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Автор: Grafiati

Опубліковано: 22 червня 2022

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1

Zhang, Jie, Abel Cherouat, and Houman Borouchaki. "Fast Point and Element Search Method in Adaptive Remeshing Procedure and Its Applications." ISRN Applied Mathematics 2011 (August 17, 2011): 1–13. http://dx.doi.org/10.5402/2011/509721.

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The FPES (fast point and element search) method is a useful and efficient strategy for node field transfer from old mesh to the new mesh in adaptive remeshing procedure. The FE mesh after adaptive remeshing with various error estimates will be refined at local region, and the mesh after adaptive remeshing has the heterogeneous density distribution. The FPES has the capacity to define the nearest search path adapting to the mesh with heterogeneous density distribution. It is a point location process which includes point searching, point location in element, and weight factor distribution. This strategy has been integrated to our finite element adaptive remeshing simulations, and it works well and rapidly. The three-dimension finite element numerical simulation of simply tensile test, orthogonal cutting, and metal milling process is given out to study its accuracy and efficiency.

2

Guo, Zaoyang, Yujie Zhao, Zhaohui Chen, Minmao Liao, Zhengliang Li, and Bo Liu. "A Mesh Adaptive Procedure for Large Increment Method." International Journal of Applied Mechanics 07, no. 04 (August 2015): 1550061. http://dx.doi.org/10.1142/s1758825115500611.

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A posteriorih-version mesh adaptive procedure is presented in the framework of large increment method (LIM) for elastic problems. In this mesh adaptive strategy, the classical Zienkiewicz–Zhu (ZZ) error estimator is adopted and a first class h-adaptive mesh refinement procedure is implemented. A major advantage of the proposed mesh adaptive procedure is that the numerical results from the previous mesh can be utilized to obtain the initial solution for the new mesh. Two-dimensional (2D) examples show that this initial solution is much closer to the real solution than the minimum norm solution used in the original LIM and the revised method can converge faster than the original method.

3

Azarenok, B. N. "Variational method for adaptive mesh generation." Computational Mathematics and Mathematical Physics 48, no. 5 (May 2008): 786–804. http://dx.doi.org/10.1134/s0965542508050084.

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4

Xu, Yan, Gao Feng Wei, and Hai Yan Chen. "Overview of High Precision Adaptive Numerical Manifold Method." Advanced Materials Research 962-965 (June 2014): 2988–91. http://dx.doi.org/10.4028/www.scientific.net/amr.962-965.2988.

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The core of the numerical manifold method (NMM) is a two-mesh problem description. Two meshes are employed in an analysis: the mathematical mesh provides the nodes for building a finite covering of the solution domain, while the physical mesh provides the domain of integration. The NMM can deal with the continuum and discontinuous problem, and has been applied to the rock mechanics and engineering widely. This paper introduces the research progress of the NMM in the basic theory and application aspects. The adaptive mesh generation of NMM is discussed. The adaptive finite cover mesh reconstruction technology is given.

5

Koohi, Mahdi, and Abbas Shakery. "An Adaptive Mesh Method for Object Tracking." International Journal of Peer to Peer Networks 2, no. 2 (April 30, 2011): 1–10. http://dx.doi.org/10.5121/ijp2p.2011.2201.

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6

Fang, F., M. D. Piggott, C. C. Pain, G. J. Gorman, and A. J. H. Goddard. "An adaptive mesh adjoint data assimilation method." Ocean Modelling 15, no. 1-2 (January 2006): 39–55. http://dx.doi.org/10.1016/j.ocemod.2006.02.002.

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7

Kim, Jeong-Hun, Hyun-Gyu Kim, Byung-Chai Lee, and Seyoung Im. "Adaptive mesh generation by bubble packing method." Structural Engineering and Mechanics 15, no. 1 (January 25, 2003): 135–49. http://dx.doi.org/10.12989/sem.2003.15.1.135.

