Adaptive Finite Volume Method Research Articles

CFD Research for Air Bearing with Gradient-Depth Recesses

Ultra-precision measurement and manufacturing need high-precision machines, just as a photolithography machine needs air bearings. In gas lubrication, the use of compound restrictors with recesses has been widely proven to be an effective method to improve stiffness, which directly affects the accuracy of the machine. However, determination of the structural parameters of recesses is lacking in theoretical models. This paper has established a mechanical property model for a small-scale guideway, which can respond to the variation in force caused by micron-level changes in the recesses’ depth. To meet the requirements of high positioning accuracy and movement accuracy, this paper puts forward a high-stiffness guideway without an air tube. In order to improve rotational stiffness and determine the structural parameters of recesses, this paper found a guideway with the optimal gradient depth of recesses. Both AFVM (adaptive finite volume method) research and experimental results show that the gradient depth of recesses could significantly improve the rotational stiffness of guideways without air tubes.

An Adaptive PML Finite Volume Algorithm for the Scattering by Periodic Gratings

In this paper, we studied the finite volume formulations for solving the diffraction grating problem that is truncated by the perfectly matched layer (PML) technique. Based on a reliable the a-posteriori error estimate, an adaptive PML finite volume method is discussed for the numerical approximation of the diffraction grating problem. The PML parameters are obtained numerically by sharp a-posteriori error estimates of the PML finite volume method such as the thickness of the layer and the medium property. It is worth mentioning in the a-posteriori error estimates that we derive the error representation formula and use a [Formula: see text]-orthogonality property of the residual similar to the Galerkin orthogonality used in the finite element method. Furthermore, the lower bound is established to demonstrate the efficiency of the a-posteriori error estimates. Numerical experiments are given to illustrate the accuracy and robustness of our adaptive algorithm.

Key Technologies and Design Methods of Ultra-Precision Aerostatic Bearings

Precision measurement technology and instruments are the basis for high-quality development in modern industry. With the specialization of supporting facilities, ultra-precision motion guidance has become a core technology to achieve ultra-precision measurement. Aerostatic bearings have the advantages of low friction and high precision, and are gradually playing an irreplaceable role in precision measurement and instruments. This paper summarizes and puts forward two contradictions for improving the loading capacity of aerostatic bearings, and investigates how to solve them via structural design. This paper analyzes the characteristics of the engineering calculation method, the gauge pressure ratio method, the gas resistance calculation method, and the AFVM (adaptive finite volume method) proposed. Finally, this paper summarizes the application scope of each method, gives solutions to the core problems, and outlines the developing trends.

Novel adaptive finite volume method on unstructured meshes for time-domain wave scattering and diffraction

Novel adaptive finite volume method on unstructured meshes for time-domain wave scattering and diffraction

An adaptive finite volume method for the diffraction grating problem with the truncated DtN boundary condition

An adaptive finite volume method for the diffraction grating problem with the truncated DtN boundary condition

On Accurately Resolving Detonation Dynamics by Adaptive Finite Volume Method on Unstructured Grids

On Accurately Resolving Detonation Dynamics by Adaptive Finite Volume Method on Unstructured Grids

Adaptive Moving Mesh Finite Volume Method Experimentation to Solving Riemann's Type Hyperbolic Problems

Adaptive Moving Mesh Finite Volume Method Experimentation to Solving Riemann's Type Hyperbolic Problems

Flood-Modeling and Risk Map Simulation for Mae Suai Dam-Break, Northern Thailand

Dam-break floods are a serious disaster. This study aims to simulate and model the Mae Suai dam-break flood using shallow water equations (SWE) with an adaptive tree grid finite volume method, and determine the relationship between the initial water levels in the dam and the simulation results set regarding arrival times and maximum water depths using a polynomial model. We used elevation data obtained from the Shuttle Radar Topography Mission. The method was evaluated using the Xe-Pian dam-break flood simulation. The numerical results of water propagation was in agreement with the satellite image. The SWE and numerical algorithm was then used for the Mae Suai dam-break flood simulation. The numerical solution sets were approximated by a polynomial function of appropriate degree for flood arrival times and maximum water depth. Comparisons showed that the polynomial model results were similar to the SWE results; however, the proposed method was more efficient and can obtain a flood risk map without the need to fully solve the SWE. The method can also be applied for dam-break flood simulations and models in other regions using information from the dam.

