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Journal articles on the topic 'Adaptive multigrid algorithms'

Author: Grafiati

Published: 4 June 2021

Last updated: 7 February 2022

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1

Bartsch, Guido, and Christian Wulf. "Adaptive Multigrid for Helmholtz Problems." Journal of Computational Acoustics 11, no. 03 (September 2003): 341–50. http://dx.doi.org/10.1142/s0218396x03001997.

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Solving Helmholtz problems for low frequency sound fields by a truncated modal basis approach is very efficient. The most time-consuming process is the calculation of the undamped modes. Using traditional FE solvers, the user has to provide a mesh which has at least six nodes per wavelength in each spatial direction to achieve acceptable results. Because the mesh size increases with the 3rd power of the highest frequency of interest, this uniform dense mesh approach is a very expensive way of creating a modal space. However, the number of modes and the accuracy of the modal basis directly influences the solution quality. It is well known that the representation of sound fields by modal basis functions φi is optimal with respect to the L2 error norm. This means that having a modal basis Φ := {φi, i = 1⋯n}, the distance between true and approximated sound field takes its minimum in the mean square. So, it is necessary to have a FE basis which also minimizes the discretization error when computing the modal basis. One can reach this goal by applying adaptive mesh refinements. Additionally, this yields the opportunity of using fast multigrid methods to solve discrete eigenvalue problems. In context of this presentation we will discuss the results of our adaptive multigrid algorithms.

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Carstensen, Carsten, and Jun Hu. "Hierarchical Argyris Finite Element Method for Adaptive and Multigrid Algorithms." Computational Methods in Applied Mathematics 21, no. 3 (June 1, 2021): 529–56. http://dx.doi.org/10.1515/cmam-2021-0083.

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Abstract The global arrangement of the degrees of freedom in a standard Argyris finite element method (FEM) enforces C 2 {C^{2}} at interior vertices, while solely global C 1 {C^{1}} continuity is required for the conformity in H 2 {H^{2}} . Since the Argyris finite element functions are not C 2 {C^{2}} at the midpoints of edges in general, the bisection of an edge for mesh-refinement leads to non-nestedness: the standard Argyris finite element space A ′ ⁢ ( 𝒯 ) {A^{\prime}(\mathcal{T})} associated to a triangulation 𝒯 {\mathcal{T}} with a refinement 𝒯 ^ {\widehat{\mathcal{T}}} is not a subspace of the standard Argyris finite element space A ′ ⁢ ( 𝒯 ^ ) {A^{\prime}(\widehat{\mathcal{T}})} associated to the refined triangulation 𝒯 ^ {\widehat{\mathcal{T}}} . This paper suggests an extension A ⁢ ( 𝒯 ) {A(\mathcal{T})} of A ′ ⁢ ( 𝒯 ) {A^{\prime}(\mathcal{T})} that allows for nestedness A ⁢ ( 𝒯 ) ⊂ A ⁢ ( 𝒯 ^ ) {A(\mathcal{T})\subset A(\widehat{\mathcal{T}})} under mesh-refinement. The extended Argyris finite element space A ⁢ ( 𝒯 ) {A(\mathcal{T})} is called hierarchical, but is still based on the concept of the Argyris finite element as a triple ( T , P 5 ⁢ ( T ) , ( Λ 1 , … , Λ 21 ) ) {(T,P_{5}(T),(\Lambda_{1},\dots,\Lambda_{21}))} in the sense of Ciarlet. The other main results of this paper are the optimal convergence rates of an adaptive mesh-refinement algorithm via the abstract framework of the axioms of adaptivity and uniform convergence of a local multigrid V-cycle algorithm for the effective solution of the discrete system.

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Stals, Linda. "Algorithm-based fault recovery of adaptively refined parallel multilevel grids." International Journal of High Performance Computing Applications 33, no. 1 (August 23, 2017): 189–211. http://dx.doi.org/10.1177/1094342017720801.

