Adaptive Mesh Method Research Articles

High-order ALE gas-kinetic scheme with WENO reconstruction

In this paper, a high-order multi-dimensional gas-kinetic scheme is presented for both inviscid and viscous flows in arbitrary Lagrangian-Eulerian (ALE) formulation. Compared with the traditional ALE method, the flow variables are updated in the finite volume framework, and the rezoning and remapping steps are not required. The two-stage fourth-order method is used for the temporal discretization, and the second-order gas-kinetic solver is applied for the flux evaluation. In the two-stage method, the spatial reconstruction is performed at the initial and intermediate stages, and the computational meshes are determined by the mesh velocity. In the moving mesh procedure, the mesh may distort severely and the mesh quality is reduced. To achieve the accuracy and improve the robustness, the newly developed WENO method [40] on quadrilateral meshes is adopted at each stage. The Gaussian quadrature is used for flux calculation. For each Gaussian point, the WENO reconstruction is performed in local moving coordinate, where the variation of mesh velocity along cell interface is taken into account. Numerical examples are presented to validate the performance of current scheme, where the mesh adaptation method and the cell centered Lagrangian method are used to provide mesh velocities.

Understanding fracture of a carbon black filled rubber compound using material force theory

This paper is based on the determination of important fracture parameters for commercially important rubber blends through experimental and finite element modeling methods. Properties of NR/BR blends, which are mainly suitable for tire applications, of various compositions have been tested under tensile load and the result is utilized to determine the suitable hyperelastic material model. A model proposed by Yeoh has been found to be more relevant to the test materials and is used to represent them in simulation. Single Edge Notched Tensile samples (SENT) are ued for the fracture analysis. The samples are tested for uniaxial Mode I fracture (Tensile), which is the most commonly observed among the three modes of fracture, and crack growth was studied using the Virtual Crack Closure Technique (VCCT). A finite-element replica of the test specimen has been created using in-house code, which uses an implicit mesh adaptive method to study the strain potential around the crack tip. The code for each specimen is executed using the commercially available third-party software ANSYS Parametric Design Language (APDL). The stress–strain behaviors from experimental and simulation responses were compared. The energy release rate otherwise known as tearing energy has been derived for both experimental and numerical tests.

Multiresolution analysis as a criterion for effective dynamic mesh adaptation – A case study for Euler equations in the SAMR framework AMROC

Dynamic mesh adaptation methods require suitable refinement indicators. In the absence of a comprehensive error estimation theory, adaptive mesh refinement (AMR) for nonlinear hyperbolic conservation laws, e.g. compressible Euler equations, in practice utilizes mainly heuristic smoothness indicators like combinations of scaled gradient criteria. As an alternative, we describe in detail an easy to implement and computationally inexpensive criterion built on a two-level wavelet transform that applies projection and prediction operators from multiresolution analysis. The core idea is the use of the amplitude of the wavelet coefficients as smoothness indicator, as it can be related to the local regularity of the solution. Implemented within the fully parallelized and structured adaptive mesh refinement (SAMR) software system AMROC (Adaptive Mesh Refinement in Object-oriented C++), the proposed criterion is tested and comprehensively compared to results obtained by applying the scaled gradient approach. A rigorous quantification technique in terms of numerical adaptation error versus used finite volume cells is developed and applied to study typical two- and three-dimensional problems from compressible gas dynamics. It is found that the proposed multiresolution approach is considerably more efficient and also identifies – besides discontinuous shock and contact waves – in particular smooth rarefaction waves and their interaction as well as small-scale disturbances much more reliably. Aside from pathological cases consisting solely of planar shock waves, the majority of realistic cases show reductions in the number of used finite volume cells between 20 to 40%, while the numerical error remains basically unaltered.

