BDDC PRECONDITIONERS FOR ISOGEOMETRIC ANALYSIS

2013 ◽  
Vol 23 (06) ◽  
pp. 1099-1142 ◽  
Author(s):  
L. BEIRÃO DA VEIGA ◽  
D. CHO ◽  
L. F. PAVARINO ◽  
S. SCACCHI
Keyword(s):  
Convergence Rate ◽  
Domain Decomposition ◽  
Elliptic Problem ◽  
Isogeometric Analysis ◽  
Numerical Experiments ◽  
Three Dimensional ◽  
Parallel Computers ◽  
Basis Functions ◽  
Polynomial Degree ◽  
Balancing Domain Decomposition

A Balancing Domain Decomposition by Constraints (BDDC) preconditioner for Isogeometric Analysis of scalar elliptic problems is constructed and analyzed by introducing appropriate discrete norms. A main result of this work is the proof that the proposed isogeometric BDDC preconditioner is scalable in the number of subdomains and quasi-optimal in the ratio of subdomain and element sizes. Another main result is the numerical validation of the theoretical convergence rate estimates by carrying out several two- and three-dimensional tests on serial and parallel computers. These numerical experiments also illustrate the preconditioner performance with respect to the polynomial degree and the regularity of the NURBS basis functions, as well as its robustness with respect to discontinuities of the coefficient of the elliptic problem across subdomain boundaries.

scholarly journals Basic principles of hp virtual elements on quasiuniform meshes

2016 ◽  
Vol 26 (08) ◽  
pp. 1567-1598 ◽  
Author(s):  
L. Beir ao da Veiga ◽  
A. Chernov ◽  
L. Mascotto ◽  
A. Russo
Keyword(s):  
Numerical Experiments ◽  
Exponential Convergence ◽  
Mesh Size ◽  
Initial Study ◽  
Analytic Solutions ◽  
Basis Functions ◽  
Basic Principles ◽  
Polynomial Degree ◽  
Convergence Results ◽  
Convergence Estimates

In the present paper we initiate the study of [Formula: see text] Virtual Elements. We focus on the case with uniform polynomial degree across the mesh and derive theoretical convergence estimates that are explicit both in the mesh size [Formula: see text] and in the polynomial degree [Formula: see text] in the case of finite Sobolev regularity. Exponential convergence is proved in the case of analytic solutions. The theoretical convergence results are validated in numerical experiments. Finally, an initial study on the possible choice of local basis functions is included.

Singularities in p-Type Finite Element Computations: Numerical Experiments on the Locality of Pollution Effects

1991 ◽  
Vol 58 (2) ◽  
pp. 586-588 ◽  
Author(s):  
G. Becker ◽  
P. Carnevali ◽  
B. Chayapathy ◽  
W. Imaino ◽  
R. B. Morris ◽  
...  
Keyword(s):  
Finite Element ◽  
Numerical Experiments ◽  
Three Dimensional ◽  
Basis Functions ◽  
Effective Wavelength ◽  
Pollution Effects ◽  
Element Size ◽  
Graded Meshes ◽  
P Type

Using a recently developed code for three-dimensional p-type finite element computations in elasticity, we have studied, with a series of numerical experiments, the behavior of the solution in the presence of singularities. Worst-cast theoretical estimates predict a global pollution of the solution, unless optimally graded meshes are used. On the other hand, we observe that except in an immediate neighborhood of a singularity, and even using uniform meshes, the quality of the solution is not degraded to any practically relevant extent. This has important practical consequences because it allows modelers to introduce artifical singularities for the purpose of simplifying models. The size of the area around a singularity where the solution is appreciably degraded can be estimated in terms of the minimum effective wavelength of the basis functions used; the latter, in turn, can be related to the element size and the polynomial order used.

scholarly journals Coupling non-conforming discretizations of PDEs by spectral approximation of the Lagrange multiplier space

2019 ◽  
Vol 53 (5) ◽  
pp. 1667-1694 ◽  
Author(s):  
Simone Deparis ◽  
Antonio Iubatti ◽  
Luca Pegolotti
Keyword(s):  
Isogeometric Analysis ◽  
Hybrid Methods ◽  
Field Method ◽  
Basis Functions ◽  
Spectral Approximation ◽  
The Finite Element Method ◽  
Mortar Method ◽  
Polynomial Degree ◽  
Multiplier Space ◽  
Conforming Meshes

This work focuses on the development of a non-conforming method for the coupling of PDEs based on weakly imposed transmission conditions: the continuity of the global solution is enforced by a finite number of Lagrange multipliers defined over the interfaces of adjacent subdomains. The method falls into the class of primal hybrid methods, which include also the well-known mortar method. Differently from the mortar method, we discretize the space of basis functions on the interface by spectral approximation independently of the discretization of the two adjacent domains. In particular, our approach can be regarded as a specialization of the three-field method in which the spaces used to enforce the continuity of the solution and its conormal derivative across the interface are taken equal. One of the possible choices to approximate the interface variational space – which we consider here – is by Fourier basis functions. As we show in the numerical simulations, the method is well-suited for the coupling of problems defined on globally non-conforming meshes or discretized with basis functions of different polynomial degree in each subdomain. We also investigate the possibility of coupling solutions obtained with incompatible numerical methods, namely the finite element method and isogeometric analysis.