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On future extreme scale computers, it is expected that faults will become an increasingly serious problem as the number of individual components grows and failures become more frequent. This is driving the interest in designing algorithms with built-in fault tolerance that can continue to operate and that can replace data even if part of the computation is lost in a failure. For fault-free computations, the use of adaptive refinement techniques in combination with finite element methods is well established. Furthermore, iterative solution techniques that incorporate information about the grid structure, such as the parallel geometric multigrid method, have been shown to be an efficient approach to solving various types of partial different equations. In this article, we present an advanced parallel adaptive multigrid method that uses dynamic data structures to store a nested sequence of meshes and the iteratively evolving solution. After a fail-stop fault, the data residing on the faulty processor will be lost. However, with suitably designed data structures, the neighbouring processors contain enough information so that a consistent mesh can be reconstructed in the faulty domain with the goal of resuming the computation without having to restart from scratch. This recovery is based on a set of carefully designed distributed algorithms that build on the existing parallel adaptive refinement routines, but which must be carefully augmented and extended.

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Yang, Yidu, Yu Zhang, and Hai Bi. "Multigrid Discretization and Iterative Algorithm for Mixed Variational Formulation of the Eigenvalue Problem of Electric Field." Abstract and Applied Analysis 2012 (2012): 1–25. http://dx.doi.org/10.1155/2012/190768.

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This paper discusses highly finite element algorithms for the eigenvalue problem of electric field. Combining the mixed finite element method with the Rayleigh quotient iteration method, a new multi-grid discretization scheme and an adaptive algorithm are proposed and applied to the eigenvalue problem of electric field. Theoretical analysis and numerical results show that the computational schemes established in the paper have high efficiency.

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Bader, M., S. Schraufstetter, C. A. Vigh, and J. Behrens. "Memory efficient adaptive mesh generation and implementation of multigrid algorithms using Sierpinski curves." International Journal of Computational Science and Engineering 4, no. 1 (2008): 12. http://dx.doi.org/10.1504/ijcse.2008.021108.

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6

Yang, Feng Wei, Chandrasekhar Venkataraman, Vanessa Styles, and Anotida Madzvamuse. "A Robust and Efficient Adaptive Multigrid Solver for the Optimal Control of Phase Field Formulations of Geometric Evolution Laws." Communications in Computational Physics 21, no. 1 (December 5, 2016): 65–92. http://dx.doi.org/10.4208/cicp.240715.080716a.

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AbstractWe propose and investigate a novel solution strategy to efficiently and accurately compute approximate solutions to semilinear optimal control problems, focusing on the optimal control of phase field formulations of geometric evolution laws. The optimal control of geometric evolution laws arises in a number of applications in fields including material science, image processing, tumour growth and cell motility. Despite this, many open problems remain in the analysis and approximation of such problems. In the current work we focus on a phase field formulation of the optimal control problem, hence exploiting the well developed mathematical theory for the optimal control of semilinear parabolic partial differential equations. Approximation of the resulting optimal control problemis computationally challenging, requiring massive amounts of computational time and memory storage. The main focus of this work is to propose, derive, implement and test an efficient solution method for such problems. The solver for the discretised partial differential equations is based upon a geometric multigrid method incorporating advanced techniques to deal with the nonlinearities in the problem and utilising adaptive mesh refinement. An in-house two-grid solution strategy for the forward and adjoint problems, that significantly reduces memory requirements and CPU time, is proposed and investigated computationally. Furthermore, parallelisation as well as an adaptive-step gradient update for the control are employed to further improve efficiency. Along with a detailed description of our proposed solution method together with its implementation we present a number of computational results that demonstrate and evaluate our algorithms with respect to accuracy and efficiency. A highlight of the present work is simulation results on the optimal control of phase field formulations of geometric evolution laws in 3-D which would be computationally infeasible without the solution strategies proposed in the present work.

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Miraçi, Ani, Jan Papež, and Martin Vohralík. "Contractive Local Adaptive Smoothing Based on Dörfler’s Marking in A-Posteriori-Steered p-Robust Multigrid Solvers." Computational Methods in Applied Mathematics 21, no. 2 (February 5, 2021): 445–68. http://dx.doi.org/10.1515/cmam-2020-0024.

