260 results on '"Biharmonic equation"'
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2. Normalized solutions for a biharmonic Choquard equation with exponential critical growth in R4.
- Author
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Chen, Wenjing and Wang, Zexi
- Subjects
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BIHARMONIC equations , *EXPONENTIAL functions , *CONTINUOUS functions - Abstract
In this paper, we study the following biharmonic Choquard-type problem Δ 2 u - β Δ u = λ u + (I μ ∗ F (u)) f (u) , in R 4 , ∫ R 4 | u | 2 d x = c 2 > 0 , u ∈ H 2 (R 4) , where β ≥ 0 , λ ∈ R , I μ = 1 | x | μ with μ ∈ (0 , 4) , F(u) is the primitive function of f(u), and f is a continuous function with exponential critical growth. By using the mountain-pass argument, we prove the existence of radial ground-state solutions for the above problem. [ABSTRACT FROM AUTHOR]
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- 2024
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3. Modified [formula omitted] interior penalty analysis for fourth order Dirichlet boundary control problem and a posteriori error estimates.
- Author
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Chowdhury, Sudipto, Garg, Divay, and Shokeen, Ravina
- Abstract
We revisit the L 2 norm error estimate for the C 0 -interior penalty analysis of fourth order Dirichlet boundary control problem. The L 2 norm error estimate for the optimal control is derived under reduced regularity assumption and this analysis can be carried out on any convex polygonal domain. Moreover, residual based a posteriori error bounds are derived for the optimal control, state and adjoint state variables under minimal regularity assumptions. The error estimator is shown to be reliable and locally efficient. The theoretical findings are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]
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- 2024
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4. COMPUTATION OF TWO-DIMENSIONAL STOKES FLOWS VIA LIGHTNING AND AAA RATIONAL APPROXIMATION.
- Author
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YIDAN XUE, WATERS, SARAH L., and TREFETHEN, LLOYD N.
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STOKES flow , *FLUID flow , *LIGHTNING , *REYNOLDS number , *ANALYTIC functions , *ANALYTICAL solutions - Abstract
Low Reynolds number fluid flows are governed by the Stokes equations. In two dimensions, Stokes flows can be described by two analytic functions, known as Goursat functions. Brubeck and Trefethen [SIAM J. Sci. Comput., 44 (2022), pp. A1205-A1226] recently introduced a lightning Stokes solver that uses rational functions to approximate the Goursat functions in polyg- onal domains. In this paper, we present the LARS algorithm (lightning-AAA rational Stokes) for computing two-dimensional (2D) Stokes flows in domains with smooth boundaries and multiply con- nected domains using lightning and AAA rational approximation [Y. Nakatsukasa, O. Sète, and L. N. Trefethen, SIAM J. Sci. Comput., 40 (2018), pp. A1494-A1522]. After validating our solver against known analytical solutions, we solve a variety of 2D Stokes flow problems with physical and engineering applications. Using these examples, we show rational approximation can now be used to compute 2D Stokes flows in general domains. The computations take less than a second and give solutions with at least 6-digit accuracy. [ABSTRACT FROM AUTHOR]
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- 2024
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5. Introducing higher-order Haar wavelet method for solving three-dimensional partial differential equations.
- Author
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Sinha, Arvind Kumar and Sahoo, Radhakrushna
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PARTIAL differential equations , *HELMHOLTZ equation , *TENSOR products , *KRONECKER products , *COLLOCATION methods , *POISSON'S equation , *BIHARMONIC equations - Abstract
In this paper, we develop a collocation method for solving three-dimensional partial differential equations using Haar wavelet and Kronecker tensor product. The approach is based on a series of Haar wavelet basis functions to approximate sixth-order mixed derivatives. The proposed method is mathematically fast, less error and straightforward for the numerical solution of many types of three-dimensional Poisson, biharmonic and Helmholtz equations. Some numerical examples verify the accuracy and efficiency of the proposed method. Finally, we conclude that numerical results computed by our proposed method are more accurate than numerical results obtained in the existing methods in the literature. We find that the CPU time consumed by the suggested approach is lesser than the CPU time of existing methods. Thus, the process is fast, efficient and has a low numerical error. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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6. Shape reconstruction for an inverse biharmonic problem from partial Cauchy data.
- Author
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Bouslah, Zineb, Hadj, Abdelhak, and Saker, Hacene
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INVERSE problems , *BIHARMONIC equations , *NONLINEAR integral equations , *GEOMETRIC shapes , *BOUNDARY value problems - Abstract
In this paper, we address the Robin inverse problem for the biharmonic equation in a 2D$$ 2D $$ simply connected domain, to reconstruct the geometric shape of a non‐accessible part of the boundary from a single measurement of Riquier–Neumann data on the accessible part of that boundary. Our approach extends the nonlinear boundary integral equation, to recover the shape of the boundary. We propose the Newton iterative technique based on the Fréchet derivatives to linearize the system and then establish an injectivity of the linearized system for certain Robin coefficients, as well as the iteration scheme to describe the inverse algorithm for recovering the shape with respect to the unknowns. The mathematical spirit of the proposed method will be presented, and to illustrate its feasibility, some numerical examples will be provided. [ABSTRACT FROM AUTHOR]
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- 2024
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7. Gradient auxiliary physics-informed neural network for nonlinear biharmonic equation.
- Author
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Liu, Yu and Ma, Wentao
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NONLINEAR equations , *PARTIAL differential equations , *BIHARMONIC equations , *INVERSE problems , *DIFFERENTIAL equations , *MACHINE learning - Abstract
Physics-informed neural network (PINN) has gained wide attention for solving forward and inverse problems of partial differential equation (PDE) from both data-driven and model-driven perspectives. In a short time, various machine learning methods based on PINN have been developed for solving a broad range of PDE. To improve the training accuracy of PINN, the popular approach is to introduce penalty parameters to correct the imbalance among different parts of loss function during model training. However, this approach generally fails to eliminate the model error generated by the boundary condition. To eliminate this issue, we propose a gradient auxiliary physics-informed neural network (GA-PINN) for nonlinear biharmonic equation. The key idea of GA-PINN is to split original biharmonic equation subjected to clamped or simply supported boundary conditions into third order or second order differential equations by introducing gradient auxiliary functions. As a consequence, we can rewrite clamped or simply supported boundary conditions as Dirichlet boundary conditions, then conveniently construct the neural network composite functions to satisfy those Dirichlet boundary conditions. We introduce the dynamic weight method to automatically balance the contributions of different loss terms during training. The capabilities of the proposed GA-PINN by solving several nonlinear biharmonic problems are demonstrated. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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8. The local boundary knots method for solution of Stokes and the biharmonic equation.
