basis and test functions. Error estimates in L 2 norm are derived and numerical experiments are provided to validate the theoretical results. As an illustrative application, we generate sample paths of the Riemann-Liouville fractional Brownian motion which is of importance in many applications ranging from geophysics to traffic flow in telecommunication networks. more...
- Published
- 2021
- Full Text
- View/download PDF
12. Space-Time Petrov--Galerkin FEM for Fractional Diffusion Problems.
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Duan, Beiping, Jin, Bangti, Lazarov, Raytcho, Pasciak, Joseph, and Zhou, Zhi
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SPACE-time configurations ,FINITE element method ,GALERKIN methods - Abstract
We present and analyze a space-time Petrov--Galerkin finite element method for a time-fractional diffusion equation involving a Riemann--Liouville fractional derivative of order α ∈(0, 1)in time and zero initial data. We derive a proper weak formulation involving different solution and test spaces and show the inf-sup condition for the bilinear form and thus itswell-posedness. Further,we develop a novel finite element formulation, show the well-posedness of the discrete problem, and derive error bounds in both energy and L² norms for the finite element solution. In the proof of the discrete inf-sup condition, a certain nonstandard L² stability property of the L² projection operator plays a key role.We provide extensive numerical examples to verify the convergence analysis. [ABSTRACT FROM AUTHOR] more...
- Published
- 2018
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13. Mesh distortion insensitive and locking‐free Petrov–Galerkin low‐order EAS elements for linear elasticity
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Peter Betsch and Robin Pfefferkorn
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Numerical Analysis ,Computer science ,Applied Mathematics ,Distortion ,Linear elasticity ,Solid mechanics ,Mesh networking ,General Engineering ,Petrov–Galerkin method ,Polygon mesh ,Topology ,Finite element method ,Stiffness matrix - Abstract
One of the most successful mixed finite element methods in solid mechanics is the enhanced assumed strain (EAS) method developed by Simo and Rifai in 1990. However, one major drawback of EAS elements is the highly mesh dependent accuracy. In fact, it can be shown that not only EAS elements, but every finite element with a symmetric stiffness matrix must either fail the patch test or be sensitive to mesh distortion in bending problems (higher order displacement modes) if the shape of the element is arbitrary. This theorem was established by MacNeal in 1992. In the present work we propose a novel Petrov–Galerkin approach for the EAS method, which is equivalent to the standard EAS method in case of regular meshes. However, in case of distorted meshes, it allows to overcome the mesh-distortion sensitivity without loosing other advantages of the EAS method. Three design conditions established in this work facilitate the construction of the element which does not only fulfill the patch test but is also exact in many bending problems regardless of mesh distortion and has an exceptionally high coarse mesh accuracy. Consequently, high quality demands on mesh topology might be relaxed. more...
- Published
- 2021
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14. The Discontinuous Petrov–Galerkin methodology for the mixed Multiscale Finite Element Method
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Witold Cecot and Marta Oleksy
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Traction (engineering) ,Stability (learning theory) ,Petrov–Galerkin method ,010103 numerical & computational mathematics ,01 natural sciences ,Finite element method ,Displacement (vector) ,Mathematics::Numerical Analysis ,010101 applied mathematics ,Computational Mathematics ,Operator (computer programming) ,Computational Theory and Mathematics ,Approximation error ,Modeling and Simulation ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
We present the application of the Discontinuous Petrov–Galerkin (DPG) methodology for the mixed Multiscale Finite Element Method (MsFEM). The MsFEM upscaling technique relies on incorporating fine-scale features through special, in a sense optimized for approximability, trial functions while the DPG methodology allows for the selection of the optimal test functions to provide stability of the FEM approximation. The special trial functions are computed online by the solution of local boundary value problems. We improved this process using the static condensation that restricted the construction of the functions to the coarse mesh skeleton (element interfaces) only. We have verified by numerical tests that the proposed improvement of MsFEM reduced both the approximation error and computational cost. Moreover, it simplified the algorithm significantly. The key component of this prolongation construction is our novel method for prolongation of both traction and displacement vectors on element edges of arbitrary shape. The proposed prolongation operator may be also used in the multigrid solver for direct analysis of composites with varying material parameters, using an arbitrary well-posed functional setting with or without the DPG methodology. more...
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- 2021
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15. A new meshless Fragile Points Method (FPM) with minimum unknowns at each point, for flexoelectric analysis under two theories with crack propagation, I: Theory and implementation
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Yue Guan, Leiting Dong, and Satya N. Atluri
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Polynomial ,Applied Mathematics ,Petrov–Galerkin method ,02 engineering and technology ,Isogeometric analysis ,021001 nanoscience & nanotechnology ,Computer Science::Numerical Analysis ,Finite element method ,Numerical integration ,Algebraic equation ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Mechanics of Materials ,Meshfree methods ,Applied mathematics ,0210 nano-technology ,Galerkin method ,Mathematics - Abstract
Flexoelectricity refers to a phenomenon which involves a coupling of the mechanical strain gradient and electric polarization. In this study, a meshless Fragile Points Method (FPM), is presented for analyzing flexoelectric effects in dielectric solids. Local, simple, polynomial and discontinuous trial and test functions are generated with the help of a local meshless Differential Quadrature approximation of derivatives. Both primal and mixed FPM are developed, based on two alternate flexoelectric theories, with or without the electric gradient effect and Maxwell stress. In the present primal as well as mixed FPM, only the displacements and electric potential are retained as explicit unknown variables at each internal Fragile Point in the final algebraic equations. Thus the number of unknowns in the final system of algebraic equations is kept to be absolutely minimal. An algorithm for simulating crack initiation and propagation using the present FPM is presented, with classic stress-based criterion as well as a Bonding-Energy-Rate(BER)-based criterion for crack development. The present primal and mixed FPM approaches represent clear advantages as compared to the current methods for computational flexoelectric analyses, using primal as well as mixed Finite Element Methods, Element Free Galerkin (EFG) Methods, Meshless Local Petrov Galerkin (MLPG) Methods, and Isogeometric Analysis (IGA) Methods, because of the following new features: they are simpler Galerkin meshless methods using polynomial trial and test functions; minimal DoFs per Point make it very user-friendly; arbitrary polygonal subdomains make it flexible for modeling complex geometries; the numerical integration of the primal as well as mixed FPM weak forms is trivially simple; and FPM can be easily employed in crack development simulations without remeshing or trial function enhancement. more...