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Abstract In this work, we study a local adaptive smoothing algorithm for a-posteriori-steered p-robust multigrid methods. The solver tackles a linear system which is generated by the discretization of a second-order elliptic diffusion problem using conforming finite elements of polynomial order p ≥ 1 {p\geq 1} . After one V-cycle (“full-smoothing” substep) of the solver of [A. Miraçi, J. Papež, and M. Vohralík, A-posteriori-steered p-robust multigrid with optimal step-sizes and adaptive number of smoothing steps, SIAM J. Sci. Comput. 2021, 10.1137/20M1349503], we dispose of a reliable, efficient, and localized estimation of the algebraic error. We use this existing result to develop our new adaptive algorithm: thanks to the information of the estimator and based on a bulk-chasing criterion, cf. [W. Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 1996, 3, 1106–1124], we mark patches of elements with increased estimated error on all levels. Then, we proceed by a modified and cheaper V-cycle (“adaptive-smoothing” substep), which only applies smoothing in the marked regions. The proposed adaptive multigrid solver picks autonomously and adaptively the optimal step-size per level as in our previous work but also the type of smoothing per level (weighted restricted additive or additive Schwarz) and concentrates smoothing to marked regions with high error. We prove that, under a numerical condition that we verify in the algorithm, each substep (full and adaptive) contracts the error p-robustly, which is confirmed by numerical experiments. Moreover, the proposed algorithm behaves numerically robustly with respect to the number of levels as well as to the diffusion coefficient jump for a uniformly-refined hierarchy of meshes.

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Sun, J., and P. Monk. "An Adaptive Algebraic Multigrid Algorithm for Micromagnetism." IEEE Transactions on Magnetics 42, no. 6 (June 2006): 1643–47. http://dx.doi.org/10.1109/tmag.2006.872004.

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9

Ji, Hua, Fue-Sang Lien, and Eugene Yee. "Parallel Adaptive Mesh Refinement Combined with Additive Multigrid for the Efficient Solution of the Poisson Equation." ISRN Applied Mathematics 2012 (March 12, 2012): 1–24. http://dx.doi.org/10.5402/2012/246491.

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Three different speed-up methods (viz., additive multigrid method, adaptive mesh refinement (AMR), and parallelization) have been combined in order to provide a highly efficient parallel solver for the Poisson equation. Rather than using an ordinary tree data structure to organize the information on the adaptive Cartesian mesh, a modified form of the fully threaded tree (FTT) data structure is used. The Hilbert space-filling curve (SFC) approach has been adopted for dynamic grid partitioning (resulting in a partitioning that is near optimal with respect to load balancing on a parallel computational platform). Finally, an additive multigrid method (BPX preconditioner), which itself is parallelizable to a certain extent, has been used to solve the linear equation system arising from the discretization. Our numerical experiments show that the proposed parallel AMR algorithm based on the FTT data structure, Hilbert SFC for grid partitioning, and additive multigrid method is highly efficient.

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10

LOPEZ, S., and R. CASCIARO. "ALGORITHMIC ASPECTS OF ADAPTIVE MULTIGRID FINITE ELEMENT ANALYSIS." International Journal for Numerical Methods in Engineering 40, no. 5 (March 15, 1997): 919–36. http://dx.doi.org/10.1002/(sici)1097-0207(19970315)40:5<919::aid-nme95>3.0.co;2-u.

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11

Livshits, I. "Multiple Galerkin Adaptive Algebraic Multigrid Algorithm for the Helmholtz Equations." SIAM Journal on Scientific Computing 37, no. 5 (January 2015): S195—S215. http://dx.doi.org/10.1137/140975310.

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12

van der Ven, H. "An adaptive multitime multigrid algorithm for time-periodic flow simulations." Journal of Computational Physics 227, no. 10 (May 2008): 5286–303. http://dx.doi.org/10.1016/j.jcp.2008.01.039.