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8

Altas, Irfan, and John W. Stephenson. "A two-dimensional adaptive mesh generation method." Journal of Computational Physics 94, no. 1 (May 1991): 201–24. http://dx.doi.org/10.1016/0021-9991(91)90143-9.

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9

Lei, Humin, Tao Liu, Deng Li, Jikun Ye, and Lei Shao. "Adaptive Mesh Iteration Method for Trajectory Optimization Based on Hermite-Pseudospectral Direct Transcription." Mathematical Problems in Engineering 2017 (2017): 1–7. http://dx.doi.org/10.1155/2017/2184658.

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An adaptive mesh iteration method based on Hermite-Pseudospectral is described for trajectory optimization. The method uses the Legendre-Gauss-Lobatto points as interpolation points; then the state equations are approximated by Hermite interpolating polynomials. The method allows for changes in both number of mesh points and the number of mesh intervals and produces significantly smaller mesh sizes with a higher accuracy tolerance solution. The derived relative error estimate is then used to trade the number of mesh points with the number of mesh intervals. The adaptive mesh iteration method is applied successfully to the examples of trajectory optimization of Maneuverable Reentry Research Vehicle, and the simulation experiment results show that the adaptive mesh iteration method has many advantages.

10

Tyranowski, Tomasz M., and Mathieu Desbrun. "R-Adaptive Multisymplectic and Variational Integrators." Mathematics 7, no. 7 (July 18, 2019): 642. http://dx.doi.org/10.3390/math7070642.

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Moving mesh methods (also called r-adaptive methods) are space-adaptive strategies used for the numerical simulation of time-dependent partial differential equations. These methods keep the total number of mesh points fixed during the simulation but redistribute them over time to follow the areas where a higher mesh point density is required. There are a very limited number of moving mesh methods designed for solving field-theoretic partial differential equations, and the numerical analysis of the resulting schemes is challenging. In this paper, we present two ways to construct r-adaptive variational and multisymplectic integrators for (1+1)-dimensional Lagrangian field theories. The first method uses a variational discretization of the physical equations, and the mesh equations are then coupled in a way typical of the existing r-adaptive schemes. The second method treats the mesh points as pseudo-particles and incorporates their dynamics directly into the variational principle. A user-specified adaptation strategy is then enforced through Lagrange multipliers as a constraint on the dynamics of both the physical field and the mesh points. We discuss the advantages and limitations of our methods. Numerical results for the Sine–Gordon equation are also presented.

11

Hsu, L. C., J. Z. Ye, and C. H. Hsu. "Simulation of Flow Past a Cylinder With Adaptive Spectral Element Method." Journal of Mechanics 33, no. 2 (September 9, 2016): 235–47. http://dx.doi.org/10.1017/jmech.2016.77.

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AbstractThe simulations of flow past a two-dimensional circular cylinder are conducted to investigate the feasibility of adaptive mesh refinement applied on curved spectral elements. The nonconforming spectral element method and adaptive meshes technique are used to the curve surfaces and observe whether any discontinuity of the solutions. The adaptive nonconforming spectral element method is implemented to compare with those obtained by conforming mesh method with respect to several existing numerical and experimental studies. Meanwhile, three kinds of estimated error base mesh adaptation are conducted to compare their accuracy and efficiency with conforming mesh method. The results show adaptive nonconforming mesh method is more efficient than the conforming method. Especially, the vorticity error based method performs highest accuracy and fastest convergence. The results show this mesh refinement technique is applicable on the curved elements with satisfactory accuracy. It releases this technique may be applied on the simulations of flow past objects with more general geometries.

12

Yuan, Si, Yiyi Dong, Qinyan Xing, and Nan Fang. "Adaptive Finite Element Method of Lines with Local Mesh Refinement in Maximum Norm Based on Element Energy Projection Method." International Journal of Computational Methods 17, no. 04 (November 29, 2019): 1950008. http://dx.doi.org/10.1142/s0219876219500087.

