Adaptive Mesh Method Research Articles

A high-order physical-constraints-preserving arbitrary Lagrangian–Eulerian discontinuous Galerkin scheme for one-dimensional compressible multi-material flows

In this paper, a high-order physical-constraints-preserving arbitrary Lagrangian–Eulerian (ALE) discontinuous Galerkin scheme is proposed for one-dimensional compressible multi-material flows. Our scheme couples a conservative equation related to the volume-fraction model with the Euler equations for describing the dynamics of fluid mixture. The mesh velocity in the ALE framework is obtained by using an adaptive mesh method that can automatically concentrate the mesh nodes near the regions with large gradient values and greatly reduce the numerical dissipation near material interfaces. Using this adaptive mesh, the resolution of solution near some special regions such as material interfaces can be improved effectively by our scheme. With the appropriate time step condition and using a bound-preserving and positivity-preserving limiter, our scheme can ensure the positivity of density and pressure and the boundness of volume-fraction, which further ensures the computational robustness and degree of confidence of simulations under large density or pressure ratios and so on. In general, our scheme can be applied to the simulations of compressible multi-material flows efficiently with the essentially non-oscillatory property and physical-constraints-preserving (bound-preserving and positivity-preserving) property, and its steps are more concise compared to some other methods such as the indirect ALE methods. Some examples are tested to demonstrate the accuracy, essentially non-oscillatory property and physical-constraints-preserving property of our scheme.

A flux-based moving mesh method applied to solving the Poisson–Nernst–Planck equations

The moving mesh method is one of the important adaptive mesh methods which is practically useful when the mesh size and overall resolution are provided and fixed as seen in many engineering computations. We employ a moving mesh technique combined with a finite element method (FEM) to solve a widely-used charged carrier transport model, the Poisson–Nernst–Planck (PNP) equations, the solutions of which often have boundary or internal layers and sharp interfaces when the convection is dominated. Considering that flux is a significant physical quantity in designing stable and accurate numerical algorithm for transport problems such as the PNP system, we start from a relatively general functional and propose a flux-based monitor function and an adaptive step size-controlling algorithm for guiding the mesh movement in the FEM solution of the PNP equations. The numerical results illustrate that the moving mesh method is effectively addressing challenges arising from solution singularities and the convection-dominated effects. It also shows that the moving mesh finite element method we propose exhibits superior performance compared to both the traditional moving mesh finite element method and fixed mesh finite element method in some scenarios. Furthermore, it demonstrates better adherence to the physical properties of energy dissipation inherent in the PNP equation.

Application of Delaunay adaptive mesh refinement in flood risk assessment of multi-bridge system with short distance

To ensure bridge safety, the flood risk analysis is significant important. However, due to the small size and large number of piers in the short-distance multi-bridge system, the extremely long calculation time and low efficiency of the numerical model are induced by the small mesh size and large mesh number. In this paper, a flood risk assessment model of the multi-bridge system with short distance was established to improve the calculation efficiency based on the finite volume method combined with the Delaunay mesh adaptive refinement method. The calculated water level with refined and non-refined mesh was compared with the experimental data of a partial failure dam break test case and Shukry experiment of open channel bend flow. The calculated water level results are in good agreement with the experimental data. In addition, the mesh refinement model improved the calculation efficiency by more than 73% with ensuring the calculation accuracy. Finally, the flood risk of a real multi-bridge system with short distance was evaluated by using the numerical model. The calculated results shown that, different from the general flow law, the water level in the upstream and downstream channel of Bridge 2 rose with a maximum difference value of 0.326 m while the water level in the far downstream channel of Bridge 2 dropped result from the construction of Bridge 2 on the basis of the Bridges 1, 3 and 4. The construction of Bridge 2 also increased the flow velocity around Bridge 3 with maximum 0.013 m/s. This study provides a new tool and technical reference for flood risk analyses of similar multi-bridge system with short distance.

An Adaptive Moving Mesh Method for Simulating Finite-time Blowup Solutions of the Landau-Lifshitz-Gilbert Equation

An Adaptive Moving Mesh Method for Simulating Finite-time Blowup Solutions of the Landau-Lifshitz-Gilbert Equation

An efficient and robust staggered scheme based on adaptive time field for phase field fracture model

The standard staggered scheme in phase field model is computationally slow, so that a large amount of iterations are required. In order to improve the computational efficiency, a novel efficient and robust solution scheme based on adaptive time field is proposed in this paper. This scheme is inspired by the adaptive mesh method for space field, and an error indicator based on phase field and history field variables is established, where the time field is coupled with the phase field and energy field in the space field, so that the time field changes adaptively with the computational iterative process, which improves the convergence rate and reduces the time cost. The numerical results show that the presented novel scheme not only significantly improves the computational efficiency, but also has accurate computational results. Compared with the standard staggered scheme, the time cost of the presented scheme is less than 4% of that of the standard scheme. At the same time, the computational efficiency of the presented solver is faster than the current monolithic BFGS scheme, and it has excellent convergence characteristics.