- Author
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Kovářík, Karel, Mužík, Juraj, Gago, Filip, and Sitányiová, Dana
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STOKES equations , *STOKES flow , *BIHARMONIC equations , *COLLOCATION methods , *SPARSE matrices - Abstract
The paper focuses on the derivation of a local variant of the boundary knot method (BKM) for the cases of Stokes flow and the biharmonic equation. Compared to the global variant, the local one leads to a sparse result matrix of the system of equations and thus makes the solution of especially large-scale problems more efficient. It is also important to keep the conditionality of the interpolation matrix within reasonable bounds. For the localization, a combination of BKM and finite collocation method was used. The results of the local variant were compared on several examples and the dependence of the solution on the density of the point network and the dimensions of the stencil used was also tested in the paper. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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9. Local parameter selection in the C0 interior penalty method for the biharmonic equation.
- Author
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Bringmann, Philipp, Carstensen, Carsten, and Streitberger, Julian
- Abstract
The symmetric 0 interior penalty method is one of the most popular discontinuous Galerkin methods for the biharmonic equation. This paper introduces an automatic local selection of the involved stability parameter in terms of the geometry of the underlying triangulation for arbitrary polynomial degrees. The proposed choice ensures a stable discretization with guaranteed discrete ellipticity constant. Numerical evidence for uniform and adaptive mesh-refinement and various polynomial degrees supports the reliability and efficiency of the local parameter selection and recommends this in practice. The approach is documented in 2D for triangles, but the methodology behind can be generalized to higher dimensions, to non-uniform polynomial degrees, and to rectangular discretizations. An appendix presents the realization of our proposed parameter selection in various established finite element software packages. a detailed documentation of C0 interior penalty method in. [ABSTRACT FROM AUTHOR]
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- 2023
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10. NONCONFORMING VIRTUAL ELEMENTS FOR THE BIHARMONIC EQUATION WITH MORLEY DEGREES OF FREEDOM ON POLYGONAL MESHES.
- Author
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CARSTENSEN, CARSTEN, KHOT, REKHA, and PANI, AMIYA K.
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A posteriori error analysis , *BIHARMONIC equations , *DEGREES of freedom - Abstract
The lowest-order nonconforming virtual element extends the Morley triangular element to polygons for the approximation of the weak solution u ∊ V := H2 0 (\Omega) to the biharmonic equation. The abstract framework allows (even a mixture of) two examples of the local discrete spaces Vh(P) and a smoother allows rough source terms F ∊ V* =H-20(Ω). The a priori and a posteriori error analysis in this paper circumvents any trace of second derivatives by some computable conforming companion operator J: Vh → V from the nonconforming virtual element space Vh. The operator J is a right-inverse of the interpolation operator and leads to optimal error estimates in piecewise Sobolev norms without any additional regularity assumptions on u ∊ V. As a smoother the companion operator modifies the discrete right-hand side and then allows a quasi-best approximation. An explicit residual-based a posteriori error estimator is reliable and efficient up to data oscillations. Numerical examples display the predicted empirical convergence rates for uniform and optimal convergence rates for adaptive mesh-refinement. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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11. Existence and nonexistence of solutions to a critical biharmonic equation with logarithmic perturbation.
- Author
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Li, Qi, Han, Yuzhu, and Wang, Tianlong
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BIHARMONIC equations , *PERTURBATION theory - Abstract
In this paper, the following critical biharmonic elliptic problem { Δ 2 u = λ u + μ u ln u 2 + | u | 2 ⁎ ⁎ − 2 u , x ∈ Ω , u = ∂ u ∂ ν = 0 , x ∈ ∂ Ω is considered, where Ω ⊂ R N is a bounded smooth domain with N ≥ 5. Some interesting phenomena occur due to the uncertainty on the sign of the logarithmic term. It is shown, mainly by using Mountain Pass Lemma, that the problem admits at least one nontrivial weak solution under some appropriate assumptions of λ and μ. Moreover, a nonexistence result is also obtained. Comparing the results in this paper with the known ones, one sees that some new phenomena occur when the logarithmic perturbation is introduced. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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12. The interior penalty virtual element method for the biharmonic problem.
- Author
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Zhao, Jikun, Mao, Shipeng, Zhang, Bei, and Wang, Fei
- Subjects
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NUMBER systems , *PROBLEM solving , *DEGREES of freedom , *LINEAR systems , *BIHARMONIC equations , *POLYGONS - Abstract
In this paper, an interior penalty virtual element method (IPVEM) is developed for solving the biharmonic problem on polygonal meshes. By modifying the existing H^2-conforming virtual element, an H^1-nonconforming virtual element is obtained with the same degrees of freedom as the usual H^1-conforming virtual element, such that it locally has H^2-regularity on each polygon in meshes. To enforce the C^1 continuity of the solution, an interior penalty formulation is adopted. Hence, this new numerical scheme can be regarded as a combination of the virtual element space and discontinuous Galerkin scheme. Compared with the existing methods, this approach has some advantages in reducing the degree of freedom and capability of handling hanging nodes. The well-posedness and optimal convergence of the IPVEM are proven in a mesh-dependent norm. We also derive a sharp estimate of the condition number of the linear system associated with IPVEM. Some numerical results are presented to verify the optimal convergence of the IPVEM and the sharp estimate of the condition number of the discrete problem. Besides, in the numerical test, the IPVEM has a good performance in computational accuracy by contrast with the other VEMs solving the biharmonic problem. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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13. An Invariance and Closed Form Analysis of the Nonlinear Biharmonic Beam Equation.
