Multi-level Adaptive Mesh Refinement Research Articles

An adaptive mesh refinement strategy for 3D phase modeling of brittle fracture

In order to improve the calculation accuracy and efficiency of the phase field method (PFM), an adaptive mesh refinement strategy suitable for 3D problems is proposed. The refined area is selected in terms of both the phase field value and the phase field increment value. The transition area is then chosen based on the transition element determination algorithm. The corresponding areas are refined using three basic and two advanced refinement templates. The new mesh is made up of hexahedrons and has no hanging nodes. The improved backtracking algorithm based on distance judgment is used in 3D adaptive mesh refinement and has proven its capacity to increase calculation efficiency. The accuracy and applicability of PFM to low-quality meshes with 3D adaptive mesh refinement are verified through benchmark tests, and the errors are all within 1%. Compared with the global refinement mesh, the number of elements in the PF model with 3D adaptive mesh refinement is reduced by more than 80%, and the computational efficiency is increased by 16 times. Combined with virtual crack insertion technology, mode I-II-III fracture is successfully simulated. The accurate and efficient calculation of cracked structures is realized. 3D multi-level adaptive mesh refinement is preliminary developed, and compared with single-level adaptive mesh, the computational efficiency is about two times higher, which provides a potential way to solve the efficient simulation of crack propagation in large-volume structures.

Multi-level adaptive mesh refinement technique for phase-field method

In order to solve the contradiction between the calculation efficiency and accuracy of the phase-field method (PFM), a multi-level adaptive mesh refinement technique suitable for PFM was developed and improved. A multi-level mesh refinement method of 2-D quadrilateral elements including three-layer transition elements without hanging nodes is designed, and the automatic refinement crack tip elements is realized by combining the phase field value. The local element coarsening technique is adopted to solve the problem of rapid elements number growth and reduce elements number in the calculation process. Combined with backtracking algorithm and “tree structure” node information storage technique, crack propagation is simulated. Through the benchmark example, the accuracy of the multi-level adaptive mesh refinement technique and the applicability to non-orthogonal mesh are verified, and the efficiency is greatly improved. For more complex crack propagation, the simulation results are also in good agreement with the experimental results. The critical gradient method proposed at last can find the refinement range more accurately and reduce elements number, which shows a great prospect in engineering applications.

A multi-level adaptive mesh refinement for an integrated finite element/level set formulation to simulate multiphase flows with surface tension

A multi-level adaptive mesh refinement technique for an integrated finite element/level set formulation was implemented to simulate the incompressible interfacial flows with surface tension by using unstructured P2P1 triangular and tetrahedral meshes. Linear/quadratic elements near the interface were dynamically refined as an interface evolves in order to produce an accurate solution with a lower computational cost. An adaptive mesh refinement algorithm for a linear element has been extended to a quadratic element by using a node-reordering. The proposed algorithm was successfully validated by solving several benchmark problems. Simulation of a static bubble problem was found to give an accurate solution with negligible spurious currents compared with existing results. Simulation results of rising bubble problems in 2D and 3D were also compared against existing numerical and experimental results in terms of terminal rising speed and terminal bubble shape. We have found that the present algorithm provided stable and accurate solutions even for a high density-ratio and that the solution accuracy strongly depended on the mesh resolution near the interface. Further, the present algorithm was shown to be very efficient because the computational overhead by mesh-refinement algorithm was very small compared to the elapsed times consumed for solving the incompressible Navier–Stokes equations and the advection/re-initialization equations of level set method. Lastly, the coalescence scenario of two oblique bubbles was well reproduced by the present 3D simulation.

XFEM with hanging nodes for two-phase incompressible flow

This study develops the h-version of the Extended Finite Element Method (XFEM) applied to the simulation of two-fluid incompressible flow in two and three dimensions. A multilevel adaptive mesh refinement realized via hanging nodes on 1-irregular meshes is employed in the vicinity of the two-fluid interface. The sign-enrichment is used for the XFEM approximation which accurately accounts for the jump in the pressure field. The level-set method is used for the implicit representation of the interface. The Laplace–Beltrami technique is employed for the modelling of the surface tension, which avoids the explicit computation of the curvature. An emphasis of this work is on how the interplay between the interface movement (in terms of a time-dependent level-set function), the adaptive refinement and the enriched XFEM approximations, is realized. This study also demonstrates that the approximation of the normal vector to the interface, required for the computation of the surface tension, can have a significant impact on the accuracy of the solver. Several two- and three-dimensional test cases are investigated.