Anisotropic mesh adaptation for high-order finite elements spaces with the log-simplex method. Application to discontinuous Galerkin methods

In this article, a high-order solution-based mesh adaptation method is investigated. This later, which is called the log-simplex method, relies on the approximation of high-order differential form of the solution by a dedicated metric-field. The method is a natural extension of Hessian-based methods to the high-order case.

Adaptive mesh methods on compact manifolds via Optimal Transport and Optimal Information Transport

Moving mesh methods were devised to redistribute a mesh in a smooth way, while keeping the number of vertices of the mesh and their connectivity unchanged. A fruitful theoretical point-of-view is to take such moving mesh methods and think of them as an application of the diffeomorphic density matching problem. Given two probability measures μ0 and μ1, the diffeomorphic density matching problem consists of finding a diffeomorphic pushforward map T such that T#μ0=μ1. Moving mesh methods are seen to be an instance of the diffeomorphic density matching problem by treating the probability density as the local density of nodes in the mesh. It is preferable that the restructuring of the mesh be done in a smooth way that avoids tangling the connections between nodes, which would lead to numerical instability when the mesh is used in computational applications. This then suggests that a diffeomorphic map T is desirable to avoid tangling. The first tool employed to solve the moving mesh problem between source and target probability densities on the sphere was Optimal Transport (OT). Recently Optimal Information Transport (OIT) was rigorously derived and developed allowing for the computation of a diffeomorphic mapping by simply solving a Poisson equation. Not only is the equation simpler to solve numerically in OIT, but with Optimal Transport there is no guarantee that the mapping between probability density functions defines a diffeomorphism for general 2D compact manifolds.In this manuscript, we perform a side-by-side comparison of using Optimal Transport and Optimal Information Transport on the sphere for adaptive mesh problems. We choose to perform this comparison with recently developed provably convergent solvers, but these are, of course, not the only numerical methods that may be used. We believe that Optimal Information Transport is preferable in computations due to the fact that the partial differential equation (PDE) solve step is simply a Poisson equation. For more general surfaces M, we show how the Optimal Transport and Optimal Information Transport problems can be reduced to solving on the sphere, provided that there exists a diffeomorphic mapping Φ:M→S2. This implies that the Optimal Transport problem on M with a special cost function can be solved with regularity guarantees, while computations for the problem are performed on the unit sphere.

Optimization of vane pump structure based on modal characteristic analysis

In order to ensure the good stability and reliability of the vane pump, the modal characteristics of the mechanical structure were studied and analyzed through finite element simulation and impact modal testing. Based on the stiffness criterion, an optimization plan was proposed for the structure of the vane pump without reducing the natural frequency. Adopting tetrahedral adaptive mesh method, the mesh quality of the modal simulation model was verified to ensure the accuracy and efficiency of the calculation. In modal testing, force sensors and acceleration sensors were arranged reasonably. The excitation signal would be transmitted to the integrated vibration testing system for processing, and modal analysis and processing would be carried out through LMS Test Lab. Based on curve fitting, the spectrum and modal parameters of the tested structure could be obtained. The influence of different curvature radius of cover plates on natural frequencies were studied, and the results show that increasing the curvature radius within a certain range can enhance stiffness and improve processability.

Investigation on close-coupled gas atomization for Fe-based amorphous powder production via simulation and industrial trials: Part I. Melt breakup behaviors during primary atomization

The primary atomization process is simulated by Volume of Fluid (VOF) model and dynamic adaptive mesh method. The influence of melt mass flow rate and atomization pressure on the breakup process is investigated, and the effect of hot gas atomization is also evaluated. The results show that the breakup is insufficient when the melt mass flow rate is larger than 0.075 kg s−1, while the liquid film breakup inducing the nozzle clogging occurs when the melt mass flow rate is too low (0.025 kg s−1). The backflow of droplets occurs at low atomization pressure (1.0 MPa), and various defects (satellite powders, hollow powders and needle-shaped powders) appear when the atomization pressure is larger than 3.0 MPa. Although the breakup efficiency can be significantly improved by increasing the gas temperature, severe deformation of the gas-liquid interface is induced due to the Kelvin–Helmholtz wave and gas-liquid interaction, easily leading to the formation of hollow powders. Besides, the gas-to-melt ratio (GMR) is identified as a simple criterion for predicting primary atomization breakup modes, with liquid film breakup occurring when GMR≥4.4 and "fountain" breakup occurring when GMR≤4.3. In this work, not only the gas-liquid interaction is systematically analyzed by establishing a flow-heat transfer-VOF coupling model, but also the GMR is proposed to predict the breakup mode in the industrial production, which can provide theoretical and methodological guidance for the optimization of atomization operational parameters.