- Author
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Masood, Y., Kara, A. H., and Zaman, F. D.
- Subjects
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TRANSFORMATION groups , *NOETHER'S theorem , *PARTIAL differential equations , *NONLINEAR analysis , *CONSERVATION laws (Physics) , *BIHARMONIC equations , *CONSERVATION laws (Mathematics) - Abstract
In this paper, we study the one-parameter Lie groups of point transformations that leave invariant the biharmonic partial differential equation (PDE) uxxxx + 2uxxyy + uyyyy = f(u). To this end, we construct the Lie and Noether symmetry generators and present reductions of biharmonic PDE. When f is arbitrary function of u, we obtain the solution of biharmonic equation in terms of Green function. The equation is further analysed when f is exponential function and for general power law. Furthermore, we use Noether's theorem and the 'multiplier approach' to construct conservation laws of the PDE. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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14. A C0 finite element method for the biharmonic problem with Navier boundary conditions in a polygonal domain.
- Author
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Li, Hengguang, Yin, Peimeng, and Zhang, Zhimin
- Subjects
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FINITE element method , *CONVEX domains , *BIHARMONIC equations , *FUNCTION spaces - Abstract
In this paper we study the biharmonic equation with Navier boundary conditions in a polygonal domain. In particular, we propose a method that effectively decouples the fourth-order problem as a system of Poisson equations. Our method differs from the naive mixed method that leads to two Poisson problems but only applies to convex domains; our decomposition involves a third Poisson equation to confine the solution in the correct function space, and therefore can be used in both convex and nonconvex domains. A |$C^0$| finite element algorithm is in turn proposed to solve the resulting system. In addition, we derive optimal error estimates for the numerical solution on both quasi-uniform meshes and graded meshes. Numerical test results are presented to justify the theoretical findings. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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15. A new mixed method for the biharmonic eigenvalue problem.
- Author
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Kosin, V., Beuchler, S., and Wick, T.
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EIGENVALUES , *BIHARMONIC equations , *CONVEX domains - Abstract
In this paper, we investigate a new mixed method proposed by Rafetseder and Zulehner for Kirchhoff plates and apply it to fourth order eigenvalue problems. Using two auxiliary variables this new mixed method makes it possible to require only H 1 regularity for the displacement and the auxiliary variables, without the demand of a convex domain. We provide a direct comparison, specifically in view of convergence orders, to the C 0 -IPG method and Ciarlet-Raviart's mixed method of vibration problems with the boundary conditions of the clamped plate and the simply supported plate. The numerical experiments are done with the open-source finite element library deal.II and include the implementation of the coupling of finite elements with elements on the boundary to incorporate non-trivial boundary conditions. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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16. Positive solutions for a class of biharmonic problems: existence, nonexistence and multiplicity.
- Author
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Feng, Meiqiang
- Abstract
The main objective of this article is to consider a biharmonic problem with Navier boundary conditions. Among others, some criteria for the existence, multiplicity and nonexistence of positive solutions are established by employed fixed point theorems in a cone. In addition, we not only consider the sublinear case, but also we will study the case of appropriate combinations of superlinearity and sublinearity. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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17. A local pointwise inequality for a biharmonic equation with negative exponents.
- Author
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Chen, Fan, Chen, Jianqing, and Ruan, Qihua
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BIHARMONIC equations , *EXPONENTS , *OPEN-ended questions - Abstract
In this paper, we are inspired by Ngô, Nguyen and Phan's (2018 Nonlinearity 31 5484–99) study of the pointwise inequality for positive C 4-solutions of biharmonic equations with negative exponent by using the growth condition of solutions. They propose an open question of whether the growth condition is necessary to obtain the pointwise inequality. We give a positive answer to this open question. We establish the following local pointwise inequality − Δ u u + α | ∇ u | 2 u 2 + β u − q + 1 2 ⩽ C R 2 for positive C 4-solutions of the biharmonic equations with negative exponent − Δ 2 u = u − q i n B R where B R denotes the ball centered at x 0 with radius R, n ⩾ 3, q > 1, and some constants α ⩾ 0, β ⩾ 0, C > 0. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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18. Representations of solutions of Lamé system with real coefficients via monogenic functions in the biharmonic algebra.
- Author
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Gryshchuk, Serhii
- Subjects
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MONOGENIC functions , *BIHARMONIC functions , *FUNCTION algebras , *BIHARMONIC equations , *CLIFFORD algebras , *PARTIAL differential equations , *ASSOCIATIVE algebras - Abstract
New representations of solutions of Lamé system with real coefficients via monogenic functions in the biharmonic algebra are found [ABSTRACT FROM AUTHOR]
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- 2023
- Full Text
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19. A serendipity fully discrete div-div complex on polygonal meshes.
- Author
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Botti, Michele, Di Pietro, Daniele A., and Salah, Marwa
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SERENDIPITY , *DEGREES of freedom , *BIHARMONIC equations , *ELASTICITY - Abstract
In this work we address the reduction of face degrees of freedom (DOFs) for discrete elasticity complexes. Specifically, using serendipity techniques, we develop a reduced version of a recently introduced two-dimensional complex arising from traces of the three-dimensional elasticity complex. The keystone of the reduction process is a new estimate of symmetric tensor-valued polynomial fields in terms of boundary values, completed with suitable projections of internal values for higher degrees. We prove an extensive set of new results for the original complex and show that the reduced complex has the same homological and analytical properties as the original one. This paper also contains an appendix with proofs of general Poincaré--Korn-type inequalities for hybrid fields. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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20. A fully discrete plates complex on polygonal meshes with application to the Kirchhoff--Love problem.