- Published
- 2021
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16. Immersed‐interface finite element method based on a nonconformal Petrov–Galerkin formulation
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Mayuresh J. Patil
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Numerical Analysis ,Interface (Java) ,Applied Mathematics ,Mathematical analysis ,General Engineering ,Petrov–Galerkin method ,Finite element method ,Mathematics - Published
- 2021
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17. Mesh sampling and weighting for the hyperreduction of nonlinear Petrov–Galerkin reduced‐order models with local reduced‐order bases
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Charbel Farhat, Sebastian Grimberg, Charbel Bou-Mosleh, and Radek Tezaur
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Numerical Analysis ,Finite volume method ,Computer science ,Applied Mathematics ,General Engineering ,Petrov–Galerkin method ,Finite difference ,Applied mathematics ,CPU time ,Galerkin method ,Finite element method ,Parametric statistics ,Weighting - Abstract
The energy-conserving sampling and weighting (ECSW) method is a hyperreduction method originally developed for accelerating the performance of Galerkin projection-based reduced-order models (PROMs) associated with large-scale finite element models, when the underlying projected operators need to be frequently recomputed as in parametric and/or nonlinear problems. In this paper, this hyperreduction method is extended to Petrov-Galerkin PROMs where the underlying high-dimensional models can be associated with arbitrary finite element, finite volume, and finite difference semi-discretization methods. Its scope is also extended to cover local PROMs based on piecewise-affine approximation subspaces, such as those designed for mitigating the Kolmogorov $n$-width barrier issue associated with convection-dominated flow problems. The resulting ECSW method is shown in this paper to be robust and accurate. In particular, its offline phase is shown to be fast and parallelizable, and the potential of its online phase for large-scale applications of industrial relevance is demonstrated for turbulent flow problems with $O(10^7)$ and $O(10^8)$ degrees of freedom. For such problems, the online part of the ECSW method proposed in this paper for Petrov-Galerkin PROMs is shown to enable wall-clock time and CPU time speedup factors of several orders of magnitude while delivering exceptional accuracy. more...
- Published
- 2021
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18. Petrov–Galerkin method for the band structure computation of anisotropic and piezoelectric phononic crystals
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Hui Zheng, Liwei Shi, Liqun Wang, Meiling Zhao, and Songming Hou
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Crystal ,Materials science ,Condensed Matter::Superconductivity ,Applied Mathematics ,Modeling and Simulation ,Mathematical analysis ,Petrov–Galerkin method ,Basis function ,Boundary value problem ,Anisotropy ,Rotation (mathematics) ,Piezoelectricity ,Finite element method - Abstract
In this paper, the Petrov-Galerkin finite element interface method is modified to the vectorial form and applied to compute the band structure of phononic crystals with complicated scatterer geometry. Value-periodic subspace and projective grid are employed in this method. Complete mathematical model together with the continuity and equilibrium conditions on the scatterer interface as well as the Bloch-periodic boundary conditions on the unit cell are presented for these systems. We then apply this method to compute the band structure of three kinds of special phononic crystals: phononic crystal with anisotropic inclusions, phononic crystal containing piezoelectric materials and phononic crystal containing nano-piezoelectric materials. Taking advantage of asymmetric basis functions and non-body-fitted meshes, our method is suitable for the calculation and analysis of phononic crystals with complicated scatterer geometries. With plenty of numerical experiments, the accuracy and convergency of the proposed method is demonstrated. The influences of the rotation angle of anisotropic inclusion, complicated scatterer shape, filling fraction of the scatterers, and the rotation angle of the scatterers to the band structure of these three kinds of phononic crystals are investigated. This will provide a new perspective for the design and manufacture of phononic crystals with specific band structures. more...
- Published
- 2021
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19. Application of Petrov-Galerkin finite element method to shallow water waves model: Modified Korteweg-de Vries equation.
- Author
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Ak, T., Karakoc, S. B. G., and Biswas, A.
- Subjects
FINITE element method ,WATER waves ,WAVES (Fluid mechanics) ,HYDRODYNAMICS ,NUMERICAL analysis - Abstract
In this article, modified Korteweg-de Vries (mKdV) equation is solved numerically by using lumped Petrov-Galerkin approach, where weight functions are quadratic and the element shape functions are cubic B-splines. The proposed numerical scheme is tested by applying four test problems including single solitary wave, interaction of two and three solitary waves, and evolution of solitons with the Gaussian initial condition. In order to show the performance of the algorithm, the error norms, L
2 , L∞, and a couple of conserved quantities are computed. For the linear stability analysis of numerical algorithm, Fourier method is also investigated. As a result, the computed results show that the presented numerical scheme is a successful numerical technique for solving the mKdV equation. Therefore, the presented method is preferable to some recent numerical methods. [ABSTRACT FROM AUTHOR] more...- Published
- 2017
- Full Text
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20. Recent Advances in Least-Squares and Discontinuous Petrov–Galerkin Finite Element Methods
- Author
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Jay Gopalakrishnan, Leszek Demkowicz, Fleurianne Bertrand, MESA+ Institute, and Mathematics of Computational Science
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Computational Mathematics ,Computational Theory and Mathematics ,Modeling and Simulation ,Petrov–Galerkin method ,Applied mathematics ,Least squares ,Finite element method ,Mathematics - Published
- 2021
21. A Petrov–Galerkin finite element method for simulating chemotaxis models on stationary surfaces
- Author
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Xinlong Feng, Jianping Zhao, Shubo Zhao, and Xufeng Xiao
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Numerical analysis ,Petrov–Galerkin method ,010103 numerical & computational mathematics ,01 natural sciences ,Signal on ,Finite element method ,010101 applied mathematics ,Computational Mathematics ,Effective mass (solid-state physics) ,Maximum principle ,Computational Theory and Mathematics ,Modeling and Simulation ,Applied mathematics ,0101 mathematics ,Laplace operator ,Conservation of mass ,Mathematics - Abstract
In this paper, we present a Petrov–Galerkin finite element method for a class of chemotaxis models defined on surfaces, which describe the movement by one community in reaction to one chemical or biological signal on manifolds. It is desired for numerical methods to satisfy discrete maximum principle and discrete mass conservation property, which is a challenge due to the singular behavior of numerical solution. Thus a Petrov–Galerkin method is combined with an effective mass conservation factor to overcome the challenge. Furthermore, we prove two facts, this method maintains positivity and discrete mass conservation property. In addition, decoupled approach is applied based on the gradient and Laplacian recoveries to solve the coupling system. The relevant stability analyses is provided. Finally, numerical simulations of blowing-up problems and pattern formulations demonstrate the effectiveness of the proposed method. more...
