Multigrid Research Articles

Multigrid Method for Solving Inverse Problems for Heat Equation

In this paper, the inverse problems for the boundary value and initial value in a heat equation are posed and solved. It is well known that those problems are ill posed. The problems are reformulated as integral equations of the first kind by using the separation-of-variables method. The discretization of the integral equation allowed us to reduce the integral equation to a system of linear algebraic equations or a linear operator equation of the first kind on Hilbert spaces. The Landweber-type iterative method was used in order to find an approximation solution. The V-cycle multigrid method is used to obtain more frequent and fast convergence for iteration. The numerical computation examples are presented to verify the accuracy and fast computing of the approximation solution.

Thermal-fluid topology optimization with unconditional energy stability and second-order accuracy via phase-field model

This paper aims to establish a novel and efficient topology optimization method for the thermal-fluid. To adaptively design the fluid–solid coupling structure and make the objective energy to dissipate, the proposed method considers several constraints, such as the volume conservation, inlet and outlet flow velocity field and fluid–solid boundary constraints. The governing system includes the phase-field model, the steady state Darcy equation and the heat transfer equation. Under the constraints of multiple physical fields, we prove the existence of minimal solutions to the optimization problem. We use a Crank–Nicolson (CN) type scheme to discretize the governing system. The multigrid method is used to solve the resulting system of discrete equations. We prove the boundedness and unconditional stability of the original energy, which implies that a large time step can be used. The proposed discrete system is both spatially and temporally second-order accurate. Various computational tests have been performed to demonstrate that the numerical approach is efficient in designing the complicated structures of thermal fluid flows.

NonInvasive Multigrid For SemiStructured Grids

Multigrid solvers for hierarchical hybrid grids (HHG) have been proposed to promote the efficient utilization of high performance computer architectures. These HHG meshes are constructed by uniformly refining a relatively coarse fully unstructured mesh. While HHG meshes provide some flexibility for unstructured applications, most multigrid calculations can be accomplished using efficient structured grid ideas and kernels. This paper focuses on generalizing the HHG idea so that it is applicable to a broader community of computational scientists, and so that it is easier for existing applications to leverage structured multigrid components. Specifically, we adapt the structured multigrid methodology to significantly more complex semi-structured meshes. Further, we illustrate how mature applications might adopt a semi-structured solver in a relatively non-invasive fashion. To do this, we propose a formal mathematical framework for describing the semi-structured solver. This formalism allows us to precisely define the associated multigrid method and to show its relationship to a more traditional multigrid solver. Additionally, the mathematical framework clarifies the associated software design and implementation. Numerical experiments highlight the relationship of the new solver with classical multigrid. We also demonstrate the generality and potential performance gains associated with this type of semi-structured multigrid.

An Augmented Lagrangian Preconditioner for the Magnetohydrodynamics Equations at High Reynolds and Coupling Numbers

The magnetohydrodynamics (MHD) equations are generally known to be difficult to solve numerically, due to their highly nonlinear structure and the strong coupling between the electromagnetic and hydrodynamic variables, especially for high Reynolds and coupling numbers. In this work, we present a scalable augmented Lagrangian preconditioner for a finite element discretization of the $\mathbf{B}$-$\mathbf{E}$ formulation of the incompressible viscoresistive MHD equations. For stationary problems, our solver achieves robust performance with respect to the Reynolds and coupling numbers in two dimensions and good results in three dimensions. We extend our method to fully implicit methods for time-dependent problems which we solve robustly in both two and three dimensions. Our approach relies on specialized parameter-robust multigrid methods for the hydrodynamic and electromagnetic blocks. The scheme ensures exactly divergence-free approximations of both the velocity and the magnetic field up to solver tolerances. We confirm the robustness of our solver by numerical experiments in which we consider fluid and magnetic Reynolds numbers and coupling numbers up to 10,000 for stationary problems and up to 100,000 for transient problems in two and three dimensions.