- Author
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Pietro, Daniele A. Di and Droniou, Jérôme
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BIHARMONIC equations - Abstract
In this work we develop a novel fully discrete version of the plates complex, an exact Hilbert complex relevant for the mixed formulation of fourth-order problems. The derivation of the discrete complex follows the discrete de Rham paradigm, leading to an arbitrary-order construction that applies to meshes composed of general polygonal elements. The discrete plates complex is then used to derive a novel numerical scheme for Kirchhoff–Love plates, for which a full stability and convergence analysis are performed. Extensive numerical tests complete the exposition. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
21. Kansa–RBF algorithms for elliptic BVPs in annular domains with mixed boundary conditions.
- Author
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Jankowska, Malgorzata A., Karageorghis, Andreas, and Chen, C.S.
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KRYLOV subspace , *NUMERICAL solutions to boundary value problems , *CIRCULANT matrices , *NEUMANN boundary conditions - Abstract
We employ a Kansa–radial basis function (RBF) method for the numerical solution of elliptic boundary value problems in annular domains with mixed Dirichlet/Neumann boundary conditions. By exploiting the circular boundaries and the properties of circulant matrices we employ, in an efficient way, the pre-conditioned Krylov subspace iterative solvers GMRES and BiCGSTAB for the solution of the resulting linear systems. In particular, we employ block circulant pre-conditioners which allow for the efficient solution of the relevant systems in the iterative solution. Moreover, by exploiting the properties of circulant matrices we perform the matrix–vector multiplications involved in the iterative solvers efficiently. The feasibility of the proposed techniques is illustrated by several numerical examples. • The Kansa-RBF method is applied to elliptic BVPs in annular domains with mixed BCs. • GMRES and BiCGSTAB are used for the solution of the resulting linear systems. • The block circulant structures of the matrices are exploited. • The efficacy of the method is illustrated by several numerical examples. • Numerical experiments are also carried out in multiple precision. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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22. A new mixed finite-element method for H2 elliptic problems.
- Author
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Farrell, Patrick E., Hamdan, Abdalaziz, and MacLachlan, Scott P.
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FINITE element method , *DIFFERENTIAL equations , *MULTIGRID methods (Numerical analysis) , *LINEAR systems , *BIHARMONIC equations , *LAGRANGE multiplier - Abstract
Fourth-order differential equations play an important role in many applications in science and engineering. In this paper, we present a three-field mixed finite-element formulation for fourth-order problems, with a focus on the effective treatment of the different boundary conditions that arise naturally in a variational formulation. Our formulation is based on introducing the gradient of the solution as an explicit variable, constrained using a Lagrange multiplier. The essential boundary conditions are enforced weakly, using Nitsche's method where required. As a result, the problem is rewritten as a saddle-point system, requiring analysis of the resulting finite-element discretization and the construction of optimal linear solvers. Here, we discuss the analysis of the well-posedness and accuracy of the finite-element formulation. Moreover, we develop monolithic multigrid solvers for the resulting linear systems. Two and three-dimensional numerical results are presented to demonstrate the accuracy of the discretization and efficiency of the multigrid solvers proposed. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
23. Solving a Class of High-Order Elliptic PDEs Using Deep Neural Networks Based on Its Coupled Scheme.
- Author
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Li, Xi'an, Wu, Jinran, Zhang, Lei, and Tai, Xin
- Subjects
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ARTIFICIAL neural networks , *DEEP learning , *BIHARMONIC equations , *PARTIAL differential equations , *RITZ method , *APPROXIMATION error - Abstract
Deep learning—in particular, deep neural networks (DNNs)—as a mesh-free and self-adapting method has demonstrated its great potential in the field of scientific computation. In this work, inspired by the Deep Ritz method proposed by Weinan E et al. to solve a class of variational problems that generally stem from partial differential equations, we present a coupled deep neural network (CDNN) to solve the fourth-order biharmonic equation by splitting it into two well-posed Poisson's problems, and then design a hybrid loss function for this method that can make efficiently the optimization of DNN easier and reduce the computer resources. In addition, a new activation function based on Fourier theory is introduced for our CDNN method. This activation function can reduce significantly the approximation error of the DNN. Finally, some numerical experiments are carried out to demonstrate the feasibility and efficiency of the CDNN method for the biharmonic equation in various cases. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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24. Solving Biharmonic Equations with Tri-Cubic C 1 Splines on Unstructured Hex Meshes.
- Author
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Youngquist, Jeremy and Peters, Jörg
- Subjects
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SPLINES , *ELLIPTIC differential equations , *ISOGEOMETRIC analysis , *POISSON'S equation , *BIHARMONIC equations - Abstract
Unstructured hex meshes are partitions of three spaces into boxes that can include irregular edges, where n ≠ 4 boxes meet along an edge, and irregular points, where the box arrangement is not consistent with a tensor-product grid. A new class of tri-cubic C 1 splines is evaluated as a tool for solving elliptic higher-order partial differential equations over unstructured hex meshes. Convergence rates for four levels of refinement are computed for an implementation of the isogeometric Galerkin approach applied to Poisson's equation and the biharmonic equation. The ratios of error are contrasted and superior to an implementation of Catmull-Clark solids. For the trivariate Poisson problem on irregularly partitioned domains, the reduction by 2 4 in the L 2 norm is consistent with the optimal convergence on a regular grid, whereas the convergence rate for Catmull-Clark solids is measured as O (h 3 ). The tri-cubic splines in the isogeometric framework correctly solve the trivariate biharmonic equation, but the convergence rate in the irregular case is lower than O( h 4 ). An optimal reduction of 2 4 is observed when the functions on the C 1 geometry are relaxed to be C 0 . [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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25. An optimal two-step quadratic spline collocation method for the Dirichlet biharmonic problem.
- Author
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Bialecki, Bernard, Fairweather, Graeme, and Karageorghis, Andreas
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COLLOCATION methods , *DIRICHLET problem , *CONJUGATE gradient methods , *FAST Fourier transforms , *PARTIAL differential equations , *BIHARMONIC equations , *SPLINES - Abstract
A two-step quadratic spline collocation method is formulated for the solution of the Dirichlet biharmonic problem on the unit square rewritten as a coupled system of two second-order partial differential equations. This method involves fast Fourier transforms and, in comparison to its one-step counterpart, it has the advantage of requiring the solution a symmetric positive definite Schur complement system rather than a nonsymmetric one. As a consequence, the corresponding step of the new method is performed using a preconditioned conjugate gradient method. The total cost of the method on a N × N partition of the unit square is O (N 2 log N) . To demonstrate the optimal accuracy of the method, the results of numerical experiments are provided. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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26. A C0-conforming DG finite element method for biharmonic equations on triangle/tetrahedron.
