Moving Mesh Equation Research Articles

Adaptively Detect and Accurately Resolve Macro-scale Shocks in an Efficient Equation-Free Multiscale Simulation

The Equation-Free approach to efficient multiscale numerical computation marries trusted micro-scale simulations to a framework for numerical macro-scale reduction -- the patch dynamics scheme. A recent novel patch scheme empowered the Equation-Free approach to simulate systems containing shocks on the macro-scale. However, the scheme did not predict the formation of shocks accurately, and it could not simulate moving shocks. This article resolves both issues, as a first step in one spatial dimension, by embedding the Equation-Free, shock-resolving patch scheme within a classic framework for adaptive moving meshes. Our canonical micro-scale problems exhibit heterogeneous nonlinear advection and heterogeneous diffusion. We demonstrate many remarkable benefits from the moving patch scheme, including efficient and accurate macro-scale prediction despite the unknown macro-scale closure. Equation-free methods are here extended to simulate moving, forming and merging shocks without a priori knowledge of the existence or closure of the shocks. Whereas adaptive moving mesh equations are typically stiff, typically requiring small time-steps on the macro-scale, the moving macro-scale mesh of patches is typically not stiff given the context of the micro-scale time-steps required for the sub-patch dynamics.

A surface moving mesh method based on equidistribution and alignment

A surface moving mesh method is presented for general surfaces with or without explicit parameterization. The method can be viewed as a nontrivial extension of the moving mesh partial differential equation method that has been developed for bulk meshes and demonstrated to work well for various applications. The main challenges in the development of surface mesh movement come from the fact that the Jacobian matrix of the affine mapping between the reference element and any simplicial surface element is not square. The development starts with revealing the relation between the area of a surface element in the Euclidean or Riemannian metric and the Jacobian matrix of the corresponding affine mapping, formulating the equidistribution and alignment conditions for surface meshes, and establishing a meshing energy function based on the conditions. The moving mesh equation is then defined as the gradient system of the energy function, with the nodal mesh velocities being projected onto the underlying surface. The analytical expression for the mesh velocities is obtained in a compact, matrix form, which makes the implementation of the new method on a computer relatively easy and robust. Moreover, it is analytically shown that any mesh trajectory generated by the method remains nonsingular if it is so initially. It is emphasized that the method is developed directly on surface meshes, making no use of any information on surface parameterization. It utilizes surface normal vectors to ensure that the mesh vertices remain on the surface while moving, and also assumes that the initial surface mesh is given. The new method can apply to general surfaces with or without explicit parameterization since the surface normal vectors can be computed based on the current mesh. A selection of two- and three-dimensional examples are presented.

Moving mesh method for simulating high-dimensional time dependent PDEs with fast propagating shock waves

When solutions of the high dimensional PDE involve fast propagating shock waves, existing variational moving mesh methods are rather unstable and time consuming. To circumvent the problem, we improve the existing variational method by adding one term which adapts the mesh to the dynamic variations of the PDEs during time iterations. This is an extension of the moving mesh strategy in (Gao, 2018) for one dimensional (1-D) problems. We prove the stability and convergence of the present moving mesh equation for arbitrary initial mesh. Consequently, by enlarging the time step sizes of mesh generation, the computational time is shortened without causing instability. The moving mesh equation and the PDE are simulated step by step applying the alternate finite difference method. Both theoretical analysis and numerical results imply that the method is highly effective and robust when simulating high dimensional time dependent PDEs with fast moving shock waves.