Unsteady flow simulation with mesh adaptation

Mesh adaptation is a reliable and effective method to improve the precision of flow simulation with computational fluid dynamics. Mesh refinement is a common technique to simulate steady flows. In order to dynamically optimize the mesh for transient flows, mesh coarsening is also required to be involved in an iterative procedure. In this paper, we propose a robust mesh adaptation method, both refinement and coarsening included. A data structure of [Formula: see text]-way tree is adopted to save and access the parent–children relationship of mesh elements. Local element subdivision is employed to refine mesh, and element mergence is devised to coarsen mesh. The unrefined elements adjacent to a refined element are converted to polyhedrons to eliminate suspending points, which can also prevent refinement diffusing from one refined element to its neighbors. Based on an adaptation detector for vortices recognizing, the mesh adaptation was integrated to simulate the unsteady flow around a tri-wedges. The numerical results show that the mesh zones where vortices located are refined in real time and the vortices are resolved better with mesh adaptation.

Experimental and numerical investigation on wear behavior of carbon-fiber-reinforced carbon matrix composite used in rotary gas seals

This paper presents a numerical method for simulating progressive wear of the carbon-fiber-reinforced carbon matrix composite used in rotary gas seals due to high experimental cost. In this study, the carbon-fiber-reinforced carbon matrix composite wear rates are experimentally measured, and a formula for the carbon-fiber-reinforced carbon matrix composite wear rates and material design parameters (orientation angles and densities) and operating parameters (load and velocities) is fitted as an input to the wear simulation. A finite element incremental wear simulation procedure for the pin-on-disk wear problem is presented by calculating the local nodal wear rate and wear direction and by introducing the Arbitrary Lagrangian Eulerian adaptive meshing method. In the procedure, the relation between the anisotropy of carbon-fiber-reinforced carbon matrix composite and the varied orientation angle is considered. Results show that the calculated wear volume agrees well with the experimental data. And then the numerical methodology is utilized to investigate the effects of time, orientation angle and operating conditions on the disk wear. The developed numerical methodology could be applied to other sliding wear problems in engineering machinery.

The Birth of a Massive First-star Binary

Abstract We study the formation of massive Population III binary stars using a newly developed radiation hydrodynamics code with the adaptive mesh refinement and adaptive ray-tracing methods. We follow the evolution of a typical primordial star-forming cloud obtained from a cosmological hydrodynamics simulation. Several protostars form as a result of disk fragmentation and grow in mass by the gas accretion, which is finally quenched by the radiation feedback from the protostars. Our code enables us, for the first time, to consider the feedback by both the ionizing and dissociating radiation from the multiple protostars, which is essential for self-consistently determining their final masses. At the final step of the simulation, we observe a very wide (≳104 au) binary stellar system consisting of 60 and 70 M ⊙ stars. One of the member stars also has two smaller mass (10 M ⊙) companion stars orbiting at 200 and 800 au, making up a mini-triplet system. Our results suggest that massive binary or multiple systems are common among Population III stars.

Numerical simulation of aerodynamic problems based on adaptive mesh refinement method

Numerical simulation of aerodynamic problems using an algorithm, which identifies shock regions and refines the mesh there using an adaptive mesh refinement method, is performed. The static mesh adaptation method is based on cell partitioning by adding new nodes to cell faces. The method of identification of the shock regions using a pressure and density gradient criterion and its verification are presented. The use and applications of the developed algorithm are illustrated, and supersonic and hypersonic flows with strong shock waves are calculated. The results computed explore the shock-wave structure of the flow, which develops near the nose cone section of a body and determines its aerodynamic performance. The aerodynamic performance of bodies with different nose cone designs, including a body with a spike and a pointed cone with an opposing gas jet injected from the nose section, are evaluated based on the proposed method. The method of static mesh adaptation provides a visually sharper picture of the flow around the body and reduces the numerical error of drag coefficient calculations in comparison with wind tunnel measurements. The results computed on meshes of similar resolution constructed with the adaptation method and automatic mesh generator are compared. The results computed on the adaptive mesh are similar to those computed on the mesh generated automatically, but the adaptive mesh has a smaller number of cells.