- Published
- 2020
- Full Text
- View/download PDF
22. A Petrov–Galerkin finite element-meshfree formulation for multi-dimensional fractional diffusion equations
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Dongdong Wang, Like Deng, Dongliang Qi, and Zeng Lin
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Weight function ,Discretization ,Applied Mathematics ,Mechanical Engineering ,Computational Mechanics ,Petrov–Galerkin method ,Ocean Engineering ,Weak formulation ,Finite element method ,Fractional calculus ,Computational Mathematics ,Nonlinear system ,Computational Theory and Mathematics ,Applied mathematics ,Meshfree methods ,Mathematics - Abstract
Meshfree methods with arbitrary order smooth approximation are very attractive for accurate numerical modeling of fractional differential equations, especially for multi-dimensional problems. However, the non-local property of fractional derivatives poses considerable difficulty and complexity for the numerical simulations of fractional differential equations and this issue becomes much more severe for meshfree methods due to the rational nature of their shape functions. In order to resolve this issue, a new weak formulation regarding multi-dimensional Riemann–Liouville fractional diffusion equations is introduced through unequally splitting the original fractional derivative of the governing equation into a fractional derivative for the weight function and an integer derivative for the trial function. Accordingly, a Petrov–Galerkin finite element-meshfree method is developed, where smooth reproducing kernel meshfree shape functions are adopted for the trial function approximation to enhance the solution accuracy, and the discretization of weight function is realized by the explicit finite element shape functions with an analytical fractional derivative evaluation to further reduce the computational complexity and improve efficiency. The proposed method enables a direct and efficient employment of meshfree approximation, and also eliminates the undesirable singular integration problem arising in the fractional derivative computation of meshfree shape functions. A nonlinear extension of the proposed method to the fractional Allen–Cahn equation is presented as well. The effectiveness of the proposed methodology is consistently demonstrated by numerical results. more...
- Published
- 2020
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23. Numerical simulation of two-dimensional fins using the meshless local Petrov – Galerkin method
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Rajul Garg, Harishchandra Thakur, and Brajesh Tripathi
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Natural convection ,Fin ,Biot number ,020209 energy ,General Engineering ,Petrov–Galerkin method ,02 engineering and technology ,Finite element method ,Computer Science Applications ,Forced convection ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Computational Theory and Mathematics ,Heat transfer ,0202 electrical engineering, electronic engineering, information engineering ,Applied mathematics ,Literature survey ,Software ,Mathematics - Abstract
Purpose The study aims to highlight the behaviour of one-dimensional and two-dimensional fin models under the natural room conditions, considering the different values of dimensionless Biot number (Bi). The effect of convection and radiation on the heat transfer process has also been demonstrated using the meshless local Petrov–Galerkin (MLPG) approach. Design/methodology/approach It is true that MLPG method is time-consuming and expensive in terms of man-hours, as it is in the developing stage, but with the advent of computationally fast new-generation computers, there is a big possibility of the development of MLPG software, which will not only reduce the computational time and cost but also enhance the accuracy and precision in the results. Bi values of 0.01 and 0.10 have been taken for the experimental investigation of one-dimensional and two-dimensional rectangular fin models. The numerical simulation results obtained by the analytical method, benchmark numerical method and the MLPG method for both the models have been compared with that of the experimental investigation results for validation and found to be in good agreement. Performance of the fin has also been demonstrated. Findings The experimental and numerical investigations have been conducted for one-dimensional and two-dimensional linear and nonlinear fin models of rectangular shape. MLPG is used as a potential numerical method. Effect of radiation is also, implemented successfully. Results are found to be in good agreement with analytical solution, when one-dimensional steady problem is solved; however, two-dimensional results obtained by the MLPG method are compared with that of the finite element method and found that the proposed method is as accurate as the established method. It is also found that for higher Bi, the one-dimensional model is not appropriate, as it does not demonstrate the appreciated error; hence, a two-dimensional model is required to predict the performance of a fin. Radiative fin illustrates more heat transfer than the pure convective fin. The performance parameters show that as the Bi increases, the performance of fin decreases because of high thermal resistance. Research limitations/implications Though, best of the efforts have been put to showcase the behaviour of one-dimensional and two-dimensional fins under nonlinear conditions, at different Bi values, yet lot more is to be demonstrated. Nonlinearity, in the present paper, is exhibited by using the thermal and material properties as the function of temperature, but can be further demonstrated with their dependency on the area. Additionally, this paper can be made more elaborative by extending the research for transient problems, with different fin profiles. Natural convection model is adopted in the present study but it can also be studied by using forced convection model. Practical implications Fins are the most commonly used medium to enhance heat transfer from a hot primary surface. Heat transfer in its natural condition is nonlinear and hence been demonstrated. The outcome is practically viable, as it is applicable at large to the broad areas like automobile, aerospace and electronic and electrical devices. Originality/value As per the literature survey, lot of work has been done on fins using different numerical methods; but to the best of authors’ knowledge, this study is first in the area of nonlinear heat transfer of fins using dimensionless Bi by the truly meshfree MLPG method. more...
- Published
- 2020
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24. Fluid–Structure Interaction Based on Meshless Local Petrov–Galerkin Method for Worm Soft Robot Analysis
- Author
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Seog Young Han, Dae Hwan Moon, Seung‑Hyun Shin, and Jong‑Beom Na
- Subjects
0209 industrial biotechnology ,Level set method ,Renewable Energy, Sustainability and the Environment ,Computer science ,Mechanical Engineering ,Petrov–Galerkin method ,02 engineering and technology ,021001 nanoscience & nanotechnology ,Industrial and Manufacturing Engineering ,Finite element method ,Nonlinear system ,020901 industrial engineering & automation ,Control theory ,Management of Technology and Innovation ,Distortion ,Hyperelastic material ,Fluid–structure interaction ,Robot ,General Materials Science ,0210 nano-technology - Abstract
The purpose of this study was to develop a two-way fluid–structure interaction (FSI) method using the meshless local Petrov–Galerkin (MLPG) method for both the structure and the fluid to accurately predict the nonlinear behavior of a worm soft robot. Previous research on soft robots has been mainly performed by finite element analysis (FEA). However, the nonlinear behavior of a soft robot causes element distortion and discontinuous stress between the adjacent elements, even when adaptive mesh is employed in the FEA. Therefore, MLPG was employed here to precisely predict the nonlinear behavior of a soft robot without using finite elements. In addition, a pneumatic soft robot simulation requires two-way FSI analysis that can transmit and receive data between the fluid and the structure, and the structure and the fluid, in sequence. To improve accuracy for the interface, the arbitrary Lagrangian–Eulerian method and the level set method were applied here. It was verified that the maximum errors of the finite element method FSI and the developed MLPG FSI method were 4.69%, and 0.77%, respectively, and the latter method required fewer nodes than FEM. more...