Towards additive manufacturing of compressor impellers: 3D modeling of multilayer laser solid freeform fabrication of nickel alloy 625 powder mixed with nano-CeO2 on AISI 4140

Gas turbine blades, turbine shafts and centrifugal compressor impellers are often damaged by erosion and/or corrosion. By laser cladding technique, a coating layer can be deposited on the base material in order to rebuild, repair and improve anti-erosion or anti-corrosion properties of the sensitive machine parts. In this paper, a three-dimensional finite element modeling of the laser solid freeform fabrication (LSFF) process for nickel alloy 625 powder mixed with nano-CeO2 on AISI 4140 steel is extensively studied. Using Comsol Multiphysics software and the finite element method (FEM), the heat transfer equation, moving mesh equation and stress tensor are numerically solved. The dynamic geometry of the molten zone is studied by a 3D moving mesh based on Arbitrary Lagrangian–Eulerian (ALE) module. Clad shape, temperature distribution and stress fields are obtained. The effects of preheating as well as addition of nano-CeO2 are investigated. Dependence of the clad height on the scanning velocity of the laser is also studied.

Moving Mesh Strategies of Adaptive Methods for Solving Nonlinear Partial Differential Equations

This paper proposes moving mesh strategies for the moving mesh methods when solving the nonlinear time dependent partial differential equations (PDEs). Firstly we analyse Huang’s moving mesh PDEs (MMPDEs) and observe that, after Euler discretion they could be taken as one step of the root searching iteration methods. We improve Huang’s MMPDE by adding one Lagrange speed term. The proposed moving mesh PDE could draw the mesh to equidistribution quickly and stably. The numerical algorithm for the coupled system of the original PDE and the moving mesh equation is proposed and the computational experiments are given to illustrate the validity of the new method.

Moving mesh discontinuous Galerkin methods for PDEs with traveling waves

In this paper, a moving mesh discontinuous Galerkin (dG) method is developed for nonlinear partial differential equations (PDEs) with traveling wave solutions. The moving mesh strategy for one dimensional PDEs is based on the rezoning approach which decouples the solution of the PDE from the moving mesh equation. We show that the dG moving mesh method is able to resolve sharp wave fronts and wave speeds accurately for the optimal, arc-length and curvature monitor functions. Numerical results reveal the efficiency of the proposed moving mesh dG method for solving Burgers’, Burgers’–Fisher and Schlögl (Nagumo) equations.

Adaptive finite-volume WENO schemes on dynamically redistributed grids for compressible Euler equations

A high-order adaptive finite-volume method is presented for simulating inviscid compressible flows on time-dependent redistributed grids. The method achieves dynamic adaptation through a combination of time-dependent mesh node clustering in regions characterized by strong solution gradients and an optimal selection of the order of accuracy and the associated reconstruction stencil in a conservative finite-volume framework. This combined approach maximizes spatial resolution in discontinuous regions that require low-order approximations for oscillation-free shock capturing. Over smooth regions, high-order discretization through finite-volume WENO schemes minimizes numerical dissipation and provides excellent resolution of intricate flow features. The method including the moving mesh equations and the compressible flow solver is formulated entirely on a transformed time-independent computational domain discretized using a simple uniform Cartesian mesh. Approximations for the metric terms that enforce discrete geometric conservation law while preserving the fourth-order accuracy of the two-point Gaussian quadrature rule are developed. Spurious Cartesian grid induced shock instabilities such as carbuncles that feature in a local one-dimensional contact capturing treatment along the cell face normals are effectively eliminated through upwind flux calculation using a rotated Hartex–Lax–van Leer contact resolving (HLLC) approximate Riemann solver for the Euler equations in generalized coordinates. Numerical experiments with the fifth and ninth-order WENO reconstructions at the two-point Gaussian quadrature nodes, over a range of challenging test cases, indicate that the redistributed mesh effectively adapts to the dynamic flow gradients thereby improving the solution accuracy substantially even when the initial starting mesh is non-adaptive. The high adaptivity combined with the fifth and especially the ninth-order WENO reconstruction allows remarkably sharp capture of discontinuous propagating shocks with simultaneous resolution of smooth yet complex small scale unsteady flow features to an exceptional detail.