- Author
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Ye, Xiu and Zhang, Shangyou
- Subjects
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FINITE element method , *YANG-Baxter equation , *TETRAHEDRA , *BIHARMONIC equations , *TRIANGLES , *VECTOR valued functions - Abstract
A C0-conforming discontinuous Galerkin (CDG) finite element method is introduced for solving the biharmonic equation. The first strong gradient of C0 finite element functions is a vector of discontinuous piecewise polynomials. The second gradient is the weak gradient of discontinuous piecewise polynomials. This method, by its name, uses nonconforming (non C1) approximations and keeps simple formulation of conforming finite element methods without any stabilizers. Optimal order error estimates in both a discrete H2-norm and the L2-norm are established for the corresponding finite element solutions. Numerical results are presented to confirm the theory of convergence. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
27. A weak divergence CDG method for the biharmonic equation on triangular and tetrahedral meshes.
- Author
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Ye, Xiu and Zhang, Shangyou
- Subjects
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BIHARMONIC equations , *FINITE element method , *CONTINUOUS functions - Abstract
A conforming discontinuous Galerkin (CDG) C 0 - P k finite element method is introduced for solving the biharmonic equation on triangular and tetrahedral meshes. A C 0 - P k finite element function is a continuous and piecewise polynomial of degree k on a triangular or tetrahedral mesh. The CDG method is based on taking weak divergence on the gradient of C 0 - P k finite elements. Optimal order error estimates in both a discrete H 2 norm and the L 2 norm are established. Numerical results are presented to verify the theory. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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28. On approximation of functions from Hölder classes by biharmonic Poisson integrals defined in the upper half-plane.
- Author
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Shutovskyi, Arsen M.
- Abstract
The paper investigates the approximate properties of biharmonic Poisson integrals for the upper half-plane on the classes of Hölder functions in the uniform metric. The exact values for the upper bounds of the deviations of the functions of the Hölder class
H 1 from the biharmonic Poisson integrals defined in the upper half-plane were found. [ABSTRACT FROM AUTHOR]- Published
- 2024
- Full Text
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29. High order Morley elements for biharmonic equations on polytopal partitions.
- Author
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Li, Dan, Wang, Chunmei, Wang, Junping, and Zhang, Shangyou
- Subjects
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BIHARMONIC equations , *FINITE element method , *SCHUR complement , *GALERKIN methods , *DEGREES of freedom - Abstract
This paper introduces an extension of the Morley element for approximating solutions to biharmonic equations. Traditionally limited to piecewise quadratic polynomials on triangular elements, the extension leverages weak Galerkin finite element methods to accommodate higher degrees of polynomials and the flexibility of general polytopal elements. By utilizing the Schur complement of the weak Galerkin method, the extension allows for fewest local degrees of freedom while maintaining sufficient accuracy and stability for the numerical solutions. The numerical scheme incorporates locally constructed weak tangential derivatives and weak second order partial derivatives, resulting in an accurate approximation of the biharmonic equation. Optimal order error estimates in both a discrete H 2 norm and the usual L 2 norm are established to assess the accuracy of the numerical approximation. Additionally, numerical results are presented to validate the developed theory and demonstrate the effectiveness of the proposed extension. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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30. Isogeometric collocation for solving the biharmonic equation over planar multi-patch domains.
- Author
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Kapl, Mario, Kosmač, Aljaž, and Vitrih, Vito
- Subjects
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BIHARMONIC equations , *ISOGEOMETRIC analysis , *PARTIAL differential equations , *COLLOCATION methods - Abstract
We present an isogeometric collocation method for solving the biharmonic equation over planar bilinearly parameterized multi-patch domains. The developed approach is based on the use of the globally C 4 -smooth isogeometric spline space (Kapl and Vitrih, 2021) to approximate the solution of the considered partial differential equation, and proposes as collocation points two different choices, namely on the one hand the Greville points and on the other hand the so-called superconvergent points. Several examples demonstrate the potential of our collocation method for solving the biharmonic equation over planar multi-patch domains, and numerically study the convergence behavior of the two types of collocation points with respect to the L 2 -norm as well as to equivalents of the H s -seminorms for 1 ≤ s ≤ 4. In the studied case of spline degree p = 9 , the numerical results indicate in case of the Greville points a convergence of order O (h p − 3) independent of the considered (semi)norm, and show in case of the superconvergent points an improved convergence of order O (h p − 2) for all (semi)norms except for the equivalent of the H 4 -seminorm, where the order O (h p − 3) is anyway optimal. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
31. A computational approach based on the Legendre-Galerkin method for solving a distributed optimal control problem constrained by the biharmonic equation.
- Author
-
Khasi, Manoochehr
- Abstract
This paper presents a Legendre-Galerkin spectral method to compute the solution of a distributed optimal control problem (OCP) constrained by the biharmonic equation on regular and irregular domains. First, the optimality system is obtained by the Karush–Kuhn–Tucker optimality conditions. Next, it is discretized by the proposed method, and a coupled matrix equation is obtained. Finally, the arising system is solved by using the Kronecker product. Using the appropriate base functions causes a system with sparse matrices. Also, one of the advantages of this approach is its high accuracy for solving fourth-order problems. The error estimate is also investigated theoretically. Many numerical examples are included to demonstrate the accuracy and efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