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The reliable and efficient self-adaptive analysis is a modern goal of various numerical computations. Most adaptivity methods, however, adopt energy norm to measure errors, which may not be the most natural and convenient means, e.g., for problems with locally singular gradient of displacement. Based on the Element Energy Projection (EEP) super-convergent technique in the Finite Element Method of Lines (FEMOL) which is a general and powerful semi-discrete method, reliable error estimates of displacements in maximum norm can be obtained anywhere on the FEMOL mesh and hence adaptive FEMOL by maximum norm becomes feasible. However, to tackle singularity problems effectively and efficiently, an automatic and flexible local mesh refinement strategy is required to generate meshes of high quality for more efficient adaptive FEMOL analysis. Taking the two-dimensional Poisson equation as the model problem, the paper firstly introduces the FEMOL and EEP methods with interface sides resulting from local mesh refinement. Then a local mesh refinement strategy and corresponding adaptive algorithm are presented. The numerical results given show that the proposed adaptive FEMOL with local mesh refinement can produce displacement solutions satisfying the specified tolerances in maximum norm and the adaptively-generated meshes reasonably reflect the local difficulties inherent in the physical problems without much redundant accuracy.

13

Cao, Zhi-Wei, Zhi-Fan Liu, Zhi-Feng Liu, and Xiao-Hong Wang. "A Self-Adaptive Numerical Method to Solve Convection-Dominated Diffusion Problems." Mathematical Problems in Engineering 2017 (2017): 1–13. http://dx.doi.org/10.1155/2017/8379609.

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Convection-dominated diffusion problems usually develop multiscaled solutions and adaptive mesh is popular to approach high resolution numerical solutions. Most adaptive mesh methods involve complex adaptive operations that not only increase algorithmic complexity but also may introduce numerical dissipation. Hence, it is motivated in this paper to develop an adaptive mesh method which is free from complex adaptive operations. The method is developed based on a range-discrete mesh, which is uniformly distributed in the value domain and has a desirable property of self-adaptivity in the spatial domain. To solve the time-dependent problem, movement of mesh points is tracked according to the governing equation, while their values are fixed. Adaptivity of the mesh points is automatically achieved during the course of solving the discretized equation. Moreover, a singular point resulting from a nonlinear diffusive term can be maintained by treating it as a special boundary condition. Serval numerical tests are performed. Residual errors are found to be independent of the magnitude of diffusive term. The proposed method can serve as a fast and accuracy tool for assessment of propagation of steep fronts in various flow problems.

14

Ejlali, Nastaran, and Seyed Mohammad Hosseini. "Adaptive control parameterization method by density functions for optimal control problems." IMA Journal of Mathematical Control and Information 37, no. 2 (April 1, 2019): 497–512. http://dx.doi.org/10.1093/imamci/dnz010.

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Abstract This paper proposes an efficient adaptive control parameterization method for solving optimal control problems. In this method, mesh density functions are used to generate mesh points. In the first step, the problem is solved by control parameterization on uniform mesh points. Then at each step, the approximate control obtained from the previous step is applied to construct a mesh density function, and consequently a new adapted set of mesh points. Several numerical examples are included to demonstrate that the adaptive control parameterization method is more accurate than a uniform control parameterization one.

15

Hill, J., E. E. Popova, D. A. Ham, M. D. Piggott, and M. Srokosz. "Adapting to life: ocean biogeochemical modelling and adaptive remeshing." Ocean Science 10, no. 3 (May 9, 2014): 323–43. http://dx.doi.org/10.5194/os-10-323-2014.