Low Order Modeling of a Biomass Throated Gasification Reactor

As first step towards the consideration of the associated control-estimator design problem, in this study the problem of drawing a low-order finite-dimensional DA model of the bi/tristable OPDE one of a spatially distributed throated biomass gasification reactors is addressed, with focus on the description of the practical steady state (SS) of highest conversion. The reactor is described by 15 nonlinear PDEs, which are numerically solved using finite differences to discretize the spatial domain. The number and placement of nodes along the axial direction of the reactor act as degrees of freedom that must be carefully chosen. In this study, a node location method, initially developed for a different class of reactors, is applied to the throated reactor. It was found that the desired SS can be effectively described by relocating some nodes using the adaptive mesh method, which results in a model order smaller than the case with an uniform node distribution. It was also observed that one must be mindful of the discretization near the air entrance, as excessive discretization in that zone can lead to poor descriptions of the desired SS or increase unnecessarily the model order (and consequently, the number of equations to be solved online). The results presented here lay the groundwork for improving the adaptive mesh location method to extend its application to throated gasifiers, thereby ensuring accurate numerical solutions for control and estimation tasks.

Inverse radiative transfer with goal-oriented hp-adaptive mesh refinement: adaptive-mesh inversion

The inverse problem for radiative transfer is important in many applications, such as optical tomography and remote sensing. Major challenges include large memory requirements and computational expense, which arise from high-dimensionality and the need for iterations in solving the inverse problem. Here, to alleviate these issues, we propose adaptive-mesh inversion: a goal-oriented hp-adaptive mesh refinement method for solving inverse radiative transfer problems. One novel aspect here is that the two optimizations (one for inversion, and one for mesh adaptivity) are treated simultaneously and blended together. By exploiting the connection between duality-based mesh adaptivity and adjoint-based inversion techniques, we propose a goal-oriented error estimator, which is cheap to compute, and can efficiently guide the mesh-refinement to numerically solve the inverse problem. We use discontinuous Galerkin spectral element methods to discretize the forward and the adjoint problems. Then, based on the goal-oriented error estimator, we propose an hp-adaptive algorithm to refine the meshes. Numerical experiments are presented at the end and show convergence speed-up and reduced memory occupation by the goal-oriented mesh adaptive method.

An Improved Visual SLAM Method with Adaptive Feature Extraction

The feature point method is the mainstream method to accomplish inter-frame estimation in visual Simultaneous Localization and Mapping (SLAM) methods, among which the Oriented FAST and Rotated BRIEF (ORB) feature-based method provides an equilibrium of accuracy as well as efficiency. However, the ORB algorithm is prone to clustering phenomena, and its unequal distribution of extracted feature points is not conducive to the subsequent camera tracking. To solve the above problems, this paper suggests an adaptive feature extraction algorithm that first constructs multiple-scale images using an adaptive Gaussian pyramid algorithm, calculates adaptive thresholds, and uses an adaptive meshing method for regional feature point detection to adapt to different scenes. The method uses Adaptive and Generic Accelerated Segment Test (AGAST) to speed up feature detection and the non-maximum suppression method to filter feature points. The feature points are then divided equally by a quadtree technique, and the orientation of those points is determined by an intensity centroid approach. Experiments were conducted on publicly available datasets, and the outcomes demonstrate the algorithm has good adaptivity and solves the problem of a large number of corner point clusters that may result from using manually set detection thresholds. The RMSE of the absolute trajectory error of SLAM applying this method on four sequences of TUM RGB-D datasets is decreased by 13.88% when compared with ORB-SLAM3. It is demonstrated that the algorithm provides high-quality feature points for subsequent image alignment, and the application to SLAM improves the reliability and accuracy of SLAM.