Numerical Studies of Cosmic‐Ray Injection and Acceleration

A numerical scheme that incorporates a injection model into a combined gasdynamics and cosmic-ray (CR) diffusion-convection code has been developed. The hydro/CR code can follow in a very cost-effective way the evolution of CR-modified planar quasi-parallel shocks by adopting subzone shock tracking and multilevel adaptive mesh refinement techniques. An additional conservative quantity, S = Pg/ργg−1, is introduced to follow the adiabatic compression accurately in the precursor region, especially in front of strong, highly modified shocks. The thermal leakage injection model is based on the nonlinear interactions of the suprathermal particles with self-generated MHD waves in quasi-parallel shocks. The particle injection is followed numerically by filtering the diffusive flux of suprathermal particles across the shock to the upstream region according to a velocity-dependent transparency function that controls the fraction of leaking particles. This function is determined by a single parameter, , which should depend on the strength of postshock wave turbulence but is modeled as a constant parameter in our simulations. We have studied CR injection and acceleration efficiencies during the evolution of CR-modified planar shocks for a wide range of initial shock Mach numbers, M0, assuming a Bohm-like diffusion coefficient. For expected values of the injection process is very efficient when the subshock is strong, leading to fast and significant modification of the shock structure. As the CR pressure increases, the subshock weakens and the injection rate decreases accordingly so that the subshock does not disappear. Although some fraction of the particles injected early in the evolution continue to be accelerated to ever higher energies, the postshock CR pressure reaches an approximate time-asymptotic value because of a balance between fresh injection/acceleration and advection/diffusion of the CR particles away from the shock. In the strong shock limit of M0 30, the injection and acceleration processes are largely independent of the initial shock Mach number for a given , while they are sensitively dependent on M0 for M0 < 30. We conclude that the injection rates in strong parallel shocks are sufficient to lead to rapid nonlinear modifications to the shock structures and that self-consistent injection and time-dependent simulations are crucial to understanding the nonlinear evolution of CR-modified shocks.

Diffusive Shock Acceleration with Self-Consistent Injection

A numerical scheme that incorporates a self-consistent cosmic-ray (CR, hereafter) injection model into the combined gas dynamics and CR diffusion-convection code has been developed. The hydro/CR code can follow in a very cos-effective way the evolution of CR modified shocks by adopting subzone shock-tracking and multi-level Adaptive Mesh Refinement techniques. The injection model is based on interactions of the suprathermal particles with self-generated MHD waves in quasi-parallel shocks. The particle injection is followed numerically by filtering the diffusive flux of suprathermal particles across the shock to upstream region according to a velocity-dependent transparency function, which represents the fraction of leaking suprathermal particles. In the strong shock limit of Mach numbers 20, significant physical processes such as the injection and acceleration seem to become independent of M, while they are sensitively dependent on M for M

Time Evolution of Cosmic‐Ray Modified Plane Shocks

We have developed a novel computer code designed to follow the evolution of cosmic-ray modified shocks, including the full momentum dependence of the particles for a realistic diffusion coefficient model. In this form the problem is technically very difficult because one needs to cover a wide range of diffusive scales, beginning with those slightly larger than the physical shock thickness. With most finite difference schemes for Euler's equations, the numerical shock thickness is at least one zone across, so this provides a lower bound on the physical scale for diffusive transport computation. Our code uses subzone shock tracking and multilevel adaptive mesh refinement to provide enhanced spatial resolution around shocks at a modest cost compared to the coarse grid and vastly improved cost effectiveness compared to a uniform, highly refined grid. We present and discuss the implications from our initial results.

A Consistent Approach to Large Eddy Simulation Using Adaptive Mesh Refinement

The large eddy simulation of turbulent flows is discussed with particular attention paid to the issue of commutation of differentiation and filtering. Multi-level adaptive mesh refinement is proposed as a means of mostly avoiding commutation errors where increased grid resolution is required to capture key flow features. The strategy is to employ multiple uniform grids in a nested hierarchy using a constant-width filter for each grid. It is shown that commutivity of fine and coarse grid filters must be enforced in order to consistently relate variables at different refinement levels. Methods for treating fine grid boundaries and walls are also discussed. It is shown that errors associated with boundary treatments are small and localized.