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Abstract. An outstanding problem in biogeochemical modelling of the ocean is that many of the key processes occur intermittently at small scales, such as the sub-mesoscale, that are not well represented in global ocean models. This is partly due to their failure to resolve sub-mesoscale phenomena, which play a significant role in vertical nutrient supply. Simply increasing the resolution of the models may be an inefficient computational solution to this problem. An approach based on recent advances in adaptive mesh computational techniques may offer an alternative. Here the first steps in such an approach are described, using the example of a simple vertical column (quasi-1-D) ocean biogeochemical model. We present a novel method of simulating ocean biogeochemical behaviour on a vertically adaptive computational mesh, where the mesh changes in response to the biogeochemical and physical state of the system throughout the simulation. We show that the model reproduces the general physical and biological behaviour at three ocean stations (India, Papa and Bermuda) as compared to a high-resolution fixed mesh simulation and to observations. The use of an adaptive mesh does not increase the computational error, but reduces the number of mesh elements by a factor of 2–3. Unlike previous work the adaptivity metric used is flexible and we show that capturing the physical behaviour of the model is paramount to achieving a reasonable solution. Adding biological quantities to the adaptivity metric further refines the solution. We then show the potential of this method in two case studies where we change the adaptivity metric used to determine the varying mesh sizes in order to capture the dynamics of chlorophyll at Bermuda and sinking detritus at Papa. We therefore demonstrate that adaptive meshes may provide a suitable numerical technique for simulating seasonal or transient biogeochemical behaviour at high vertical resolution whilst minimising the number of elements in the mesh. More work is required to move this to fully 3-D simulations.

16

Guo, Wei, Yufeng Nie, and Weiwei Zhang. "Parallel adaptive mesh refinement method based on bubble-type local mesh generation." Journal of Parallel and Distributed Computing 117 (July 2018): 37–49. http://dx.doi.org/10.1016/j.jpdc.2018.02.008.

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17

SU, Xinrong, Satoru YAMAMOTO, and Kazuhiro NAKAHASHI. "Extending the Building Cube Method to Curvilinear Mesh with Adaptive Mesh Refinement." Journal of Fluid Science and Technology 9, no. 5 (2014): JFST0074. http://dx.doi.org/10.1299/jfst.2014jfst0074.

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18

KINUTA, Tetsuzo, Masashi YAMAKAWA, and Kenichi MATSUNO. "G616 Adaptive Mesh Generation Refinement Method with High Resolution Using Polygonal Mesh." Proceedings of the Fluids engineering conference 2012 (2012): 555–56. http://dx.doi.org/10.1299/jsmefed.2012.555.

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19

NOJIMA, Shishin, Tomotsugu SHIMOKAWA, Toshiyasu KINARI, and Sukenori SHINTAKU. "1113 Adaptive Mesh Refinement in the Quasicontinuum Method." Proceedings of Conference of Hokuriku-Shinetsu Branch 2007.44 (2007): 423–24. http://dx.doi.org/10.1299/jsmehs.2007.44.423.

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20

Mavriplis, Catherine. "Adaptive mesh strategies for the spectral element method." Computer Methods in Applied Mechanics and Engineering 116, no. 1-4 (January 1994): 77–86. http://dx.doi.org/10.1016/s0045-7825(94)80010-3.

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21

ITO, Hisao, and Osamu WATANABE. "Adaptive Elasto-Plastic Analysis Using Free Mesh Method." Transactions of the Japan Society of Mechanical Engineers Series A 67, no. 658 (2001): 955–63. http://dx.doi.org/10.1299/kikaia.67.955.

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22

Azarenok, Boris N., Sergey A. Ivanenko, and Tao Tang. "Adaptive Mesh Redistibution Method Based on Godunov's Scheme." Communications in Mathematical Sciences 1, no. 1 (2003): 152–79. http://dx.doi.org/10.4310/cms.2003.v1.n1.a10.

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23

Sikwila, Stephen T., and Stanford Shateyi. "AN ADAPTIVE MESH METHOD FOR BOUNDARY LAYER PROBLEMS." Far East Journal of Mathematical Sciences (FJMS) 113, no. 2 (April 30, 2019): 169–91. http://dx.doi.org/10.17654/ms113020169.

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24

YAMAKAWA, Masashi, Eiji KONISHI, Kenichi MATSUNO, and Shinichi ASAO. "Adaptive Polyhedral Mesh Generation Method for Compressible Flows." Journal of Computational Science and Technology 7, no. 2 (2013): 278–85. http://dx.doi.org/10.1299/jcst.7.278.