Stability of Newton TVD Runge–Kutta scheme for one-dimensional Euler equations with adaptive mesh

In this paper, we propose a moving mesh method with a Newton total variation diminishing (TVD) Runge–Kutta scheme for the Euler equations. Our scheme improves time discretization in the moving mesh algorithms. By analyzing the semi-discrete Euler equations with the discrete moving mesh equations as constraints, the stability of the Newton TVD Runge–Kutta scheme is proved. Thus, we can conclude that the proposed algorithm can generate a weak solution to the Euler equations. Finally, numerical examples are presented to verify the theoretical results and demonstrate the accuracy of the proposed scheme.

A study of moving mesh PDE methods for numerical simulation of blowup in reaction diffusion equations

A new concept called the dominance of equidistribution is introduced for analyzing moving mesh partial differential equations for numerical simulation of blowup in reaction diffusion equations. Theoretical and numerical results show that a moving mesh method works successfully when the employed moving mesh equation has the dominance of equidistribution. The property can be verified using dimensional analysis. In several aspects the current work generalizes previous work where a moving mesh equation is shown to have this dominance of equidistribution if it preserves the scaling invariance of the underlying physical partial differential equation and uses a small, constant value for τ (a parameter used for adjusting response time of the mesh movement to the change in the physical solution). Also, cases with both constant and variable τ are considered here.

A moving mesh finite element algorithm for the adaptive solution of time-dependent partial differential equations with moving boundaries

A moving mesh finite element algorithm is proposed for the adaptive solution of nonlinear diffusion equations with moving boundaries in one and two dimensions. The moving mesh equations are based upon conserving a local proportion, within each patch of finite elements, of the total “mass” that is present in the projected initial data. The accuracy of the algorithm is carefully assessed through quantitative comparison with known similarity solutions, and its robustness is tested on more general problems. Applications are shown to a variety of problems involving time-dependent partial differential equations with moving boundaries. Problems which conserve mass, such as the porous medium equation and a fourth order nonlinear diffusion problem, can be treated by a simplified form of the method, while problems which do not conserve mass require the full theory.

A Moving Mesh Method for One-dimensional Hyperbolic Conservation Laws

We develop an adaptive method for solving one-dimensional systems of hyperbolic conservation laws that employs a high resolution Godunov-type scheme for the physical equations, in conjunction with a moving mesh PDE governing the motion of the spatial grid points. Many other moving mesh methods developed to solve hyperbolic problems use a fully implicit discretization for the coupled solution-mesh equations, and so suffer from a significant degree of numerical stiffness. We employ a semi-implicit approach that couples the moving mesh equation to an efficient, explicit solver for the physical PDE, with the resulting scheme behaving in practice as a two-step predictor-corrector method. In comparison with computations on a fixed, uniform mesh, our method exhibits more accurate resolution of discontinuities for a similar level of computational work.

Moving Mesh Partial Differential Equations (MMPDES) Based on the Equidistribution Principle

This paper considers several moving mesh partial differential equations that are related to the equidistribution principle. Several of these are new, and some correspond to discrete moving mesh equations that have been used by others. Their stability is analyzed and it is seen that a key term for most of these moving mesh PDEs is a source-like term that measures the level of equidistribution. It is shown that under weak assumptions mesh crossing cannot occur for most of them. Finally, numerical experiments for these various moving mesh PDEs are performed to study their relative properties.

Dynamic rezone methods for partial differential equations in one space dimension

Moving mesh equations are derived to reduce the truncation errors and improve the efficiency of numerical methods for solving partial differential equations. We propose methods based on a blend of mesh directions and velocities chosen to reduce errors in the space and time discretizations. We use numerical calculations to demonstrate the effetiveness of the approach.

Observations on an adaptive moving grid method for one-dimensional systems of partial differential equations

Recently a scheme has been proposed for choosing a moving mesh based on minimizing the time rate of change of the solution in the moving coordinates for one-dimensional systems of PDEs. In this paper we show how to apply this idea to systems where the time derivatives cannot be solved for explicitly, writing the moving mesh equations in an implicit form. We give a geometrical interpretation of the scheme which exposes some of its weaknesses, and suggest some modifications based on this interpretation which increase the efficiency of the scheme. Finally, we present some numerical experiments which illustrate how well the resulting method works.