32. Four-order superconvergent CDG finite elements for the biharmonic equation on triangular meshes.
- Author
-
Ye, Xiu and Zhang, Shangyou
- Subjects
- *
FINITE element method , *BIHARMONIC equations , *DISCONTINUOUS functions , *BILINEAR forms , *GALERKIN methods - Abstract
In a conforming discontinuous Galerkin (CDG) finite element method, discontinuous P k polynomials are employed. To connect discontinuous functions, the inter-element traces, { u h } and { ∇ u h } , are usually defined as some averages in discontinuous Galerkin finite element methods. But in this CDG finite element method, they are defined as projections of a lifted P k + 4 polynomial from four P k polynomials on neighboring triangles. With properly chosen weak Hessian spaces, when tested by discontinuous polynomials, the variation form can have no inter-element integral, neither any stabilizer. That is, the bilinear form is the same as that of conforming finite elements for solving the biharmonic equation. Such a conforming discontinuous Galerkin finite element method converges four orders above the optimal order, i.e., the P k solution has an O (h k + 5) convergence in L 2 -norm, and an O (h k + 3) convergence in H 2 -norm. A local post-process is defined, which lifts the P k solution to a P k + 4 quasi-optimal solution. Numerical tests are provided, confirming the theory. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
33. Green's functions of some boundary value problems for the biharmonic equation.
- Author
-
Karachik, Valery
- Subjects
- *
BIHARMONIC equations , *BOUNDARY value problems , *GREEN'S functions , *DIRICHLET problem , *NEUMANN problem , *INTEGRAL representations - Abstract
In this paper the Green's functions for three boundary value problems for the biharmonic equation are investigated. First, an integral representation of solutions to the inhomogeneous biharmonic equation is given. Then the Green's function of the Dirichlet problem is found and an integral representation of the solution to the Dirichlet problem in terms of the Green's function is given. After that, the Green's function of the Navier problem and the integral representation of the solution to the Navier problem are presented. To study the Neumann-2 problem, the Green's function of the Neumann problem for the Poisson equation is discussed and on its basis the Green's function of the Neumann-2 problem is constructed. To illustrate the results obtained, solutions of the three considered homogeneous problems for the polynomial right-hand side of the equation are found. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
34. LIGHTNING STOKES SOLVER.
- Author
-
BRUBECK, PABLO D. and TREFETHEN, LLOYD N.
- Subjects
- *
STOKES flow , *BIHARMONIC equations , *ANALYTIC functions , *EDDIES - Abstract
Gopal and Trefethen recently introduced "lightning solvers" for the 2D Laplace and Helmholtz equations, based on rational functions with poles exponentially clustered near singular corners. Making use of the Goursat representation in terms of analytic functions, we apply these methods to the biharmonic equation, specifically to 2D Stokes flow. Solutions to model problems are computed to 10-digit accuracy in less than a second of laptop time. As an illustration of the high accuracy, we resolve two or more counterrotating Moffatt eddies near a singular corner. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
35. Hausdorff Analytic Functions in a Three-Dimensional Associative Noncommutative Algebra.
- Author
-
Shpakivskyi, Vitalii and Kuzmenko, Tetiana
- Subjects
- *
ANALYTIC functions , *ASSOCIATIVE algebras , *INTEGRAL representations , *NONCOMMUTATIVE algebras , *INTEGRAL functions , *BIHARMONIC equations - Abstract
A class of H-analytic (differentiable by Hausdorff) functions in a three-dimensional noncommutative algebra e A ~ 2 over the field ℂ is considered. All H-analytic functions are described in the explicit form. The obtained description is applied to the integral representation of these functions, and the mentioned functions are also applied when solving some PDEs. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
36. h-, p-, and hp-Versions of the Least-Squares Collocation Method for Solving Boundary Value Problems for Biharmonic Equation in Irregular Domains and Their Applications.
- Author
-
Belyaev, V. A., Bryndin, L. S., Golushko, S. K., Semisalov, B. V., and Shapeev, V. P.
- Subjects
- *
BIHARMONIC equations , *BOUNDARY value problems , *COLLOCATION methods , *NUMERICAL solutions to boundary value problems , *FINITE differences , *CHEBYSHEV polynomials - Abstract
New h-, p-, and hp-versions of the least-squares collocation method are proposed and implemented. They yield approximate solutions of boundary value problems for an inhomogeneous biharmonic equation in irregular and multiply-connected domains. Formulas for the extension operation in the transition from coarse to finer grids on a multigrid complex are given in the case of applying various spaces of polynomials. It is experimentally shown that numerical solutions of boundary value problems produced by the developed versions of the method have a higher order of convergence to analytical solutions with no singularities. The results are compared with those of other authors produced by applying finite difference, finite element, and other methods based on Chebyshev polynomials. Examples of problems with singularities are considered. The developed versions of the method are used to simulate the bending of an elastic isotropic plate of irregular shape subjected to transverse loading. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
37. Four Boundary Value Problems for a Nonlocal Biharmonic Equation in the Unit Ball.
- Author
-
Karachik, Valery, Turmetov, Batirkhan, and Yuan, Hongfen
- Subjects
- *
BOUNDARY value problems , *BIHARMONIC equations , *UNIT ball (Mathematics) , *EXISTENCE theorems , *GREEN'S functions , *INTEGRAL representations - Abstract
Solvability issues of four boundary value problems for a nonlocal biharmonic equation in the unit ball are investigated. Dirichlet, Neumann, Navier and Riquier–Neumann boundary value problems are studied. For the problems under consideration, existence and uniqueness theorems are proved. Necessary and sufficient conditions for the solvability of all problems are obtained and an integral representations of solutions are given in terms of the corresponding Green's functions. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
38. Ground state solution to the biharmonic equation.
- Author
-
Feng, Zhaosheng and Su, Yu
- Subjects
- *
BIHARMONIC equations , *CONTINUUM mechanics , *ELASTIC plates & shells , *VISCOUS flow , *FLUID flow - Abstract
The biharmonic equation arises in areas of continuum mechanics, including mechanics of elastic plates and the slow flow of viscous fluids. In this paper, we make an effort to establish the generalized versions of Lions-type theorem under various conditions and then apply them to study the existence of ground state solutions for the biharmonic equation. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