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25

SHIMOKAWA, Tomotsugu, Toshiyasu KINARI, and Sukenori SHINTAKU. "1229 Adaptive mesh refinement in the quasicontinuum method." Proceedings of the JSME annual meeting 2007.1 (2007): 37–38. http://dx.doi.org/10.1299/jsmemecjo.2007.1.0_37.

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26

Razbani, M. A. "Global root bracketing method with adaptive mesh refinement." Applied Mathematics and Computation 268 (October 2015): 628–35. http://dx.doi.org/10.1016/j.amc.2015.06.121.

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27

Soheili, Ali R., and John M. Stockie. "An adaptive mesh method with variable relaxation time." Journal of the Franklin Institute 344, no. 5 (August 2007): 757–64. http://dx.doi.org/10.1016/j.jfranklin.2006.02.014.

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28

Abedi, Reza, Omid Omidi, and Saeid Enayatpour. "A mesh adaptive method for dynamic well stimulation." Computers and Geotechnics 102 (October 2018): 12–27. http://dx.doi.org/10.1016/j.compgeo.2018.05.006.

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29

Zhang, Yang, Jiahua Xie, Xiaoyue Li, Zhenghai Ma, Jianfeng Zou, and Yao Zheng. "A multi-block adaptive solving technique based on lattice Boltzmann method." Modern Physics Letters B 32, no. 12n13 (May 10, 2018): 1840052. http://dx.doi.org/10.1142/s0217984918400523.

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In this paper, a CFD parallel adaptive algorithm is self-developed by combining the multi-block Lattice Boltzmann Method (LBM) with Adaptive Mesh Refinement (AMR). The mesh refinement criterion of this algorithm is based on the density, velocity and vortices of the flow field. The refined grid boundary is obtained by extending outward half a ghost cell from the coarse grid boundary, which makes the adaptive mesh more compact and the boundary treatment more convenient. Two numerical examples of the backward step flow separation and the unsteady flow around circular cylinder demonstrate the vortex structure of the cold flow field accurately and specifically.

30

Hill, J., E. E. Popova, D. A. Ham, M. D. Piggott, and M. Srokosz. "Adapting to life: ocean biogeochemical modelling and adaptive remeshing." Ocean Science Discussions 10, no. 6 (November 5, 2013): 1997–2051. http://dx.doi.org/10.5194/osd-10-1997-2013.

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Abstract. An outstanding problem in biogeochemical modelling of the ocean is that many of the key processes occur intermittently at small scales, such as the sub-mesoscale, that are not well represented in global ocean models. As an example, state-of-the-art models give values of primary production approximately two orders of magnitude lower than those observed in the ocean's oligotrophic gyres, which cover a third of the Earth's surface. This is partly due to their failure to resolve sub-mesoscale phenomena, which play a significant role in nutrient supply. Simply increasing the resolution of the models may be an inefficient computational solution to this problem. An approach based on recent advances in adaptive mesh computational techniques may offer an alternative. Here the first steps in such an approach are described, using the example of a~simple vertical column (quasi 1-D) ocean biogeochemical model. We present a novel method of simulating ocean biogeochemical behaviour on a vertically adaptive computational mesh, where the mesh changes in response to the biogeochemical and physical state of the system throughout the simulation. We show that the model reproduces the general physical and biological behaviour at three ocean stations (India, Papa and Bermuda) as compared to a high-resolution fixed mesh simulation and to observations. The simulations capture both the seasonal and inter-annual variations. The use of an adaptive mesh does not increase the computational error, but reduces the number of mesh elements by a factor of 2–3, so reducing computational overhead. We then show the potential of this method in two case studies where we change the metric used to determine the varying mesh sizes in order to capture the dynamics of chlorophyll at Bermuda and sinking detritus at Papa. We therefore demonstrate adaptive meshes may provide a~suitable numerical technique for simulating seasonal or transient biogeochemical behaviour at high spatial resolution whilst minimising computational cost.