scholarly journals Alternating directions parallel hybrid memory iGRM direct solver for non-stationary simulations

2020 ◽  
Vol 21 (4) ◽  
Author(s):  
Maciej Woźniak ◽  
Anna Janina Bukowska
Keyword(s):  
Isogeometric Analysis ◽  
Parallel Machines ◽  
Sparse Matrix ◽  
Computational Cost ◽  
Three Dimensional ◽  
Modern Method ◽  
Basis Functions ◽  
Computational Domain ◽  
Hybrid Memory ◽  
Hybrid Parallelization

The three-dimensional isogeometric analysis (IGA-FEM) is a modern method for simulation. The idea is to utilize B-splines or NURBS basis functions for both computational domain descriptions and the engineering computations. Refined isogeometric analysis (rIGA) employs a mixture of patches of elements with B-spline basis functions, and $C^0$ separators between them. It enables a reduction of the computational cost of direct solvers. Both IGA and rIGA come with challenging sparse matrix structure, that is expensive to generate. In this paper, we show a hybrid parallelization method to reduce the computational cost of the integration phase using hybrid-memory parallel machines. The two-level parallelization includes the partitioning of the computational mesh into sub-domains on the first level (MPI), and loop parallelization on the second level (OpenMP). We show that hybrid parallelization of the integration reduces the contribution of this phase significantly. Thus, alternative algorithms for fast isogeometric integration are not necessary.

scholarly journals Optimal solution of the fractional order breast cancer competition model

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
H. Hassani ◽  
J. A. Tenreiro Machado ◽  
Z. Avazzadeh ◽  
E. Safari ◽  
S. Mehrabi
Keyword(s):  
Breast Cancer ◽  
Fractional Order ◽  
Legendre Polynomials ◽  
Numerical Experiments ◽  
Optimal Solution ◽  
Optimization Method ◽  
Competition Model ◽  
Basis Functions ◽  
Cell Populations ◽  
Operational Matrices

AbstractIn this article, a fractional order breast cancer competition model (F-BCCM) under the Caputo fractional derivative is analyzed. A new set of basis functions, namely the generalized shifted Legendre polynomials, is proposed to deal with the solutions of F-BCCM. The F-BCCM describes the dynamics involving a variety of cancer factors, such as the stem, tumor and healthy cells, as well as the effects of excess estrogen and the body’s natural immune response on the cell populations. After combining the operational matrices with the Lagrange multipliers technique we obtain an optimization method for solving the F-BCCM whose convergence is investigated. Several examples show that a few number of basis functions lead to the satisfactory results. In fact, numerical experiments not only confirm the accuracy but also the practicability and computational efficiency of the devised technique.

scholarly journals A Three-Dimensional Numerical and Multi-Objective Optimal Design of Wavy Plate-Fins Heat Exchangers

Processes ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 9
Author(s):  
Chao Yu ◽  
Xiangyao Xue ◽  
Kui Shi ◽  
Mingzhen Shao
Keyword(s):  
Heat Exchangers ◽  
Cfd Simulation ◽  
Three Dimensional ◽  
Basis Functions ◽  
Approximation Model ◽  
Multi Objective Optimization ◽  
Multi Objective ◽  
Multi Objective Genetic Algorithm ◽  
Colburn Factor ◽  
Heat Exchange Efficiency

This paper presents a method for optimizing wavy plate-fin heat exchangers accurately and efficiently. It combines CFD simulation, Radical Basis Functions (RBF) with multi-objective optimization to improve the performance. The optimization of the Colburn factor j and the friction coefficient f is regarded as a multi-objective optimization problem, due to the existence of two contradictory goals. The approximation model was obtained by Radical Basis Functions, and the shape of the heat exchanger was optimized by multi-objective genetic algorithm (MOGA). The optimization results showed that j increased by 17.62% and f decreased by 20.76%, indicating that the heat exchange efficiency was significantly enhanced and the fluid structure resistance reduced. Then, from the aspects of field synergy and tubulence energy, the performance advantage of the optimized structure was further confirmed.

scholarly journals An efficient algorithm for estimating the parameters of superimposed exponential signals in multiplicative and additive noise

2013 ◽  
Vol 23 (1) ◽  
pp. 117-129 ◽  
Author(s):  
Jiawen Bian ◽  
Huiming Peng ◽  
Jing Xing ◽  
Zhihui Liu ◽  
Hongwei Li
Keyword(s):  
Parameter Estimation ◽  
Convergence Rate ◽  
Efficient Algorithm ◽  
Numerical Experiments ◽  
Additive Noise ◽  
Mean Squared Errors ◽  
Least Squares Estimators ◽  
The Mean ◽  
Newton Raphson ◽  
Raphson Algorithm

This paper considers parameter estimation of superimposed exponential signals in multiplicative and additive noise which are all independent and identically distributed. A modified Newton-Raphson algorithm is used to estimate the frequencies of the considered model, which is further used to estimate other linear parameters. It is proved that the modified Newton- Raphson algorithm is robust and the corresponding estimators of frequencies attain the same convergence rate with Least Squares Estimators (LSEs) under the same noise conditions, but it outperforms LSEs in terms of the mean squared errors. Finally, the effectiveness of the algorithm is verified by some numerical experiments.

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