A Multiresolution Discrete Element Method for Triangulated Objects with Implicit Time stepping

Simulations of many rigid bodies colliding with each other sometimes yield particularly interesting results if the colliding objects differ significantly in size and are non-spherical. The most expensive part within such a simulation code is the collision detection. We propose a family of novel multiscale collision detection algorithms that can be applied to triangulated objects within explicit and implicit time stepping methods. They are well-suited to handle objects that cannot be represented by analytical shapes or assemblies of analytical objects. Inspired by multigrid methods and adaptive mesh refinement, we determine collision points iteratively over a resolution hierarchy, and combine a functional minimisation plus penalty parameters with the actual comparision-based geometric distance calculation. Coarse surrogate geometry representations identify "no collision" scenarios early on and otherwise yield an educated guess which triangle subsets of the next finer level potentially yield collisions. They prune the search tree, and furthermore feed conservative contact force estimates into the iterative solve behind an implicit time stepping. Implicit time stepping and non-analytical shapes often yield prohibitive high compute cost for rigid body simulations. Our approach reduces these cost algorithmically by one to two orders of magnitude. It also exhibits high vectorisation efficiency due to its iterative nature.

A matrix-free multilevel preconditioner for the generalized Stokes problem with discontinuous viscosity

In this paper we present a matrix-free multilevel solver for the generalized Stokes problem with discontinuous viscosity. The algorithm is based on the FGMRES iteration preconditioned with a block-smoothed multigrid method. Numerical experiments on unstructured grids indicate that for a broad range of smoother parameters, the convergence speed is only weakly dependent on both the number of unknowns and the jumps of the viscosity coefficient.

Neural Green’s function for Laplacian systems

Solving linear system of equations stemming from Laplacian operators is at the heart of a wide range of applications. Due to the sparsity of the linear systems, iterative solvers such as Conjugate Gradient and Multigrid are usually employed when the solution has a large number of degrees of freedom. These iterative solvers can be seen as sparse approximations of the Green’s function for the Laplacian operator. In this paper we propose a machine learning approach that regresses a Green’s function from boundary conditions. This is enabled by a Green’s function that can be effectively represented in a multi-scale fashion, drastically reducing the cost associated with a dense matrix representation. Additionally, since the Green’s function is solely dependent on boundary conditions, training the proposed neural network does not require sampling the right-hand side of the linear system. We show results that our method outperforms state of the art Conjugate Gradient and Multigrid methods.

A multi-grid sampling multi-scale method for crack initiation and propagation

In this study, a multi-grid sampling multi-scale (MGSMS) method is proposed by coupling with finite element (FEM), extended finite element (XFEM) and molecular dynamics (MD) methods. Crack is studied comprehensively from microscopic initiations to macroscopic propagation by MGSMS method. In order to establish the coupling relationship between macroscopic and microscopic model, the multi-grid FEM is used to transmit the macroscopic displacement boundary conditions to the atomic model and the multi-grid XFEM is used to feedback the microscopic crack initiations to the macroscopic model. Moreover, an image recognition based crack extracting method is proposed to extract the crack coordinate from the MD result files efficiently and the Latin hypercube sampling method is used to reduce the computational cost of MD. Numerical results show that MGSMS method can be used to calculate micro-crack initiations and transmit it to the macro-crack model. The crack initiation and propagation simulation of plate under mode I loading is completed.

Boosting vision transformer for low-resolution borehole image stitching through algebraic multigrid

Boosting vision transformer for low-resolution borehole image stitching through algebraic multigrid

Neural network–based pore flow field prediction in porous media using super resolution

Predicting the pore flow velocity directly from the sub-sampled pore structure is an ill-conditioned problem. Inspired by multi-grid methods for solving systems of linear equations, we use velocity fields simulated on coarse meshes to remedy such ill-conditioning. This leads to a super-resolution-assisted geometry-to-velocity mapping for porous media.

A thermal network model for hydrocarbon heat pump systems: A coupling analysis of configuration, working condition, and refrigeration distribution

The ambition of zero emission neighborhood has been set out to operate the building components with zero life cycle greenhouse gas emissions. Hydrocarbon heat pumps as the building components with nearly zero greenhouse gas emissions are faced with the problem of simultaneously reducing refrigerant charges and maintaining competitive capacities. These must be resolved by formulating an effective modeling tool for detailed design and optimization. To develop a resistance–capacitance network model, interdisciplinary knowledge (including graph theory, heat and mass transfer, local moving boundary modeling, and algebraic multigrid) is integrated. The model simultaneously considers heat exchanger circuitries with a high detail level and component behaviors at a holistic system level. Based on experimental validation, the model’s error is ∼ 10%. In an R290 heat pump case study, the coupling mechanism of the system’s configuration, working conditions, and refrigerant distribution are analyzed. Results indicate that the refrigerant charge can be reduced by 11.6% at the expense of a 4.2% reduction in the coefficient of performance, by varying the compressor displacement alone. In response to the variation in the system’s refrigerant, 46.8%, 7.85%, and 44.2% of the total amount of refrigerants are allocated to the condenser, evaporator, and compressor, respectively. The distribution of the refrigerant quality in the heat exchanger is also visualized to indicate the direction of refrigerant reduction.