31

He, Peng, and Huazhong Tang. "An Adaptive Moving Mesh Method for Two-Dimensional Relativistic Hydrodynamics." Communications in Computational Physics 11, no. 1 (January 2012): 114–46. http://dx.doi.org/10.4208/cicp.291010.180311a.

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AbstractThis paper extends the adaptive moving mesh method developed by Tang and Tang [36] to two-dimensional (2D) relativistic hydrodynamic (RHD) equations. The algorithm consists of two “independent” parts: the time evolution of the RHD equations and the (static) mesh iteration redistribution. In the first part, the RHD equations are discretized by using a high resolution finite volume scheme on the fixed but nonuniform meshes without the full characteristic decomposition of the governing equations. The second part is an iterative procedure. In each iteration, the mesh points are first redistributed, and then the cell averages of the conservative variables are remapped onto the new mesh in a conservative way. Several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed method.

32

KC, Gokul, and Ram Prasad Dulal. "Adaptive Finite Element Method for Solving Poisson Partial Differential Equation." Journal of Nepal Mathematical Society 4, no. 1 (May 14, 2021): 1–18. http://dx.doi.org/10.3126/jnms.v4i1.37107.

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Poisson equation is an elliptic partial differential equation, a generalization of Laplace equation. Finite element method is a widely used method for numerically solving partial differential equations. Adaptive finite element method distributes more mesh nodes around the area where singularity of the solution happens. In this paper, Poisson equation is solved using finite element method in a rectangular domain with Dirichlet and Neumann boundary conditions. Posteriori error analysis is used to mark the refinement area of the mesh. Dorfler adaptive algorithm is used to refine the marked mesh. The obtained results are compared with exact solutions and displayed graphically.

33

Gu, Yaguang, and Guanghui Hu. "A Third Order Adaptive ADER Scheme for One Dimensional Conservation Laws." Communications in Computational Physics 22, no. 3 (July 6, 2017): 829–51. http://dx.doi.org/10.4208/cicp.oa-2016-0088.

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AbstractWe introduce a third order adaptive mesh method to arbitrary high order Godunov approach. Our adaptive mesh method consists of two parts, i.e., mesh-redistribution algorithm and solution algorithm. The mesh-redistribution algorithm is derived based on variational approach, while a new solution algorithm is developed to preserve high order numerical accuracy well. The feature of proposed Adaptive ADER scheme includes that 1). all simulations in this paper are stable for large CFL number, 2). third order convergence of the numerical solutions is successfully observed with adaptive mesh method, and 3). high resolution and non-oscillatory numerical solutions are obtained successfully when there are shocks in the solution. A variety of numerical examples show the feature well.

34

Hsu, Li-Chieh, and Guo-Jhih Gao. "Simulation of Vortex Shedding behind a Flat Plate with Vorticity Based Adaptive Spectral Element Method." Mathematical Problems in Engineering 2014 (2014): 1–17. http://dx.doi.org/10.1155/2014/959615.

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Vorticity error based adaptive meshes refinement scheme is developed and employed using spectral element method to simulate flow past object problems. In general, it is hard to predict and enhance meshes effectively in a region where the error is larger in the computational domain by using the conforming mesh method. Employing finer meshes throughout the whole domain leads to lengthy computational time and excessive storage. Therefore, an indicator is used to predict the regions where larger errors exist and mesh refinement is needed. To compare the efficiency of indicators, three kinds of properties are used as mesh refinement indicators, including the synthesis of velocity and pressure estimated error, vorticity estimated error, and estimated error decay rate. Simulations of the cavity flow in Re = 100 and 1000 and the cases of flow past an inclined flat plate in Re = 100 to 1000 are performed with the adaptive mesh method and conforming mesh method. The results show that the adaptive mesh method can provide the same accuracy as that of the conforming mesh method with only 62% of the elements.