- Published
- 2020
- Full Text
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25. Numerical Upscaling of Perturbed Diffusion Problems
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Axel Målqvist, Tim Keil, and Fredrik Hellman
- Subjects
35J15, 65N12, 65N30 ,Applied Mathematics ,Computation ,Numerical analysis ,MathematicsofComputing_NUMERICALANALYSIS ,Petrov–Galerkin method ,Perturbation (astronomy) ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,01 natural sciences ,Finite element method ,Computational Mathematics ,Elliptic partial differential equation ,FOS: Mathematics ,Applied mathematics ,Orthogonal decomposition ,Mathematics - Numerical Analysis ,0101 mathematics ,Domain mapping ,Mathematics - Abstract
In this paper we study elliptic partial differential equations with rapidly varying diffusion coefficient that can be represented as a perturbation of a reference coefficient. We develop a numerical method for efficiently solving multiple perturbed problems by reusing local computations performed with the reference coefficient. The proposed method is based on the Petrov-Galerkin localized orthogonal decomposition (PG-LOD), which allows for straightforward parallelization with low communication overhead and memory consumption. We focus on two types of perturbations: local defects, which we treat by recomputation of multiscale shape functions, and global mappings of a reference coefficient for which we apply the domain mapping method. We analyze the proposed method for these problem classes and present several numerical examples. more...
- Published
- 2020
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26. High degree discontinuous Petrov–Galerkin immersed finite element methods using fictitious elements for elliptic interface problems
- Author
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Qiao Zhuang and Ruchi Guo
- Subjects
Computational Mathematics ,Degree (graph theory) ,Rate of convergence ,Function space ,Group (mathematics) ,Applied Mathematics ,Petrov–Galerkin method ,Applied mathematics ,Space (mathematics) ,Least squares ,Finite element method ,Mathematics - Abstract
We propose a new strategy for constructing the p th degree immersed finite element (IFE) spaces by applying the least squares framework in Adjerid et al. (2017) on fictitious elements. This new construction method significantly reduces the ill-conditioning, caused by the small subelement issue in Adjerid et al. (2017), of solving the local IFE shape functions. The proposed IFE spaces are employed in a discontinuous Petrov–Galerkin (DPG) scheme to solve the second order elliptic interface problems. We present a group of numerical examples to show that the DPGIFE method with the new p th degree IFE space as the trial function space has the optimal convergence rate, which improves the numerical results reported in Adjerid et al. (2017). more...
- Published
- 2019
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27. Quasi-static analysis of mixed-mode crack propagation using the meshless local Petrov–Galerkin method
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Amin Memari and Hamed Mohebalizadeh
- Subjects
Applied Mathematics ,Mathematical analysis ,General Engineering ,Petrov–Galerkin method ,Fracture mechanics ,02 engineering and technology ,01 natural sciences ,Finite element method ,010101 applied mathematics ,Computational Mathematics ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Test functions for optimization ,Cylinder stress ,Boundary value problem ,0101 mathematics ,Galerkin method ,Boundary element method ,Analysis ,Mathematics - Abstract
The efficiency of classical truly meshless local Petrov–Galerkin method with linear test function approximation in the crack growth problems of complex configurations is investigated in this article. Several unique and multiple (two) cracks are examined to show the accuracy of the proposed meshfree method by comparing the crack path with boundary element, finite element, element-free Galerkin and experimental results. Crack propagation under both mechanical and thermal loads and also mode I and mixed-mode conditions are evaluated. Effect of the functionality of material properties on crack paths and sensitivity of crack growth to boundary conditions are investigated. Classical maximum circumferential stress criterion is employed into the formulation of the MLPG method to predict crack growth direction. Despite the common numerical studies, a simple and straightforward automatic increment size determination method is introduced and implemented in the MLPG method. Besides, regular domain point distribution in the vicinity of the crack tip without the increased density of points is used. Various test problems show the computational accuracy and applicability of this method to linear elastic fracture mechanics problems. more...
- Published
- 2019
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28. A Petrov-Galerkin finite element interface method for interface problems with Bloch-periodic boundary conditions and its application in phononic crystals
- Author
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Liqun Wang, Xin Lu, Liwei Shi, and Hui Zheng
- Subjects
Physics ,Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Basis (linear algebra) ,Applied Mathematics ,Mathematical analysis ,Petrov–Galerkin method ,Transverse wave ,010103 numerical & computational mathematics ,01 natural sciences ,Finite element method ,Computer Science Applications ,010101 applied mathematics ,Piecewise linear function ,Computational Mathematics ,Matrix (mathematics) ,Modeling and Simulation ,Periodic boundary conditions ,Boundary value problem ,0101 mathematics - Abstract
In this paper, we propose a Petrov-Galerkin finite element interface method (PGFEIM) to solve the elliptic and elastic interface problems with Bloch-periodic boundary conditions. The main idea of this method is to choose the standard finite element basis independent of the interface to be the test function basis, and choose a piecewise linear function satisfying the jump conditions across the interface to be the solution basis. The grid we use is a non-body-fitted grid. The PGFEIM is capable of dealing with sharp-edged interface problems with matrix coefficients and nonhomogeneous jump conditions. Further, we extend this method to compute the band structure of anti-plane transverse waves and in-plane elastic waves in phononic crystals. Different acoustic impedance ratios, arbitrary complex scatterer geometries and various material properties are considered and discussed. Numerical experiments demonstrate that the PGFEIM is nearly second order accurate in the L ∞ norm for interface problems, and it is accurate and efficient for computing the band structure of phononic crystals. more...