39. A novel approach of random packing generation of complex-shaped 3D particles with controllable sizes and shapes.
- Author
-
Wang, Xiang, Yin, Zhen-Yu, Su, Dong, Wu, Xiaoxin, and Zhao, Jidong
- Subjects
- *
RANDOM numbers , *BIHARMONIC equations , *MANUFACTURING processes , *CURVE fitting , *GRANULAR materials , *CONVEX surfaces - Abstract
This paper presents a novel computational-geometry-based approach to generating random packing of complex-shaped 3D particles with quantitatively controlled sizes and shapes for discrete modeling of granular materials. The proposed method consists of the following five essential steps: (1) partitioning of the packing domain into a prescribed number of random polyhedrons with desired sizes and form-scale shapes using the constrained Voronoi tessellation; (2) extraction of key points from the edges and facets of each polyhedron; (3) construction of a freeform curve network in each polyhedron based on Bézier curve fitting; (4) generation of solid particles with smooth, convex surfaces using the biharmonic-based surface interpolation of the constructed network; and (5) creation of concavity by superimposing spherical harmonic-based random noise. To ensure that the obtained shape descriptors (e.g., the elongation, flatness, roundness and convexity ratio) match the hypothesized values, an inverse Monte Carlo algorithm is employed to iteratively fine-tune the control parameters during particle generation. The ability of the proposed approach to generate granular particles with the desired geometric properties and packing is demonstrated through several examples. This study paves a viable pathway for realistic modeling of granular media pertaining to various engineering and industrial processes. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
40. 一类热弹性板的空间衰减估计.
- Author
-
石金诚
- Subjects
- *
DIFFERENTIAL inequalities , *DERIVATIVES (Mathematics) , *ELASTIC plates & shells , *BIHARMONIC equations - Abstract
The spatial properties of solutions for a class of thermoelastic plates with biharmonic operators were studied in a semi-infinite strip in R2. Firstly, an energy expression was constructed. Then, by means of the differential inequality technique, a differential inequality whose energy expression can be controlled with its 1st derivative was derived. Finally, the spatial decay estimates of the solution were obtained. The result can be regarded as an application of the Saint-Venant principle to hyperbolic parabolic coupled biharmonic equations. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
41. RBF–DQ algorithms for elliptic problems in axisymmetric domains.
- Author
-
Chen, C. S., Jankowska, Malgorzata A., and Karageorghis, Andreas
- Subjects
- *
FAST Fourier transforms , *NUMERICAL solutions to boundary value problems , *BIHARMONIC equations , *MATRIX decomposition - Abstract
A radialbasis function (RBF)–differentialquadrature (DQ) method is applied for the numerical solution of elliptic boundary value problems (BVPs) in three-dimensional axisymmetric domains. By appropriately selecting the collocation points, for any choice of RBF, the proposed discretization leads to linear systems in which the coefficient matrices possess block circulant structures. Matrix decomposition algorithms (MDAs) and fast Fourier transforms (FFTs) are employed for the efficient solution of these systems. Three types of BVPs are considered, namely ones governed by the Poisson equation, the inhomogeneous biharmonic equation, and the inhomogeneous Cauchy–Navier equations of elasticity. The high accuracy of the proposed technique as well as its ability to solve large-scale problems is demonstrated on several numerical examples. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
42. A comparison of smooth basis constructions for isogeometric analysis.
- Author
-
Verhelst, H.M., Weinmüller, P., Mantzaflaris, A., Takacs, T., and Toshniwal, D.
- Subjects
- *
ISOGEOMETRIC analysis , *BIHARMONIC equations , *SPLINES - Abstract
In order to perform isogeometric analysis with increased smoothness on complex domains, trimming, variational coupling or unstructured spline methods can be used. The latter two classes of methods require a multi-patch segmentation of the domain, and provide continuous bases along patch interfaces. In the context of shell modelling, variational methods are widely used, whereas the application of unstructured spline methods on shell problems is rather scarce. In this paper, we therefore provide a qualitative and a quantitative comparison of a selection of unstructured spline constructions, in particular the D-Patch, Almost- C 1 , Analysis-Suitable G 1 and the Approximate C 1 constructions. Using this comparison, we aim to provide insight into the selection of methods for practical problems, as well as directions for future research. In the qualitative comparison, the properties of each method are evaluated and compared. In the quantitative comparison, a selection of numerical examples is used to highlight different advantages and disadvantages of each method. In the latter, comparison with weak coupling methods such as Nitsche's method or penalty methods is made as well. In brief, it is concluded that the Approximate C 1 and Analysis-Suitable G 1 converge optimally in the analysis of a bi-harmonic problem, without the need of special refinement procedures. Furthermore, these methods provide accurate stress fields. On the other hand, the Almost- C 1 and D-Patch provide relatively easy construction on complex geometries. The Almost- C 1 method does not have limitations on the valence of boundary vertices, unlike the D-Patch, but is only applicable to biquadratic local bases. Following from these conclusions, future research directions are proposed, for example towards making the Approximate C 1 and Analysis-Suitable G 1 applicable to more complex geometries. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
43. Schwartz-type boundary-value problems for canonical domains in a biharmonic plane.
- Author
-
Gryshchuk, Serhii V. and Plaksa, Sergiy A.
- Subjects
- *
BOUNDARY value problems , *BIHARMONIC equations , *MONOGENIC functions , *COMMUTATIVE algebra , *CARTESIAN plane , *ALGEBRA - Abstract
A commutative algebra B over the complex field with a basis {e1, e2} satisfying the conditions e 1 2 + e 2 2 2 = 0 , e 1 2 + e 2 2 ≠ 0 is considered. This algebra is associated with the 2-D biharmonic equation. We consider Schwartz-type boundary-value problems on finding a monogenic function of the type Φ (xe1+ye2) = U1(x; y) e1 + U2(x; y) ie1 + U3(x; y) e2 + U4(x; y) ie2, (x; y) ∈ D, when the values of two components—either U1, U3 or U1, U4—are given on the boundary of a domain D lying in the Cartesian plane xOy. For solving those boundary-value problems for a half-plane and for a disk, we develop methods that are based on solution expressions via Schwartz-type integrals and obtain solutions in the explicit form. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
44. Schwartz-type boundary-value problems for canonical domains in a biharmonic plane.
- Author
-
Gryshchuk, Serhii V. and Plaksa, Sergiy A.