A fast unsmoothed aggregation algebraic multigrid framework for the large-scale simulation of incompressible flow

Multigrid methods are quite efficient for solving the pressure Poisson equation in simulations of incompressible flow. However, for viscous liquids, geometric multigrid turned out to be less efficient for solving the variational viscosity equation. In this contribution, we present an Unsmoothed Aggregation Algebraic MultiGrid (UAAMG) method with a multi-color Gauss-Seidel smoother, which consistently solves the variational viscosity equation in a few iterations for various material parameters. Moreover, we augment the OpenVDB data structure with Intel SIMD intrinsic functions to perform sparse matrix-vector multiplications efficiently on all multigrid levels. Our framework is 2.0 to 14.6 times faster compared to the state-of-the-art adaptive octree solver in commercial software for the large-scale simulation of both non-viscous and viscous flow. The code is available at http://computationalsciences.org/publications/shao-2022-multigrid.html.

RPINNs: Rectified-physics informed neural networks for solving stationary partial differential equations

Due to the development of high performance computing, deep learning algorithm has made a significant progress in many fields such as computational mathematics. The physics-informed neural networks have put forward a innovative idea for tackling a broad range of forward and inverse problems of partial differential equations. Motivated by the philosophy of physics-informed neural network and the multigrid method, we introduce the gradient information of numerical solution of physics-informed neural network into the new neural networks and propose the rectified-physics informed neural network for solving stationary partial differential equations. And for solving multi-objective optimization of neural networks, the dynamic weight strategy is adopted to balance numerical difference among terms in the loss function, and effectively alleviate the gradient ill-conditioned phenomenon. Finally, we perform a series of numerical experiments to demonstrate effectiveness of the RPINNs method which is combined with the dynamic weight strategy to improve calculation accuracy.

The Multigrid Method for the Combined Hybrid Element of Linear Elasticity Problem

Combined hybrid element method is one kind of stable finite element discrete method in which the famous Babuska–Brezzi condition is satisfied automatically. So, the method is more widely used, compared with other kinds of mixed/hybrid element methods. In this paper, we develop a non-nested multigrid algorithm for combined hybrid quadrilateral or hexahedron elements of linear elasticity problem. The critical ingredient in the algorithm is a proper intergrid transfer operator. We establish such an operator on quadrilateral or hexahedron meshes and prove the mesh-independent convergence of the k t h level iteration and full multigrid algorithm in L 2 norm. Numerical experiments are reported to support our theoretical results and illustrate the efficiency of the developed methods. We also give the numerical experiments showing the convergence of the developed method as Poisson’s ratio is close to 0.5.

Inexact and primal multilevel FETI‐DP methods: a multilevel extension and interplay with BDDC

We study a framework that allows to solve the coarse problem in the FETI-DP method approximately. It is based on the saddle-point formulation of the FETI-DP system with a block-triangular preconditioner. One of the blocks approximates the coarse problem, for which we use the multilevel BDDC method as the main tool. This strategy then naturally leads to a version of multilevel FETI-DP method, and we show that the spectra of the multilevel FETI-DP and BDDC preconditioned operators are essentially the same. The theory is illustrated by a set of numerical experiments, and we also present a few experiments when the coarse solve is approximated by algebraic multigrid.

How Do Help “Multigrid Principles based on Numerical Solutions of Partial Differential Equations” for Smoothing Process (Concepts)?