- Published
- 2019
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29. Raviart–Thomas finite elements of Petrov–Galerkin type
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François Dubois, Charles Pierre, and Isabelle Greff
- Subjects
Numerical Analysis ,Finite volume method ,Applied Mathematics ,Computation ,Duality (mathematics) ,Petrov–Galerkin method ,Context (language use) ,010103 numerical & computational mathematics ,01 natural sciences ,Finite element method ,Mathematics::Numerical Analysis ,010101 applied mathematics ,Computational Mathematics ,Orthogonality ,Modeling and Simulation ,Applied mathematics ,0101 mathematics ,Scalar field ,Analysis ,Mathematics - Abstract
Finite volume methods are widely used, in particular because they allow an explicit and local computation of a discrete gradient. This computation is only based on the values of a given scalar field. In this contribution, we wish to achieve the same goal in a mixed finite element context of Petrov–Galerkin type so as to ensure a local computation of the gradient at the interfaces of the elements. The shape functions are the Raviart–Thomas finite elements. Our purpose is to define test functions that are in duality with these shape functions: precisely, the shape and test functions will be asked to satisfy some orthogonality property. This paradigm is addressed for the discrete solution of the Poisson problem. The general theory of Babuška brings necessary and sufficient stability conditions for a Petrov–Galerkin mixed problem to be convergent. In order to ensure stability, we propose specific constraints for the dual test functions. With this choice, we prove that the mixed Petrov–Galerkin scheme is identical to the four point finite volume scheme of Herbin, and to the mass lumping approach developed by Baranger, Maitre and Oudin. Convergence is proven with the usual techniques of mixed finite elements. more...
- Published
- 2019
- Full Text
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30. Numerical approximation of the generalized regularized long wave equation using Petrov–Galerkin finite element method
- Author
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Seydi Battal Gazi Karakoç, Samir Kumar Bhowmik, and Nevşehir Hacı Bektaş Veli Üniversitesi/fen-edebiyat fakültesi/matematik bölümü/uygulamalı matematik anabilim dalı
- Subjects
Numerical Analysis ,GRLW equation ,Applied Mathematics ,Mathematical analysis ,Petrov–Galerkin method ,Numerical Analysis (math.NA) ,Wave equation ,Finite element method ,Computational Mathematics ,Numerical approximation ,Petrov–Galerkin ,FOS: Mathematics ,Cubic b splines ,Cubic B-spline ,Mathematics - Numerical Analysis ,Soliton ,Analysis ,Mathematics - Abstract
The generalized regularized long wave (GRLW) equation has been developed to model a variety of physical phenomena such as ion-acoustic and magnetohydrodynamic waves in plasma, nonlinear transverse waves in shallow water and phonon packets in nonlinear crystals. This paper aims to develop and analyze a powerful numerical scheme for the nonlinear generalized regularized long wave (GRLW) equation by Petrov--Galerkin method in which the element shape functions are cubic and weight functions are quadratic B-splines. The suggested method is performed to three test problems involving propagation of the single solitary wave, interaction of two solitary waves and evolution of solitons with the Maxwellian initial condition. The variational formulation and semi-discrete Galerkin scheme of the equation are firstly constituted. We estimate accuracy of such a spatial approximation. Then Fourier stability analysis of the linearized scheme shows that it is unconditionally stable. To verify practicality and robustness of the new scheme error norms $L_{2}$, $L_{\infty }$ and three invariants $I_{1},I_{2}$ and $I_{3}$ are calculated. The obtained numerical results are compared with other published results and shown to be precise and effective., 28 pages more...
- Published
- 2019
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31. The application of the meshless local Petrov-Galerkin method for the analysis of heat conduction and residual stress due to welding
- Author
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Shahram Shahrooi, Ali Moarrefzadeh, and Mahdi Jalali Azizpour
- Subjects
0209 industrial biotechnology ,Quadrature domains ,Mechanical Engineering ,Weak solution ,Numerical analysis ,Mathematical analysis ,Petrov–Galerkin method ,02 engineering and technology ,Welding ,Thermal conduction ,Industrial and Manufacturing Engineering ,Finite element method ,Computer Science Applications ,law.invention ,020901 industrial engineering & automation ,Control and Systems Engineering ,Residual stress ,law ,Software ,Mathematics - Abstract
The structural analysis of welding based on the thermal-elastic-plastic finite element method can lead to acceptable accuracy for the short time of welding. But if the welding time is long and the thermal source is moving, problems arise in the analysis due to the mesh re-generation. In this paper, in order to overcome the weak points of the finite element method, the meshless local Petrov-Galerkin (MLPG) method by using moving least square (MLS) has been used, based on the weak solution of equations governing the temperature field and the residual stress caused by welding. Without increasing the computational time, accuracy of the MLPG method has increased by developing this numerical method by taking into account the influence of MLPG parameters. To investigate the distance and density of nodal distribution, the relation between support domain size and quadrature domain size with distance between nodes is optimized. A good agreement is seen between the outputs of this numerical method to the hole-drilling strain-gauge method results. Therefore, it can be concluded that a new application based on thermoelastic-plastic equation for the MLPG method to numerical analysis of the residual stress due to welding is introduced. more...
- Published
- 2019
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32. A Bilinear Petrov-Galerkin Finite Element Method for Solving Elliptic Equation with Discontinuous Coefficients
- Author
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Liqun Wang
- Subjects
Physics ,Elliptic curve ,Applied Mathematics ,Mechanical Engineering ,Petrov–Galerkin method ,Bilinear interpolation ,Applied mathematics ,Finite element method - Published
- 2019
- Full Text
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33. Goal-Oriented Adaptive Mesh Refinement for Discontinuous Petrov--Galerkin Methods
- Author
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Ali Vaziri Astaneh, Brendan Keith, and Leszek Demkowicz
- Subjects
Numerical Analysis ,Goal orientation ,Adaptive mesh refinement ,Applied Mathematics ,Duality (mathematics) ,Petrov–Galerkin method ,010103 numerical & computational mathematics ,01 natural sciences ,Finite element method ,Computational Mathematics ,Applied mathematics ,A priori and a posteriori ,0101 mathematics ,Mathematics - Abstract
This article lays a mathematical foundation for goal-oriented adaptive mesh refinement and a posteriori error estimation with discontinuous Petrov--Galerkin (DPG) finite element methods. A goal-ori... more...