Weak Local Residuals as Smoothness Indicators in Adaptive Mesh Methods for Shallow Water Flows

This paper proposes some formulations of weak local residuals of shallow-water-type equations, namely, one-, one-and-a-half-, and two-dimensional shallow water equations. Smooth parts of numerical solutions have small absolute values of weak local residuals. Rougher parts of numerical solutions have larger absolute values of weak local residuals. This behaviour enables the weak local residuals to detect parts of numerical solutions which are smooth and rough (non-smooth). Weak local residuals that we formulate are implemented successfully as refinement or coarsening indicators for adaptive mesh finite volume methods used to solve shallow water equations.

Introduction to the non-orthogonal gradient-based finite volume method for three-dimensional neutron diffusion calculations with application to the rectangular reactor core configurations

The objective of this research is to introduce a new adaptive non-orthogonal gradient-based Finite Volume Method (FVM) for the accurate solution of the multi-group three-dimensional neutron diffusion equations. This method of calculation is developed for the first time, based on a new method of mesh movement and Control Volume (CV) deformation using the spatial gradient variations of the reactor power. The mesh movement and CV deformation is developed for rectangular three-dimensional reactor core configurations and different boundary conditions.For verification and evaluation of the proposed method, five benchmark problems have been considered and the results of calculations compared with the reference solutions. The results of solutions involved in obtaining reactor power as well as the corresponding effective multiplication factor showed that the proposed non-orthogonal gradient-based FVM is more accurate than the conventional FVM.

Vorticity dynamics and control of the turning locomotion of 3D bionic fish

Numerical simulations of the turning manoeuvre of a 3D bionic fish in a viscous flow are conducted in the present study using a 3D computational fluid dynamics package, which includes the adaptive multi-grid finite volume method, the immersed boundary method, the volume of fluid method, and the control strategy of fish swimming. The 3D bionic fish can turn quickly primarily using the directional control strategy of the swing of the head, in which the impacts of the swing of the fish body and the caudal fin on the directional control are also taken into account. This result demonstrates that the directional control strategy of the swing of the head is effective. An in-depth analysis of the fluid mechanism of the swimming manoeuvre of the 3D bionic fish reveals that the rotation moment in favour of turning motion is obtained by the pressure around the body of the fish, which is induced by the wake vortex, and the swing of the fish can manipulate the location and the strength of vortices shed from the fish body.

Adaptive vertex-centered finite volume methods for general second-order linear elliptic partial differential equations

We prove optimal convergence rates for the discretization of a general second-order linear elliptic PDE with an adaptive vertex-centered finite volume scheme. While our prior work Erath and Praetorius [SIAM J. Numer. Anal., 54 (2016), pp. 2228--2255] was restricted to symmetric problems, the present analysis also covers non-symmetric problems and hence the important case of present convection.

A posteriori error estimates for mixed finite volume solution of elliptic boundary value problems

Abstract The major emphasis of this work is the derivation of a posteriori error estimates for the mixed finite volume discretization of second-order elliptic equations. The estimates are established for meshes consisting of simplices on unstructured grids. We consider diffusion problems with nonhomogeneous diffusion coefficients. The error estimates are of residual types and are formulated in the energy semi-norm for a locally postprocessed approximate solutions. The estimates are fully computable and locally efficient that they can serve as indicators for adaptive refinement and for the actual control of the error. Numerical results are shown for two test examples in two space dimensions. It is found that the proposed adaptive mixed finite volume method offers a robust and accurate approach for solving second-order elliptic equations, even when highly nonhomogeneous diffusion coefficients are used in the simulations.

A numerical model for design and optimization of surface textures for tilting pad thrust bearings

A numerical model based on the Reynolds equation to study textured tilting pad thrust bearings considering mass-conserving cavitation and thermal effects is presented. A non-uniform and adaptive finite volume method is utilized and two methods are compared and selected regarding their efficiency in handling discontinuities; specifically placing additional nodes closely around discontinuities and directly incorporating discontinuities in the discrete system. Multithreading is applied to improve the computational performance and three root-finding methods to evaluate the bearing equilibrium are compared; namely Newton-Raphson method, Broyden's method with Sherman-Morrison formula and a continuation approach with fourth-order Runge-Kutta method. Results from the equivalent untextured bearing are utilized to accelerate the computation of the textured bearing and results are validated by comparison with CFD data.