Anisotropic mesh quality measures and adaptation for polygonal meshes

Anisotropic mesh quality measures and adaptation are studied for convex polygonal meshes. Three sets of alignment and equidistribution measures are developed, based on least squares fitting, generalized barycentric mappings, and the singular value decomposition, respectively. Numerical tests show that all three sets of mesh quality measures provide good measurements for the quality of polygonal meshes under given anisotropic metrics. Based on the second set of quality measures and using a moving mesh partial differential equation, an anisotropic adaptive polygonal mesh method is constructed for the numerical solution of second-order elliptic equations. Numerical examples are presented to demonstrate the effectiveness of the method.

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.

The application of the moving mesh method in a thermal hydraulic system code

Mesh distribution is an important part of the calculation in many thermal hydraulic system codes. Changing the mesh size is a common way to improve the calculation accuracy and efficiency. Automatic adjustment of the mesh size is an essential feature of adaptive mesh methods, among which the moving mesh method is a typical one. With the moving mesh method, mesh number can stay the same when adjusting the mesh distribution, which effectively benefits the calculation accuracy. In this research, the moving mesh method is added into a simplified one-dimension thermal hydraulic system code. Three numerical simulations are performed to validate the feasibility and validity of the moving mesh method added into the one-dimension thermal hydraulic system code. And an applicable way to selecting the coefficient in the weight function is proposed. The calculation results show that the accuracy in these cases has been improved, without reducing the efficiency.

Moving mesh version of wave propagation algorithm based on augmented Riemann solver

Shallow water equations are a nonlinear system of hyperbolic conservation laws. This system admits discontinuous solutions. In addition, a discontinuous bathymetry can lead to the discontinuous solutions. The accuracy of numerical methods depends on the accuracy of the numerical solutions around these discontinuities.In order to increase the accuracy, the adaptive mesh methods are more cost-effective than the methods based on the global refinement, In terms of computational cost. Moving mesh method is an adaptive mesh method. In this method, an increase of the number of mesh points is not needed for increasing the accuracy and efficiency of numerical solutions. Highly accurate solutions only can be obtained by concentrating points in areas where more accuracy is needed and decreasing the number of points in smooth areas.In this paper, LeVeque wave propagation algorithm is improved by the moving mesh method, for shallow water equations with variable bathymetry. Moreover, augmented Riemann solver is used for solving Riemann problems in each time step. Finally, moving mesh version of wave propagation algorithm based on augmented Riemann solver is proposed. The numerical solutions based on the moving mesh method and fixed mesh method have a significant difference and application of a moving mesh method yields highly accurate solutions around shocks and discontinuities.

Numerical simulation of aircraft wake vortex evolution and wake encounters based on adaptive mesh method

The fast and accurate simulation of aircraft wake vortices evolution and safety assessment of wake encounters are important in identifying the hazard zone of wake vortices, reducing wake separation, and increasing the capacity of airports. However, numerical simulation of wake vortices often takes a lot of time due to a large number of grids. In order to reduce the computation time, Solution-Based Dynamic adaptive mesh method is applied to compute the wake vortex evolution using FLUENT. Three cases with different ambient turbulence intensities are carried out with the large eddy simulation (LES) based on adaptive mesh. The numerical result shows that refiner meshes in the region of the vortex core generated by adaptive mesh method can more effectively capture the dynamics of vortices, identify more secondary vortices and reduce the numerical dissipation due to more compact vortex core resolution. Then, the rolling moment of the following aircraft is calculated and the hazard zone is identified for the aircraft pairing of A340 and A320. The results show that the safety of wake encounters has a close relationship with ambient turbulent intensity and development of vortex instability.