- Published
- 2019
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34. The Slope Stability Solution Using Meshless Local Petrov-Galerkin Method
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Filip Gago, Juraj Mužík, and Roman Bulko
- Subjects
050210 logistics & transportation ,Numerical analysis ,05 social sciences ,0211 other engineering and technologies ,Petrov–Galerkin method ,02 engineering and technology ,General Medicine ,Computer Science::Computational Geometry ,Weak formulation ,Computer Science::Numerical Analysis ,Finite element method ,Mathematics::Numerical Analysis ,Computer Science::Computational Engineering, Finance, and Science ,Slope stability ,021105 building & construction ,0502 economics and business ,Meshfree methods ,Applied mathematics ,Limit (mathematics) ,Slope stability analysis ,Mathematics - Abstract
The paper deals with use of the meshless method for slope stability analysis. There are many formulations of the meshless methods. The article presents the Meshless Local Petrov-Galerkin method (MLPG) – local weak formulation of the equilibrium equations. The main difference between meshless methods and the conventional finite element method (FEM) is that meshless shape functions are constructed using randomly scattered set of points without any relation between points. The shape function construction is the crucial part of the meshless numerical analysis. The numerical example of the slope stability was calculated using meshless computer code and compared with FEM and limit equilibrium (LEM) results. more...
- Published
- 2019
- Full Text
- View/download PDF
35. Local maximum entropy approximation-based streamline upwind Petrov–Galerkin meshfree method for convection–diffusion problem
- Author
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Sreehari Peddavarapu and Raghuraman Srinivasan
- Subjects
0209 industrial biotechnology ,Mechanical Engineering ,Applied Mathematics ,Principle of maximum entropy ,General Engineering ,Petrov–Galerkin method ,Aerospace Engineering ,Boundary (topology) ,Basis function ,02 engineering and technology ,Industrial and Manufacturing Engineering ,Finite element method ,Mathematics::Numerical Analysis ,symbols.namesake ,020901 industrial engineering & automation ,Kronecker delta ,Automotive Engineering ,symbols ,Applied mathematics ,Boundary value problem ,Convection–diffusion equation ,Mathematics - Abstract
Local maximum entropy (LME) meshfree basis functions are among the few approximants that possess the Kronecker delta property on the boundary, which enable to impose the essential boundary conditions directly like FEM. This study presents the potential of a meshfree method based on LME approximation for the Convection–Diffusion problem via well-celebrated SUPG. The present LME based Streamline Upwind Petrov–Galerkin meshfree method (SUPGM) benefits from the advantages emanated from both methods. LME approximants accounts for the disposal of the elements and direct imposition of boundary conditions and the SUPGM technique to deal with the issues associated with the non-self-adjoint convective term. Two standard benchmark problems are considered to validate the LME-SUPGM. The effect of different priors and radius of support that defines the LME basis are studied to test their importance in the present method. It is found that a balance between stability and accuracy is possible with ease by tuning the effective radius of support for LME for the given problem. The present method converges faster than FEM-SUPG and provides a smooth solution relatively. more...
- Published
- 2021
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36. Numerical Solution of Ninth Order Boundary Value Problems by Petrov-Galerkin Method with Quintic B-splines as Basis Functions and Septic B-splines as Weight Functions.
- Author
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Viswanadham, K.N.S. Kasi and Reddy, S.M.
- Subjects
NUMERICAL solutions to boundary value problems ,GALERKIN methods ,SPLINES ,RADIAL basis functions ,FINITE element method - Abstract
In this paper a finite element method involving Petrov-Galerkin method with quintic B-splines as basis functions and septic B-splines as weight functions has been developed to solve a general ninth order boundary value problem with a particular case of boundary conditions. The basis functions are redefined into a new set of basis functions which vanish on the boundary where the Dirichlet, the Neumann and second order derivative type of boundary conditions are prescribed. The weight functions are also redefined into a new set of weight functions which in number match with the number of redefined basis functions. The proposed method was applied to solve several examples of linear and nonlinear ninth order boundary value problems. The obtained numerical results were found to be in good agreement with the exact solutions available in the literature. [ABSTRACT FROM AUTHOR] more...
- Published
- 2015
- Full Text
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37. An $$h$$ - $$p$$ Version of the Continuous Petrov-Galerkin Finite Element Method for Nonlinear Volterra Integro-Differential Equations.
- Author
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Yi, Lijun
- Published
- 2015
- Full Text
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38. Construction of Weight Functions of the Petrov-Galerkin Method for Convection-Diffusion-Reaction Equations in the Three-Dimensional Case.
- Author
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Salnikov, N. and Siryk, S.
- Subjects
- *
OSCILLATION theory of differential equations , *ACCURACY , *COMPUTER simulation , *POLYNOMIAL approximation , *PIECEWISE linear topology - Abstract
A method is proposed for constructing continuous piecewise-polynomial weight functions for the Petrov-Galerkin method in the three-dimensional domain. The form of such functions is determined by a finite number of variable parameters associated with edges of a grid partition. The choice of these parameters allows one to obtain numerical approximations for the original equation without non-physical oscillations with preserving an adequate accuracy. The results of the investigation are illustrated by several numerical examples. [ABSTRACT FROM AUTHOR] more...
- Published
- 2014
- Full Text
- View/download PDF
39. SOLVING FRACTIONAL DIFFUSION AND FRACTIONAL DIFFUSION-WAVE EQUATIONS BY PETROV-GALERKIN FINITE ELEMENT METHOD.
- Author
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ESEN, A., UCAR, Y., YAGMURLU, M., and TASBOZAN, O.
- Subjects
DIFFUSION ,WAVE equation ,GALERKIN methods - Abstract
In the last few years, it has become highly evident that fractional calculus has been widely used in several areas of science. Because of this fact, their numerical solutions also have become urgently important. In this manuscript, numerical solutions of both the fractional diffusion and fractional diffusion-wave equations have been obtained by a Petrov-Galerkin finite element method using quadratic B-spline base functions as trial functions and linear B-spline base functions as the test functions. In those equations, fractional derivatives are used in terms of the Caputo sense. While the L1 discretizaton formula has been applied to fractional diffusion equation, the L2 discretizaton formula has been applied to the fractional diffusion-wave equation. Finally, the error norms L2 and L8 have been calculated for testing the accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR] more...
- Published
- 2014
40. Extension of a second order velocity slip/temperature jump boundary condition to simulate high speed micro/nanoflows.