An Adaptive Nonsymmetric Finite Volume and Boundary Element Coupling Method for a Fluid Mechanics Interface Problem

We consider an interface problem often arising in transport problems: a coupled system of partial differential equations with one (elliptic) transport equation on a bounded domain and one equation (in this case the Laplace problem) on the complement, an unbounded domain. Based on the nonsymmetric coupling of the finite volume method and boundary element method of [C. Erath, G. Of, and F.-J. Sayas, Numer. Math., 135 (2017), pp. 895--922] we introduce a semirobust residual error estimator. The upper bound of the error in an energy (semi)norm is robust against variation of the model data. The lower bound, however, additionally depends on the Péclet number and is therefore only semirobust. In several examples we use the local contributions of the a posteriori error estimator to steer an adaptive mesh-refining algorithm. The adaptive finite volume method--boundary element method coupling turns out to be an efficient method especially for solving problems from fluid mechanics, mainly because of the local flux conservation and the stable approximation of convection dominated problems.

A numerical study of 2D detonation waves with adaptive finite volume methods on unstructured grids

In this paper, a framework of adaptive finite volume solutions for the reactive Euler equations on unstructured grids is proposed. The main ingredients of the algorithm include a second order total variation diminishing Runge–Kutta method for temporal discretization, and the finite volume method with piecewise linear solution reconstruction of the conservative variables for the spatial discretization in which the least square method is employed for the reconstruction, and weighted essentially nonoscillatory strategy is used to restrain the potential numerical oscillation. To resolve the high demanding on the computational resources due to the stiffness of the system caused by the reaction term and the shock structure in the solutions, the h-adaptive method is introduced. OpenMP parallelization of the algorithm is also adopted to further improve the efficiency of the implementation. Several one and two dimensional benchmark tests on the ZND model are studied in detail, and numerical results successfully show the effectiveness of the proposed method.

Adjoint-based an adaptive finite volume method for steady Euler equations with non-oscillatory k -exact reconstruction

Abstract In [ An adaptive finite volume solver for steady Euler equations with non-oscillatory k-exact reconstruction, G. H. Hu and N. Y. Yi, J. Comput. Phys., Vol. 312, pp. 235–251 ], a high order finite volume method was proposed for steady Euler equations. The method consists of a Newton iteration method to linearize the governing equations, and a geometrical multigrid method to solve the derived system. The non-oscillatory k -exact reconstruction was developed towards the high quality solution reconstruction, and h -adaptive method was introduced to handle the efficiency issue. To further improve the numerical method, we study the following two issues in detail in this paper. First, we test the new reconstruction patch for the element which locates on the domain boundary, which could improve the convergence of the steady state significantly. Second, the adjoint-based a posteriori error estimation is introduced for the h -adaptive method, which could deliver more efficient adaptive refinement of the mesh. The numerical tests show the effectiveness of the proposed method.

An adaptive finite volume solver for steady Euler equations with non-oscillatory k-exact reconstruction

In this paper, we present an adaptive finite volume method for steady Euler equations with a non-oscillatory k-exact reconstruction on unstructured mesh. The numerical framework includes a Newton method as an outer iteration to linearize the Euler equations, and a geometrical multigrid method as an inner iteration to solve the derived linear system. A non-oscillatory k-exact reconstruction of the conservative solution in each element is proposed for the high order and non-oscillatory behavior of the numerical solutions. The importance on handling the curved boundary in an appropriate way is also studied with the numerical experiments. The h-adaptive method is introduced to enhance the efficiency of the algorithm. The numerical tests show successfully that the quality solutions can be obtained smoothly with the proposed algorithm, i.e., the expected convergence order of the numerical solution with the mesh refinement can be reached, while the non-oscillation shock structure can be obtained. Furthermore, the mesh adaptive method with the appropriate error indicators can effectively enhance the implementation efficiency of numerical method, while the steady state convergence and numerical accuracy are kept in the meantime.

Adaptive Vertex-Centered Finite Volume Methods with Convergence Rates

We consider the vertex-centered finite volume method with first-order conforming ansatz functions. The adaptive mesh-refinement is driven by the local contributions of the weighted-residual error estimator. We prove that the adaptive algorithm leads to linear convergence with generically optimal algebraic rates for the error estimator and the sum of energy error plus data oscillations. While similar results have been derived for finite element methods and boundary element methods, the present work appears to be the first for adaptive finite volume methods, where the lack of the classical Galerkin orthogonality leads to new challenges.

Adaptive Finite Volume Method for the Shallow Water Equations on Triangular Grids

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.