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13

Chang, Qianshun, and Goubin Wang. "Multigrid and adaptive algorithm for solving the nonlinear schrodinger equation." Journal of Computational Physics 85, no. 2 (December 1989): 504. http://dx.doi.org/10.1016/0021-9991(89)90172-1.

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14

Chang, Qianshun, and Guobin Wang. "Multigrid and adaptive algorithm for solving the nonlinear Schrödinger equation." Journal of Computational Physics 88, no. 2 (June 1990): 362–80. http://dx.doi.org/10.1016/0021-9991(90)90184-3.

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15

Li, Feiyan, and Hai Bi. "A Type of Multigrid Method Based on the Fixed-Shift Inverse Iteration for the Steklov Eigenvalue Problem." Advances in Mathematical Physics 2016 (2016): 1–13. http://dx.doi.org/10.1155/2016/4691759.

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For the Steklov eigenvalue problem, we establish a type of multigrid discretizations based on the fixed-shift inverse iteration and study in depth its a priori/a posteriori error estimates. In addition, we also propose an adaptive algorithm on the basis of the a posteriori error estimates. Finally, we present some numerical examples to validate the efficiency of our method.

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16

Hemker, P. W., and F. Sprengel. "On the Representation of Functions and Finite Difference Operators on Adaptive Dyadic Grids." Computational Methods in Applied Mathematics 1, no. 3 (2001): 222–41. http://dx.doi.org/10.2478/cmam-2001-0016.

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Abstract In this paper we describe methods to approximate functions and dif- ferential operators on adaptive sparse (dyadic) grids. We distinguish between several representations of a function on the sparse grid and we describe how finite difference (FD) operators can be applied to these representations. For general variable coefficient equations on sparse grids, genuine finite element (FE) discretizations are not feasible and FD operators allow an easier operator evaluation than the adapted FE operators. However, the structure of the FD operators is complex. With the aim to construct an efficient multigrid procedure, we analyze the structure of the discrete Laplacian in its hierarchical representation and show the relation between the full and the sparse grid case. The rather complex relations, that are expressed by scaling matrices for each separate coordinate direction, make us doubt about the pos- sibility of constructing efficient preconditioners that show spectral equivalence. Hence, we question the possibility of constructing a natural multigrid algorithm with optimal O(N) efficiency. We conjecture that for the efficient solution of a general class of adaptive grid problems it is better to accept an additional condition for the dyadic grids (condition L) and to apply adaptive hp-discretization.

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17

Zeyao, Mo, Shen Longjun, and Gabriel Wittum. "Parallel adaptive multigrid algorithm for 2-d 3-t diffusion equations." International Journal of Computer Mathematics 81, no. 3 (March 2004): 361–74. http://dx.doi.org/10.1080/00207160410001661735.

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18

Sterck, Hans De, and Killian Miller. "An Adaptive Algebraic Multigrid Algorithm for Low-Rank Canonical Tensor Decomposition." SIAM Journal on Scientific Computing 35, no. 1 (January 2013): B1—B24. http://dx.doi.org/10.1137/110855934.

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Wise, S. M., J. S. Lowengrub, and V. Cristini. "An adaptive multigrid algorithm for simulating solid tumor growth using mixture models." Mathematical and Computer Modelling 53, no. 1-2 (January 2011): 1–20. http://dx.doi.org/10.1016/j.mcm.2010.07.007.

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20

Huber, Markus, Ulrich Rüde, and Barbara Wohlmuth. "Adaptive control in roll-forward recovery for extreme scale multigrid." International Journal of High Performance Computing Applications 33, no. 5 (December 25, 2018): 817–37. http://dx.doi.org/10.1177/1094342018817088.