- Author
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Rooholghdos, Seyed Ali and Roohi, Ehsan
- Subjects
- *
TEMPERATURE , *BOUNDARY value problems , *FINITE element method , *GALERKIN methods , *DISCRETIZATION methods , *NUMBER theory - Abstract
Abstract: In the current work, for the first time, we extend the application of a second order slip/jump equations introduced by Karniadakis et al. for the simulation of high speed, high Knudsen ( ) number flows over a nano-scale flat plate and a micro-scale cylinder. The NS equations subject to a second order slip/jump boundary conditions are solved using the Petrov–Galerkin Finite Element discretization. We compare our numerical solution for flow and thermal field with the solution of the DSMC and Generalized Hydrodynamic (GH) techniques, as well as a recently developed slip/jump boundary condition, i.e., Paterson equation. Current results demonstrate the suitable accuracy of the employed boundary conditions for different set of test cases. Our numerical solutions are obtained with much less numerical costs compared to alternative boundary conditions. [Copyright &y& Elsevier] more...
- Published
- 2014
- Full Text
- View/download PDF
41. Accuracy and Stability of the Petrov-Galerkin Method for Solving the Stationary Convection-Diffusion Equation.
- Author
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Siryk, S.
- Subjects
- *
NUMERICAL solutions to convection-diffusion equations , *STABILITY theory , *GALERKIN methods , *FINITE element method , *PARAMETERS (Statistics) , *PARAMETER estimation , *STOCHASTIC convergence - Abstract
The accuracy and stability of numerical solution of the stationary convection-diffusion equation by the finite element Petrov-Galerkin method are analyzed with the use of weight functions with different stabilization parameters as test functions, and estimates are obtained for the accuracy of the method depending on the choice of a collection of stabilization parameters. The convergence of the method is shown. [ABSTRACT FROM AUTHOR] more...
- Published
- 2014
- Full Text
- View/download PDF
42. A discontinuous Petrov-Galerkin method for compressible viscous flows in three dimensions
- Author
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Waldemar Rachowicz, Adam Zdunek, and Witold Cecot
- Subjects
Compressibility ,Petrov–Galerkin method ,Stability (learning theory) ,Applied mathematics ,Perturbation (astronomy) ,A priori and a posteriori ,Polygon mesh ,Weak formulation ,Finite element method ,Mathematics - Abstract
The Discontinuous Petrov-Galerkin (DPG) method allows one to construct stable finite element schemes for some classes of singularily perturbed problems like, for instance, convection-dominated diffusion. The central ingredient of the method is a special weak formulation characterized by a relaxed interelement continuity of approximation. It satisfies the inf-sup stability condition with the stability constant independent of the small parameter of perturbation. This technology was applied by Chan et al. in [1] to develop a scheme for simulation of the two-dimensional compressible Navier-Stokes equations. In this paper we extend that work to three dimensions. We present the necessary modifications, the details of the functional setting, the discrete trial and test finite element spaces, and the final form of the algorithm. The issues of a posteriori error estimation and h-adaptivity of finite element meshes are addressed. The technique is illustrated with a few preliminary numerical examples. more...
- Published
- 2020
- Full Text
- View/download PDF
43. On a relation of discontinuous Petrov–Galerkin and least-squares finite element methods
- Author
-
Johannes Storn
- Subjects
Helmholtz equation ,Linear elasticity ,Petrov–Galerkin method ,Residual ,Finite element method ,Mathematics::Numerical Analysis ,Computational Mathematics ,Computational Theory and Mathematics ,Modeling and Simulation ,Norm (mathematics) ,Modelling and Simulation ,Applied mathematics ,Minification ,Condition number ,Mathematics - Abstract
The discontinuous Petrov–Galerkin (DPG) method minimizes a residual in a non-standard norm. This paper shows that the minimization of this residual is equivalent to the minimization of a residual in a L 2 norm. Since such residuals are well known from least squares finite element methods, this novel interpretation allows to extend results for least squares methods to the DPG method and vice versa. This paper exemplifies the benefits of this possibility by the verification of an asymptotic exactness result for a DPG method for the Helmholtz equation, the design of a locking-free DPG method for linear elasticity, and an investigation of the spectral condition number. more...
- Published
- 2020
44. Mixed FEM and the discontinuous Petrov-Galerkin (DPG) methodology in numerical homogenization
- Author
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Witold Cecot and Marta Oleksy
- Subjects
symbols.namesake ,Piola transformation ,Cauchy stress tensor ,Lagrange multiplier ,symbols ,Petrov–Galerkin method ,A priori and a posteriori ,Applied mathematics ,Mixed finite element method ,Homogenization (chemistry) ,Finite element method ,Mathematics - Abstract
The mixed finite element method uses an approximation of at least two fields (e.g. displacements and stresses). Even though it is difficult to construct stable approximation for such formulations, the mixed methods provide much better convergence of the flux than the standard methods, even for heterogeneous materials with a high contrast of component parameters. Furthermore, the mixed formulations are well-posed for incompressible materials. In this work a novel application of the mixed FEM to compu- tational homogenization is presented and examined. Conformity with the H(div) space is provided by the Piola transformation and the approximation orders are assumed according to the exact sequence of energy spaces. The modified Hellinger-Reissner principle with weakly imposed stress tensor symmetry through the introduction of the Lagrange multiplier is used and the Discontinuous Petrov-Galerkin approach (DPG), with built-in a posteriori error estimation, provides stability of approximation. more...
- Published
- 2020
- Full Text
- View/download PDF
45. Performance of interpolating approximation schemes in MLPG meshless solver
- Author
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Rituraj Singh and Roman Trobec
- Subjects
Laplace's equation ,Petrov–Galerkin method ,Applied mathematics ,Boundary value problem ,Solver ,Finite element method ,Mathematics::Numerical Analysis ,Mathematics - Abstract
An interpolating moving least square (MLS) approximation was applied in the Meshless Petrov Galerkin method for a solution of Laplace equation. Accuracy and calculation complexity have been analyzed. Results show that original and interpolating MLS schemes are comparable in accuracy, with a small advantage for MLS. The interpolating MLS supports the direct imposition of essential boundary conditions, like in FEM. more...