In Twenty century, Partial Differential Equations (PDE) are solved by Numerical Approach such as Adam – Smith Methods, Numerical Simulation Methods, Finite Difference methods etc., In this article we discuss about difficulties of solve either Differential equations or PDE in 3 – Dimensional cases. So that we discuss only 1 – Dimensional Multigrid Methods (M. M) and we discuss about M. M. Errors, Corrections, Type of grid such as Coarse Grid (C. G) and Fine Grid (F. G), Smoothing, non – smoothing approximation of C. G. Finally, we explain that M. G. M. works by decomposing problem into separate length scale and also using an iterative method. This method optimizes errors deduction in the length scales globally. In Multigrid Methods (MgM) several sub – routines must be developed to pass the data from C. G to F. G (Interpolation) from F. G to C. G (Reduction) and correction of the error at each grid interval (Smoothing), simply we have results reaction as (Reduction)C.G⇄F.G (Interpolation).

Computation and Analysis of an Offshore Wind Power Forecast: Towards a Better Assessment of Offshore Wind Power Plant Aerodynamics

For the first time, the Weather Research and Forecast (WRF) model with the Wind Farm Parameterization (WFP) modeling method is utilized for a short-range wind power forecast simulation of 48 h of an offshore wind farm with 100 turbines located on the east coast of the China Yellow Sea. The effects of the horizontal multi-grid downsize method were deployed and investigated on this simulation computation. The simulation was validated with the field data from the Supervisory Control and Data Acquisition (SCADA) system, and the results showed that the horizontal mesh downsize method improved the accuracy of wind speed and then wind power forecast. Meanwhile, the wind power plant aerodynamics with turbine wake and sea–land shore effects were investigated, where the wake effects from the wind farm prolonged several miles downstream, evaluated at two wind speeds of 7 m/s and 10 m/s instances captured from the 48 h of simulation. At the same time, it was interesting to find some sea–land atmospheric effects with wind speed oscillation, especially at the higher wind speed condition. Finally, the research results show that the WRF + WFP model for the wind power forecast for production operation may not be ready at this stage; however, they show that the methodology helps to evaluate the wind power plant aerodynamics with wake effects and micrometeorology of the sea–land interconnection region. This plant aerodynamics study set the stage for a wake turbine interaction study in the future, such as one utilizing the NREL FAST.FARM tool.

Convergence and optimality of adaptive multigrid method for multiple eigenvalue problems

A new type of adaptive multigrid method is presented for multiple eigenvalue problems based on multilevel correction scheme and adaptive multigrid method. Different from the classical adaptive finite element method which requires to solve eigenvalue problems on the adaptively refined triangulations, with our approach we just need to solve several linear boundary value problems in the current refined space and an eigenvalue problem in a very low dimensional space. Further, the involved boundary value problems are solved by an adaptive multigrid iteration. Since there is no eigenvalue problem to be solved on the refined triangulations, which is quite time-consuming, the proposed method can achieve the same efficiency as that of the adaptive multigrid method for the associated linear boundary value problems. Besides, the corresponding convergence and optimal complexity are verified theoretically and demonstrated numerically.

Extrapolation Cascadic Multigrid Method for Cell-Centered FV Discretization of Diffusion Equations with Strongly Discontinuous and Anisotropic Coefficients

Extrapolation Cascadic Multigrid Method for Cell-Centered FV Discretization of Diffusion Equations with Strongly Discontinuous and Anisotropic Coefficients

Development and Evaluation of a Real-Time Hourly One-Kilometre Gridded Multisource Fusion Air Temperature Dataset in China Based on Remote Sensing DEM

High-resolution gridded 2 m air temperature datasets are important input data for global and regional climate change studies, agrohydrologic model simulations and numerical weather predictions, etc. In this study, the digital elevation model (DEM) is used to correct temperature forecasts produced by ECMWF. The multi-grid variation formulation method is then used to fuse the data from corrected temperature forecasts and ground automatic station observations. The fused dataset covers the area over (0–60°N, 70–140°S), where different underlying surfaces exist, such as plains, basins, plateaus, and mountains. The spatial and temporal resolutions are 1 km and 1 h, respectively. The comparison of the fusion data with the verification observations, including 2400 weather stations, indicates that the accuracy of the gridded temperature is superior to European Centre for Medium-Range Weather Forecasts (ECMWF) data. This is because a more significant number of stations and high-resolution terrain data are used to generate the fusion data than are utilized in the ECMWF. The obtained dataset can describe the temperature feature of peaks and valleys more precisely. Due to its continuous temporal coverage and consistent quality, the fusion dataset is one of China’s most widely used temperature datasets. However, data uncertainty will increase for areas with sparse observations and high mountains, and we must be cautious when using data from these areas.