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With the increasing number of compute components, failures in future exa-scale computer systems are expected to become more frequent. This motivates the study of novel resilience techniques. Here, we extend a recently proposed algorithm-based recovery method for multigrid iterations by introducing an adaptive control. After a fault, the healthy part of the system continues the iterative solution process, while the solution in the faulty domain is reconstructed by an asynchronous online recovery. The computations in both the faulty and the healthy subdomains must be coordinated in a sensitive way, in particular, both under- and over-solving must be avoided. Both of these waste computational resources and will therefore increase the overall time-to-solution. To control the local recovery and guarantee an optimal recoupling, we introduce a stopping criterion based on a mathematical error estimator. It involves hierarchically weighted sums of residuals within the context of uniformly refined meshes and is well-suited in the context of parallel high-performance computing. The recoupling process is steered by local contributions of the error estimator before the fault. Failure scenarios when solving up to 6.9 × 1011 unknowns on more than 245,766 parallel processes will be reported on a state-of-the-art peta-scale supercomputer demonstrating the robustness of the method.

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Thompson, Christopher P., Wayne R. Cowell, and Gary K. Leaf. "On the parallelization of an adaptive multigrid algorithm for a class of flow problems." Parallel Computing 18, no. 4 (April 1992): 449–66. http://dx.doi.org/10.1016/0167-8191(92)90140-3.

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Zhou, Wu, and Yaoqin Xie. "Interactive Multigrid Refinement for Deformable Image Registration." BioMed Research International 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/532936.

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Deformable image registration is the spatial mapping of corresponding locations between images and can be used for important applications in radiotherapy. Although numerous methods have attempted to register deformable medical images automatically, such as salient-feature-based registration (SFBR), free-form deformation (FFD), and demons, no automatic method for registration is perfect, and no generic automatic algorithm has shown to work properly for clinical applications due to the fact that the deformation field is often complex and cannot be estimated well by current automatic deformable registration methods. This paper focuses on how to revise registration results interactively for deformable image registration. We can manually revise the transformed image locally in a hierarchical multigrid manner to make the transformed image register well with the reference image. The proposed method is based on multilevel B-spline to interactively revise the deformable transformation in the overlapping region between the reference image and the transformed image. The resulting deformation controls the shape of the transformed image and produces a nice registration or improves the registration results of other registration methods. Experimental results in clinical medical images for adaptive radiotherapy demonstrated the effectiveness of the proposed method.

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Thompson, C. P., G. K. Leaf, and J. Van Rosendale. "A dynamically adaptive multigrid algorithm for the incompressible Navier-Stokes equations—validation and model problems." Applied Numerical Mathematics 9, no. 6 (May 1992): 511–32. http://dx.doi.org/10.1016/0168-9274(92)90005-x.

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24

Lee, Y. C., H. M. Thompson, and P. H. Gaskell. "An efficient adaptive multigrid algorithm for predicting thin film flow on surfaces containing localised topographic features." Computers & Fluids 36, no. 5 (June 2007): 838–55. http://dx.doi.org/10.1016/j.compfluid.2006.08.006.

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Ju, Liwei, Zhongfu Tan, Huanhuan Li, Xiaobao Yu, and Huijuan Zhang. "Multiobjective Synergistic Scheduling Optimization Model for Wind Power and Plug-In Hybrid Electric Vehicles under Different Grid-Connected Modes." Mathematical Problems in Engineering 2014 (2014): 1–15. http://dx.doi.org/10.1155/2014/179583.

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In order to promote grid’s wind power absorptive capacity and to overcome the adverse impacts of wind power on the stable operation of power system, this paper establishes benefit contrastive analysis models of wind power and plug-in hybrid electric vehicles (PHEVs) under the optimization goal of minimum coal consumption and pollutant emission considering multigrid connected modes. Then, a two-step adaptive solving algorithm is put forward to get the optimal system operation scheme with the highest membership degree based on the improvedεconstraints method and fuzzy decision theory. Thirdly, the IEEE36 nodes 10-unit system is used as the simulation system. Finally, the sensitive analysis for PHEV’s grid connected number is made. The result shows the proposed algorithm is feasible and effective to solve the model. PHEV’s grid connection could achieve load shifting effect and promote wind power grid connection. Especially, the optimization goals reach the optimum in fully optimal charging mode. As PHEV’s number increases, both abandoned wind and thermal power generation cost would decrease and the peak and valley difference of load curve would gradually be reduced.