35

Wu, Yirong, and Heyu Wang. "Moving Mesh Finite Element Method for Unsteady Navier-Stokes Flow." Advances in Applied Mathematics and Mechanics 9, no. 3 (January 17, 2017): 742–56. http://dx.doi.org/10.4208/aamm.2016.m1457.

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AbstractIn this paper, we use moving mesh finite element method based upon 4P1–P1 element to solve the time-dependent Navier-Stokes equations in 2D. Two-layer nested meshes are used including velocity mesh and pressure mesh, and velocity mesh can be obtained by globally refining pressure mesh. We use hierarchy geometry tree to store the nested meshes. This data structure make convienence for adaptive mesh method and the construction of multigrid preconditioning. Several numerical problems are used to show the effect of moving mesh.

36

Zhi, Meipeng, and Yuesheng Xu. "Adaptive display images." Analysis and Applications 18, no. 01 (December 6, 2019): 1–23. http://dx.doi.org/10.1142/s0219530519410112.

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We develop a numerical method for construction of an adaptive display image from a given display image which is an artificial scene displayed in a computer screen. The adaptive display image is encoded on an adaptive pixel mesh obtained by a merging scheme from the original pixel mesh. The cardinality of the adaptive pixel mesh is significantly less than that of the original pixel mesh. The resulting adaptive display image is the best [Formula: see text] piecewise constant approximation of the original display image. Under the assumption that a natural image, the real scene that we see, belongs to a Besov space, we provide the optimal [Formula: see text] error estimate between the adaptive display image and its original natural image. Experimental results are presented to demonstrate the visual quality, the approximation accuracy and the computational complexity of the adaptive display image.

37

Mungkasi, Sudi. "Adaptive Finite Volume Method for the Shallow Water Equations on Triangular Grids." Advances in Mathematical Physics 2016 (2016): 1–7. http://dx.doi.org/10.1155/2016/7528625.

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This paper presents a numerical entropy production (NEP) scheme for two-dimensional shallow water equations on unstructured triangular grids. We implement NEP as the error indicator for adaptive mesh refinement or coarsening in solving the shallow water equations using a finite volume method. Numerical simulations show that NEP is successful to be a refinement/coarsening indicator in the adaptive mesh finite volume method, as the method refines the mesh or grids around nonsmooth regions and coarsens them around smooth regions.

38

Canuto, C., and I. Cravero. "A Wavelet-Based Adaptive Finite Element Method for Advection-Diffusion Equations." Mathematical Models and Methods in Applied Sciences 07, no. 02 (March 1997): 265–89. http://dx.doi.org/10.1142/s0218202597000165.

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We propose a wavelet-based procedure for adapting a finite element mesh to the structure of the solution. After a finite element solution is computed on a given unstructured mesh, it is wavelet-analyzed on a superimposed regular dyadic grid; the analysis leads to an adapted distribution of grid points, which defines the new unstructured mesh via a Delaunay triangulation. Several examples of discretizations of steady convection-diffusion problems in the convection-dominated regime indicate the feasibility of our approach.

39

Liu, Zhiqi, Jianhan Liang, and Yu Pan. "Efficient Thermodynamic Properties Reconstruction Method with Adaptive Triangular Mesh." Journal of Thermophysics and Heat Transfer 29, no. 1 (January 2015): 83–89. http://dx.doi.org/10.2514/1.t4422.

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40

NAGAOKA, Shinsuke, and Genki YAGAWA. "708 Dynamic adaptive analysis using Enriched Free Mesh Method." Proceedings of the Materials and Mechanics Conference 2007 (2007): 524–25. http://dx.doi.org/10.1299/jsmemm.2007.524.

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Tsuboi, Hajime, and Szabolcs Gyimothy. "Adaptive mesh generation for edge-element finite element method." Journal of Applied Physics 89, no. 11 (June 2001): 6713–15. http://dx.doi.org/10.1063/1.1363703.

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42

Carey, G. F., and S. Kennon. "Adaptive mesh redistribution for a boundary element (panel) method." International Journal for Numerical Methods in Engineering 24, no. 12 (December 1987): 2315–25. http://dx.doi.org/10.1002/nme.1620241206.