Simple and effective approach to modeling crack propagation in the framework of extended finite element method

In this study, the extended finite element method (XFEM) coupled with the hierarchical mesh adaptation method and direct method is developed to model crack propagation. First, we observe that crack tip stresses can be approximated with high accuracy by using the stress intensity factor (SIF) terms and T-stress term within 0.1% of the crack length away from the crack tip, without considering other high order terms of the crack-tip stresses. Next, based on the XFEM, the hierarchical mesh adaptation method is developed to obtain the stresses around the location which is very close to the crack tip. Then, the SIFs are derived directly from the crack-tip stresses based on the least square fitting without considering the special algorithms and procedures. Finally, the numerical results reveal that the proposed method can obtain SIFs accurately and it can predict a better crack trajectory for crack simulation. In addition, the proposed method has the advantage over the J-integral method and it is benefit for the simulation of large-scale engineering problems.

Numerical study of slug flow heat transfer in microchannels

In the present paper, the heat transfer of slug flow in rectangular microchannels is studied numerically. The Al2O3-water nanofluid with the volume fraction of 1% is used as a homogeneous liquid. Three different configurations are considered to investigate the hydrodynamics and heat transfer in the microchannels. Boundary mesh adaptation method is used to capture the liquid film around gas bubbles. Slug flow characteristics are simulated with a very low computational time compared to previous researches. Slug and bubble lengths, liquid film properties, pressure drop, gas void fraction and heat transfer characteristics are studied. Results show that different inlet configurations impact on slug and bubble lengths. Gas hold-up is influenced by vortices and high frequency fluctuations that are created in the liquid film due to the difference between gas and slug acceleration. It is found that the Nusselt number is an increasing function of slug length and unit cell length. The results reveal that the hottest point on the wall is where the back tail vortex is generated. The nanofluid enhances the heat transfer coefficient by 10% in comparison with the pure water.

Novel quadtree algorithm for adaptive analysis based on cell-based smoothed finite element method

For finite element analysis of structures with irregular geometry, unsatisfactory meshing quality could be the major concern. Adaptive meshing technique is usually applied to modulate the mesh to approach the geometry. The stress solution from triangular elements is understood less accurate than that of quadrilateral elements. Therefore, adaptive technique for quadrilateral elements is desirable for more proficient analysis of complex geometry. This paper develops a novel adaptive technique that combines a newly developed quadtree algorithm with the cell-based smoothed finite element method (CS-FEM) for automatic mesh adaptation using quadrilateral elements. Since the quadtree algorithm generates a grid with different sizes of quadrilaterals, a number of new CS-FEM elements of n-sided polygon are concomitantly constituted. In the adaptation process, the energy error norm and the geometrical feature are applied as the calibrations for the adaptive refinement. The elements on curved boundaries are either cut or merged with the surrounding elements automatically. It is found that the present new quadtree algorithm works very effectively with the CS-FEM formulation in view of stability and convergence. The results by the current S-FEM are found generally more accurate than those of the general finite element method (FEM) counterpart, especially in terms of strain energy solutions.

An adaptive modeling method for multi-throttle aerostatic thrust bearing

For searching higher carrying capacity of aerostatic thrust bearing (ATB), micro-structures such as recesses and grooves have been recently used to solve the mutual tradeoff between carrying capacity and air film stability. In order to obtain the quantitative relationship between the micro-structure and the carrying capacity, this paper puts forward a mesh adaptation method for small size multi-throttle ATB with micro-structure and thin air film. Contrary to previous methods that lessened or even ignored them, the dimensionless thickness y+ pertaining to the depth is transformed and incorporated into each grids. The adaptation procedures employ the y+ as a variable to refine local mesh quantitatively, meeting the requirements of enhanced wall treatment (EWT) for further turbulence calculation by finite volume method (FVM). By mesh adaptation method, 4 parameters of the micro-structure are studied quantitatively to obtain the regulation of their effects on the carrying capacity and discharge coefficient. By means of previous experimental results, the effectiveness of the mesh adaptation method is verified in this paper. In addition, the obtained discharge coefficient from adaptation could be employed to improve traditional analytical method, making it applicable for small size multi-throttle ATB and available to reflect the effects of micro-structure on carrying capacity.