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26

Fahmy, Mohamed Abdelsabour. "Boundary Element Algorithm for Nonlinear Modeling and Simulation of Three-Temperature Anisotropic Generalized Micropolar Piezothermoelasticity with Memory-Dependent Derivative." International Journal of Applied Mechanics 12, no. 03 (April 2020): 2050027. http://dx.doi.org/10.1142/s1758825120500271.

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The main aim of this paper is to introduce a new memory-dependent derivative theory to contribute for increasing development of technological and industrial applications of anisotropic smart materials. This theory is called three-temperature anisotropic generalized micropolar piezothermoelasticity. The governing equations of the proposed theory are very difficult to solve analytically because of material anisotropy and its nonlinear properties. Therefore, we propose a new boundary element formulation for solving such equations. The efficiency of our proposed technique has been developed by using an adaptive smoothing and prolongation algebraic multigrid (aSP-AMG) preconditioner to reduce the computation time. The numerical results are presented highlighting the effects of the kernel function and time delay on the temperature and displacements. The numerical results also verify the validity and accuracy of the proposed methodology. It can be concluded from the numerical results of our current complex and general study that some well-known uncoupled, coupled and generalized theories of anisotropic micropolar piezothermoelasticity can be connected with the three-temperature radiative heat conduction to characterize the deformation of anisotropicmicropolar piezothermoelasticstructures in the context of memory-dependent derivative.

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27

Wille, Svenøivind. "ADAPTIVE LINEARIZATION AND GRID ITERATIONS WITH THE TRI-TREE MULTIGRID REFINEMENT-RECOARSEMENT ALGORITHM FOR THE NAVIER-STOKES EQUATIONS." International Journal for Numerical Methods in Fluids 24, no. 2 (January 30, 1997): 155–68. http://dx.doi.org/10.1002/(sici)1097-0363(19970130)24:2<155::aid-fld484>3.0.co;2-1.

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Wu, Jun, Zhipeng Guo, and Chao Luo. "Development of a parallel adaptive multigrid algorithm for solving the multi-scale thermal-solute 3D phase-field problems." Computational Materials Science 142 (February 2018): 89–98. http://dx.doi.org/10.1016/j.commatsci.2017.09.045.

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Meca, Esteban, Andreas Münch, and Barbara Wagner. "Thin-film electrodes for high-capacity lithium-ion batteries: influence of phase transformations on stress." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472, no. 2193 (September 2016): 20160093. http://dx.doi.org/10.1098/rspa.2016.0093.

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In this study, we revisit experiments by Sethuraman et al. (2010 J. Power Sources , 195 , 5062–5066. ( doi:10.1016/j.jpowsour.2010.02.013 )) on the stress evolution during the lithiation/delithiation cycle of a thin film of amorphous silicon. Based on recent work that show a two-phase process of lithiation of amorphous silicon, we formulate a phase-field model coupled to elasticity in the framework of Larché-Cahn. Using an adaptive nonlinear multigrid algorithm for the finite-volume discretization of this model, our two-dimensional numerical simulations show the formation of a sharp phase boundary between the lithiated and the amorphous silicon that continues to move as a front through the thin layer. We show that our model captures the non-monotone stress loading curve and rate dependence, as observed in recent experiments and connects characteristic features of the curve with the structure formation within the layer. We take advantage of the thin film geometry and study the corresponding one-dimensional model to establish the dependence on the material parameters and obtain a comprehensive picture of the behaviour of the system.

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Liu, Xingwei, Qiulan Zhang, and Tangpei Cheng. "Accelerating Contaminant Transport Simulation in MT3DMS Using JASMIN-Based Parallel Computing." Water 12, no. 5 (May 22, 2020): 1480. http://dx.doi.org/10.3390/w12051480.