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43

Morrell, J. M., P. K. Sweby, and A. Barlow. "A cell by cell anisotropic adaptive mesh ALE method." International Journal for Numerical Methods in Fluids 56, no. 8 (2008): 1441–47. http://dx.doi.org/10.1002/fld.1599.

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44

Hussain, Ishfaq, Huiping Li, and Qunsheng Cao. "Multiscale Structure Simulation Using Adaptive Mesh in DGTD Method." IEEE Journal on Multiscale and Multiphysics Computational Techniques 2 (2017): 115–23. http://dx.doi.org/10.1109/jmmct.2017.2723261.

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45

Howell, Louis H., and John B. Bell. "An Adaptive Mesh Projection Method for Viscous Incompressible Flow." SIAM Journal on Scientific Computing 18, no. 4 (July 1997): 996–1013. http://dx.doi.org/10.1137/s1064827594270555.

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46

Bin, Jonghoon, Ali Uzun, and M. Yousuff Hussaini. "Adaptive mesh redistribution method for domains with complex boundaries." Journal of Computational Physics 230, no. 8 (April 2011): 3178–204. http://dx.doi.org/10.1016/j.jcp.2011.01.021.

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47

Duan, Xian-Bao, Fei-Fei Li, and Xin-Qiang Qin. "Adaptive mesh method for topology optimization of fluid flow." Applied Mathematics Letters 44 (June 2015): 40–44. http://dx.doi.org/10.1016/j.aml.2014.12.016.

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48

Hu, Fuxing, Rong Wang, Xueyong Chen, and Hui Feng. "An adaptive mesh method for 1D hyperbolic conservation laws." Applied Numerical Mathematics 91 (May 2015): 11–25. http://dx.doi.org/10.1016/j.apnum.2014.10.008.

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49

Zhang, Z. H., Z. J. Yang, and J. H. Li. "An Adaptive Polygonal Scaled Boundary Finite Element Method for Elastodynamics." International Journal of Computational Methods 13, no. 02 (March 2016): 1640015. http://dx.doi.org/10.1142/s0219876216400156.

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An adaptive polygonal scaled boundary finite element method (APSBFEM) is developed for elastodynamics. Flexible polygonal meshes are generated from background Delaunay triangular meshes and used to calculate structure’s dynamic responses. In each time step, a posteriori-type energy error estimator is employed to locate the polygonal subdomains with exceeding spatial discretization error, then edge midpoints of the corresponding triangles are inserted into the background. A new Delaunay triangular mesh and a polygonal mesh are regenerated successively. The state variables, including displacement, velocity and acceleration are mapped from the old polygonal mesh to the new one by a simple algorithm. A benchmark elastodynamic problem is modeled to validate the developed method. The results show that the adaptive meshes are capable of capturing the steep stress regions, and the dynamic responses agree well with those from the adaptive finite element method and the polygonal scaled boundary finite element method without adaptivity using fine meshes.

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DEGTYAREV, L. M., and V. V. DROZDOV. "Adaptive Mesh Computation of Magnetic Hydrodynamic Equilibrium." International Journal of Modern Physics C 02, no. 01 (March 1991): 30–38. http://dx.doi.org/10.1142/s0129183191000056.

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In stationary problems of mathematical physics an error of the finite-difference method ||z||=||u−uh|| is determined by the number of grid nodes N so that as N→∞, asymptotically [Formula: see text] The accuracy order m depends on the approximation of an original differential problem by a difference problem, while the constant C depends on the solution derivatives and the grid step h distribution. The value of ||z|| may be decreased by redistributing the grid points. An optimal computational grid is determined by both the region in which the original differential problem is solved and by the solution structure. Such technique will be called the method of grids adaptive to the solution. In this paper the ideology of the method of adaptive grids is presented for one-dimensional problems. A presentation of the method for two-dimensional MHD equilibrium problems is given. The main points of the method for three-dimensional MHD problems is discussed.