A combined immersed body and adaptive mesh method for simulating neutron transport within complex structures

This article describes a new adaptive immersed body method for the efficient and accurate modelling of neutron transport within geometrically complex domains. It combines two techniques of immersed body projections and self-adaptive mesh refinement to form a unique method that can efficiently resolve problems that contain complex internal structures. The approach allows complete freedom of where mesh resolution is placed and the meshes, which do not need to conform to the problem structure, are optimised for resolving the physics of the problem.Importantly the benefits can be found for problems with many internal structures such as fuel pins, control rods or cooling pipes and channels. Standard finite element meshes typically become large as they capture the geometrical detail and as a result solving times typically increase. This method overcomes this issue and allows for smaller meshes that are optimised to increase solution accuracy, as demonstrated in the numerical examples.

A positivity preserving adaptive moving mesh method for cancer cell invasion models

In this paper, we present an efficient adaptive moving mesh finite difference method for the modeling of early stage cancer cell invasion of tissue or extracellular matrix (ECM). The cancer cell invasion model is a nonlinear system of reaction–diffusion-taxis partial differential equations (PDEs) describing the time evolution of the cancer cells density and concentrations of proteins of the ECM. The solution of the model exhibits very rapid variations at the boundary of the healthy and cancer cells. Thus, using a uniform grid method to solve the cancer cell invasion model requires a very large number of grid points in order to resolve the steep gradients of the solution precisely. As a result, the computation can become prohibitively expensive and inefficient. In this effort, we propose a positivity preserving scheme for the spatial discretization of the model on an adaptive mesh to accurately capture the sharp structures of the solution. The adaptive mesh is generated by a coordinate transformation obtained as the solution of the optimal mass transfer problem. Several numerical experiments are presented to demonstrate the performance of the adaptive mesh method for solving the cancer cell invasion model. The numerical results show that the proposed adaptive mesh method is effective in capturing and predicting the correct behavior of the cancer cells movement within the ECM.

Numerical study on wave phenomena produced by the super high-speed evacuated tube maglev train

With the increasing demand of faster, safer, and more comfortable rail travel, the evacuated tube maglev train (ETMT) has gradually attracted hot concerns in recent years. According to Kantrowitz limit theory, ETMT may encounter chocked flows, leading to more complex generation and development of a series of waves as ETMT moves rapidly in the tube. To study the wave phenomena produced by ETMT running at the super high-speed, the computational domain geometry model was simplified and then 2-D axisymmetric compressible N-S equation was established in the paper. Dynamic mesh method and dynamic adaptive mesh method were used to simulate the real motion of ETMT and improve the capture accuracy of the waves respectively. Results show that the wave structure is mainly composed of expansion waves, reflected shock waves and normal shock waves as ETMT moves for enough time at the speed of 1250 km/h. The intensity of the normal shock wave in front of the head car experiences four states consisting of rapid increase, initial stability, sudden drop and final stability. In the wake, the flow velocity attenuates in a fluctuating way along the opposite motion direction of ETMT due to the complicated interaction between shock waves and expansion waves.

A goal-oriented anisotropic [formula omitted]-mesh adaptation method for linear convection–diffusion–reaction problems

We deal with the numerical solution of linear convection–diffusion–reaction equations using the hp-variant of the discontinuous Galerkin method on triangular grids. We develop a mesh adaptive algorithm which modifies the size and shape of mesh elements and the corresponding polynomial approximation degrees in order to decrease the error in a target functional under the given tolerance. We recall some theoretical results and describe in detail several technical aspects of this approach. Numerical experiments demonstrating the accuracy, efficiency and robustness of the algorithm are presented.