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To overcome the large time and memory consumption problems in large-scale high-resolution contaminant transport simulations, an efficient approach was presented to parallelize the modular three-dimensional transport model for multi-species (MT3DMS) (University of Alabama, Tuscaloosa, AL, USA) program on J adaptive structured meshes applications infrastructures (JASMIN). In this approach, a domain decomposition method and a stencil-based method were used to accomplish parallel implementation, while a ghost cell strategy was used for communication. The MODFLOW-MT3DMS coupling mode was optimized to achieve the parallel coupling of flow and contaminant transport. Five types of models were used to verify the correctness and test the parallel performance of the method. The developed parallel program JMT3D (China University of Geosciences (Beijing), Beijing, China) can increase the speed by up to 31.7 times, save memory consumption by 96% with 46 processors, and ensure that the solution accuracy and convergence do not decrease as the number of domains increases. The BiCGSTAB (Bi-conjugate gradient variant algorithm) method required the least amount of time and achieved high speedup in most cases. Coupling the flow and contaminant transport further improved the efficiency of the simulations, with a 33.45 times higher speedup achieved on 46 processors. The AMG (algebraic multigrid) method achieved a good scalability, with an efficiency above 100% on hundreds of processors for the simulation of tens of millions of cells.

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Brannick, J., R. C. Brower, M. A. Clark, J. C. Osborn, and C. Rebbi. "Adaptive Multigrid Algorithm for Lattice QCD." Physical Review Letters 100, no. 4 (January 28, 2008). http://dx.doi.org/10.1103/physrevlett.100.041601.

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Babich, R., J. Brannick, R. C. Brower, M. A. Clark, T. A. Manteuffel, S. F. McCormick, J. C. Osborn, and C. Rebbi. "Adaptive Multigrid Algorithm for the Lattice Wilson-Dirac Operator." Physical Review Letters 105, no. 20 (November 11, 2010). http://dx.doi.org/10.1103/physrevlett.105.201602.

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Li, Liping, and David B. Bogy. "A Local Adaptive Multigrid Control Volume Method for the Air Bearing Problem in Hard Disk Drives." Journal of Tribology 135, no. 3 (March 28, 2013). http://dx.doi.org/10.1115/1.4023804.

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A new, local-adaptive, grid-generating algorithm is developed and integrated with the multigrid control volume method to simulate the steady flying state of the air bearing sliders in hard disk drives (HDDs) accurately and efficiently. Local finer meshes (mesh dimension decreases to half) are created on the nodes of the current finest grids that have pressure gradients or geometry gradients larger than a predefined tolerance after the pressure distribution has been obtained on the initial uniform mesh. In this way, the pressure- or geometry-sensitive regions have higher resolution, leading to more accurate results without inefficiently larger meshes. Two sliders are used to demonstrate the applicability of this method. It is found that this new, local-adaptive, grid-generating method improves the stability and efficiency of the simulation scheme.

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Ng, Chin F., and Hermann B. Frieboes. "Simulation of Multispecies Desmoplastic Cancer Growth via a Fully Adaptive Non-linear Full Multigrid Algorithm." Frontiers in Physiology 9 (July 12, 2018). http://dx.doi.org/10.3389/fphys.2018.00821.

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Hook, Krish S. L., and Sergii Veremieiev. "Modelling of droplet impacts on dry and wet surfaces using depth-averaged form." Journal of Engineering Mathematics 126, no. 1 (February 2021). http://dx.doi.org/10.1007/s10665-020-10081-4.

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AbstractAn efficient time-adaptive multigrid algorithm is used to solve a range of normal and oblique droplet impacts on dry surfaces and liquid films using the Depth-Averaged Form (DAF) method of the governing unsteady Navier–Stokes equations. The dynamics of a moving three-phase contact line on dry surfaces is predicted by a precursor film model. The method is validated against a variety of experimental results for droplet impacts, looking at factors such as crown height and diameter, spreading diameter and splashing for a range of Weber, Reynolds and Froude numbers along with liquid film thicknesses and impact angles. It is found that, while being a computationally inexpensive methodology, the DAF method produces accurate predictions of the crown and spreading diameters as well as conditions for splash, however, underpredicts the crown height as the vertical inertia is not included in the model.

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