- Published
- 2020
- Full Text
- View/download PDF
46. A Petrov-Galerkin finite element method for 2D transient and steady state highly advective flows in porous media
- Author
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Lidija Zdravković, Wenjie Cui, David M. Potts, Klementyna A. Gawecka, David M. G. Taborda, Engineering and Physical Sciences Research Council, and Geotechnical Consulting Group
- Subjects
Finite element methods ,Technology ,Porous media ,0211 other engineering and technologies ,Petrov–Galerkin method ,CONVECTIVE-TRANSPORT-EQUATION ,2D highly advectiveflows ,010103 numerical & computational mathematics ,02 engineering and technology ,Computational fluid dynamics ,0915 Interdisciplinary Engineering ,Geological & Geomatics Engineering ,01 natural sciences ,0905 Civil Engineering ,Engineering ,Quadratic equation ,Robustness (computer science) ,Applied mathematics ,Engineering, Geological ,Geosciences, Multidisciplinary ,0101 mathematics ,FORMULATION ,021101 geological & geomatics engineering ,Science & Technology ,business.industry ,Advection ,COMPUTATIONAL FLUID-DYNAMICS ,DIFFUSION-PROBLEMS ,SCHEME ,Geology ,0914 Resources Engineering and Extractive Metallurgy ,Geotechnical Engineering and Engineering Geology ,Finite element method ,TIME ,Computer Science Applications ,Weighting ,Physical Sciences ,Computer Science ,Computer Science, Interdisciplinary Applications ,Porous medium ,business - Abstract
A new Petrov-Galerkin finite element method for two-dimensional (2D) highly advective flows in porous media, which removes numerical oscillations and retains its precision compared to the conventional Galerkin finite element method, is presented. A new continuous weighting function for quadratic elements is proposed. Moreover, a numerical scheme is developed to ensure the weighting factors are accurately determined for 2D non-uniform flows and 2D distorted elements. Finally, a series of numerical examples are performed to demonstrate the capability of the approach. Comparison against existing methods in the simulation of a benchmark problem further verifies the robustness of the proposed method. more...
- Published
- 2018
- Full Text
- View/download PDF
47. An h-p version of the continuous Petrov-Galerkin finite element method for Riemann-Liouville fractional differential equation with novel test basis functions
- Author
-
Aiguo Xiao and Weiping Bu
- Subjects
Applied Mathematics ,Numerical analysis ,Petrov–Galerkin method ,Boundary (topology) ,Basis function ,010103 numerical & computational mathematics ,Riemann liouville ,01 natural sciences ,Finite element method ,Dirichlet distribution ,010101 applied mathematics ,symbols.namesake ,symbols ,0101 mathematics ,Fractional differential ,Mathematical physics ,Mathematics - Abstract
In this paper, we develop an h-p version of finite element method for one-dimensional fractional differential equation ?0Dx?u+Au=f(x)$-_{0}D_{x}^{\alpha }u+Au=f(x)$ with Dirichlet boundary conditio... more...
- Published
- 2018
- Full Text
- View/download PDF
48. Inf-sup stability of Petrov-Galerkin immersed finite element methods for one-dimensional elliptic interface problems
- Author
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Bin Zhang, Haifeng Ji, and Qian Zhang
- Subjects
Numerical Analysis ,Interface (Java) ,Applied Mathematics ,Mathematical analysis ,Petrov–Galerkin method ,010103 numerical & computational mathematics ,01 natural sciences ,Stability (probability) ,Finite element method ,010101 applied mathematics ,Computational Mathematics ,0101 mathematics ,Analysis ,Mathematics - Published
- 2018
- Full Text
- View/download PDF
49. A low-order discontinuous Petrov–Galerkin method for the Stokes equations
- Author
-
Carsten Carstensen and Sophie Puttkammer
- Subjects
Discretization ,Applied Mathematics ,Numerical analysis ,Petrov–Galerkin method ,010103 numerical & computational mathematics ,01 natural sciences ,Finite element method ,Mathematics::Numerical Analysis ,010101 applied mathematics ,Computational Mathematics ,Piecewise ,Applied mathematics ,Direct proof ,Affine transformation ,0101 mathematics ,Ansatz ,Mathematics - Abstract
This paper introduces a low-order discontinuous Petrov-Galerkin (dPG) finite element method (FEM) for the Stokes equations. The ultra-weak formulation utilizes piecewise constant and affine ansatz functions and piecewise affine and discontinuous lowest-order Raviart–Thomas test search functions. This low-order discretization for the Stokes equations allows for a direct proof of the discrete inf-sup condition with explicit constants. The general framework of Carstensen et al. (SIAM J Numer Anal 52(3):1335–1353, 2014) then implies a complete a priori and a posteriori error analysis of the dPG FEM in the natural norms. Numerical experiments investigate the performance of the method and underline its quasi-optimal convergence. more...
- Published
- 2018
- Full Text
- View/download PDF
50. Error analysis of projection methods for non inf-sup stable mixed finite elements. The transient Stokes problem
- Author
-
Javier de Frutos, Julia Novo, Bosco García-Archilla, and UAM. Departamento de Matemáticas
- Subjects
Matemáticas ,Petrov–Galerkin method ,010103 numerical & computational mathematics ,01 natural sciences ,Projection (linear algebra) ,Euler method ,symbols.namesake ,Non inf-sup stable elements ,FOS: Mathematics ,Projection method ,Mathematics - Numerical Analysis ,0101 mathematics ,Mathematics ,Applied Mathematics ,Semi-implicit Euler method ,65M12 ,Mathematical analysis ,PSPG stabilization ,Numerical Analysis (math.NA) ,Projection methods ,Backward Euler method ,Finite element method ,010101 applied mathematics ,Computational Mathematics ,Euler's formula ,symbols - Abstract
A modified Chorin–Teman (Euler non-incremental) projection method and a modified Euler incremental projection method for non inf-sup stable mixed finite elements are analyzed. The analysis of the classical Euler non-incremental and Euler incremental methods are obtained as a particular case. We first prove that the modified Euler non-incremental scheme has an inherent stabilization that allows the use of non inf-sup stable mixed finite elements without any kind of extra added stabilization. We show that it is also true in the case of the classical Chorin–Temam method. For the second scheme, we study a stabilization that allows the use of equal-order pairs of finite elements. The relation of the methods with the so-called pressure stabilized Petrov Galerkin method (PSPG) is established. The influence of the chosen initial approximations in the computed approximations to the pressure is analyzed. Numerical tests confirm the theoretical results, Research sup-ported by Spanish MINECO under grants MTM2013-42538-P (MINECO, ES) and MTM2016-78995-P (AEI/FEDER UE) more...
- Published
- 2018
- Full Text
- View/download PDF
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