- Subjects
- *
BOUNDARY value problems , *BIHARMONIC equations , *MONOGENIC functions , *COMMUTATIVE algebra , *CARTESIAN plane , *ALGEBRA - Abstract
A commutative algebra B over the complex field with a basis {e1, e2} satisfying the conditions e 1 2 + e 2 2 2 = 0 , e 1 2 + e 2 2 ≠ 0 is considered. This algebra is associated with the 2-D biharmonic equation. We consider Schwartz-type boundary-value problems on finding a monogenic function of the type Φ (xe1+ye2) = U1(x; y) e1 + U2(x; y) ie1 + U3(x; y) e2 + U4(x; y) ie2, (x; y) ∈ D, when the values of two components—either U1, U3 or U1, U4—are given on the boundary of a domain D lying in the Cartesian plane xOy. For solving those boundary-value problems for a half-plane and for a disk, we develop methods that are based on solution expressions via Schwartz-type integrals and obtain solutions in the explicit form. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
45. Propagating and Standing Rayleigh Waves near Rivet Chains Connecting Kirchhoff Plates.
- Author
-
Nazarov, S. A.
- Subjects
- *
RAYLEIGH waves , *STANDING waves , *RIVETED joints , *RIVETS & riveting , *BIHARMONIC equations - Abstract
We show that near periodic rivet chains that connect two Kirchhoff plates and are modeled by point Sobolev transmission conditions, Rayleigh waves arise, propagate along the chains, and decay exponentially in the orthogonal direction. Under additional geometric conditions we discover the standing (periodic) waves that carry no energy. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
46. UNIFORM AUXILIARY SPACE PRECONDITIONING FOR HDG METHODS FOR ELLIPTIC OPERATORS WITH A PARAMETER DEPENDENT LOW ORDER TERM.
- Author
-
GUOSHENG FU
- Subjects
- *
ELLIPTIC operators , *REACTION-diffusion equations , *UNIFORM spaces , *BIHARMONIC equations - Abstract
The auxiliary space preconditioning (ASP) technique is applied to the hybridizable discontinuous Galerkin (HDG) schemes for three different elliptic problems with a parameter dependent low order term, namely, a symmetric interior penalty HDG scheme for the scalar reaction-diffusion equation, a divergence-conforming HDG scheme for a vectorial reaction-diffusion equation, and a C0-continuous interior penalty HDG scheme for the generalized biharmonic equation with a low order term. Uniform preconditioners are obtained for each case and the general ASP theory by J. Xu [Computing, 56 (1996), pp. 215--235] is used to prove the optimality with respect to the mesh size and uniformity with respect to the low order parameter. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
47. Existence of solutions to fourth order problems with sign-changing weights.
- Author
-
Alharbi, Hessa and Dammak, Makkia
- Subjects
- *
SOBOLEV spaces , *BIHARMONIC equations - Abstract
In this article, we investigate the existence of solutions for the indefinite nonlinear fourth order elliptic problem with Navier boundary conditions: Δ²v - Δv + Vλ(x)v = f(x, v) in Ω, v = Δv = 0 on ∂Ω, where Ω is a bounded open domain in ℝN, N ≥ 2, Vλ(x) is sign-changing weight function. Also, the reaction source term f is not necessary positive. We will prove that for λ large enough, there exists a nontrivial solution. Our method is a variational one. The most delicate point is to choose the appropriate Sobolev space and the suitable norm. [ABSTRACT FROM AUTHOR]
- Published
- 2021
48. On Lavrent'ev-Type Integral Equations in Coefficient Inverse Problems for Wave Equations.
- Author
-
Kozlov, A. I. and Kokurin, M. Yu.
- Subjects
- *
INVERSE problems , *INTEGRAL equations , *NONLINEAR equations , *LINEAR equations , *BIHARMONIC equations , *WAVE equation - Abstract
Coefficient inverse problems for second- and third-order equations with one or two unknown coefficients are investigated. The initial data are specified as the solution of an equation for a set of sounding sources averaged over time with power-law weights. It is shown that the original nonlinear inverse problems can be equivalently reduced to integral equations that are linear or nonlinear depending on the averaging method. It is proved that these equations have a unique solution determining the desired solution of the inverse problems. The results of a numerical experiment concerning the solution of a linear integral equation with a special kernel are presented. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
49. A generalized finite difference method for solving biharmonic interface problems.
- Author
-
Xing, Yanan, Song, Lina, and Li, Po-Wei
- Subjects
- *
FINITE difference method , *BIHARMONIC equations , *TAYLOR'S series - Abstract
In this paper, a meshless discrete scheme based on the generalized finite difference method (GFDM) is proposed to solve the biharmonic interface problem. This scheme turns the interface problem to be two non-interface subproblems coupled with the interface conditions. The interface conditions act as the boundary conditions of the subproblems because the interface plays the role of their boundary. Therefore, the proposed method has the advantage of dealing with problems with complex geometrical interfaces. Due to the GFDM approximates the derivatives of the unknown variables by using a linear combination of the values for nearby nodes, the proposed method also has the advantage in dealing with the interface condition with the jump of derivatives. Four numerical examples are provided to verify the accuracy and the stability of the GFDM for biharmonic interface problems. The numerical results show that the H 1 errors can reach almost the same convergence rate of the L ∞ and L 2 errors. The convergence rates are almost and even higher than fourth-order when the fourth-order Taylor Series expansion is adopted in the GFDM. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
50. Barycentric rational method for solving biharmonic equation by depression of order.
- Author
-
Li, Jin and Cheng, Yongling
- Subjects
- *
BIHARMONIC equations , *LINEAR equations - Abstract
Two‐dimensional biharmonic boundary‐value problems are considered by the linear barycentric rational method, the unknown function was approximated by the barycentric rational function. For the biharmonic equation, we change the biharmonic equation into the two Poisson equations by depression of order. The linear equations of discrete the biharmonic equation was changed into matrix form. For the basis of barycentric rational function, we present the convergence rate of linear barycentric rational method for biharmonic equation by depression of order. At last, several numerical examples are provided to validate the theoretical analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
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