An adaptive hybrid grids finite-element approach for plane wave three-dimensional electromagnetic modeling

In unstructured finite element (FE) plane wave electromagnetic modeling, the goal-oriented mesh adaptive method based on posterior errors is often used to optimize the mesh to a state where a high-precision numerical solution can be obtained with fewer elements. However, for high-frequency data, this method tends to generate a large number of small elements in the shallow near-surface area, which increases the computational burden of FE solution. To solve this problem, we developed an improved goal-oriented hybrid mesh adaptive method. This method realizes the coupling of prismatic elements and tetrahedral elements in FE analysis and can establish FE system equations without suspension nodes. Numerical example studies illustrate how the proposed method uses triangular prism elements to divide the near-surface area and greatly improves the computational efficiency of plane wave electromagnetic modeling. For high-frequency modeling, the proposed hybrid grid reduces the number of required elements, improves the numerical accuracy, reduces the iterations of adaptive grids refinement, and improves the efficiency of the adaptive numerical solution. For low-frequency forward calculations, the hybrid grid maintains the same solution efficiency as traditional adaptive numerical simulation based on pure tetrahedrons.

Prediction of the End-Bearing Capacity of Axially Loaded Piles in Saturated and Unsaturated Soils Based on the Stress Characteristics Method

This study proposed an analytical approach for predicting the end-bearing capacity of driven piles that are subjected to axial loads in saturated and unsaturated soils. This is a generalized approach in which the stress characteristic method is employed successfully for both saturated and unsaturated soils. An iterative technique computer code was developed for the proposed analytical approach extending the finite difference method to develop solutions with the aid of MATLAB (version 2019a) that provides graphical output to visualize the results. The results from the proposed approach were compared against measurements for 13 pile load tests that include 11 in saturated soils and 2 in unsaturated soils, with good agreement. In addition, numerical analyses were performed using ABAQUS (version 6.14) to simulate the driven pile penetration and pile loading by employing the arbitrary Lagrangian–Eulerian adaptive mesh methods. The comparisons between the numerical predictions and measurements from a published model pile test suggested that the ultimate bearing capacity is well predicted by the finite-element model in comparison to the proposed analytical method. However, the proposed analytical method was simple for use in engineering practice applications to estimate the end-bearing capacity of pile foundations in both saturated and unsaturated soils.

Free vibration disturbance and local mesh refinement induced by microcrack damage in circularly curved beams

PurposeThis study aimed to solve the engineering problem of free vibration disturbance and local mesh refinement induced by microcrack damage in circularly curved beams. The accurate identification of the crack damage depth, number and location depends on high-precision frequency and vibration mode solutions; therefore, it is critical to obtain these reliable solutions. The high-precision finite element method for the free vibration of cracked beams needs to be developed to grasp and control error information in the conventional solutions and the non-uniform mesh generation near the cracks. Moreover, the influence of multi-crack damage on the natural frequency and vibration mode of a circularly curved beam needs to be detected.Design/methodology/approachA scheme for cross-sectional damage defects in a circularly curved beam was established to simulate the depth, location and the number of multiple cracks by implementing cross-section reduction induced by microcrack damage. In addition, the h-version finite element mesh adaptive analysis method of the Timoshenko beam was developed. The superconvergent solution of the vibration mode of the cracked curved beam was obtained using the superconvergent patch recovery displacement method to determine the finite element solution. The superconvergent solution of the frequency was obtained by computing the Rayleigh quotient. The superconvergent solution of the eigenfunction was used to estimate the error of the finite element solution in the energy norm. The mesh was then subdivided to generate an improved mesh based on the error. Accordingly, the final optimised meshes and high-precision solution of natural frequency and mode shape satisfying the preset error tolerance can be obtained. Lastly, the disturbance behaviour of multi-crack damage on the vibration mode of a circularly curved beam was also studied.FindingsNumerical results of the free vibration and damage disturbance of cracked curved beams with cracks were obtained. The influences of crack damage depth, crack damage number and crack damage distribution on the natural frequency and mode of vibration of a circularly curved beam were quantitatively analysed. Numerical examples indicate that the vibration mode and frequency of the beam would be disturbed in the region close to the crack damage, and a greater crack depth translates to a larger frequency change. For multi-crack beams, the number and distribution of cracks also affect the vibration mode and natural frequency. The adaptive method can use a relatively dense mesh near the crack to adapt to the change in the vibration mode near the crack, thus verifying the efficacy, accuracy and reliability of the method.Originality/valueThe proposed combination of methodologies provides an extremely robust approach for free vibration of beams with cracks. The non-uniform mesh refinement in the adaptive method can adapt to changes in the vibration mode caused by crack damage. Moreover, the proposed method can adaptively divide a relatively fine mesh at the crack, which is applied to investigating free vibration under various curved beam angles and crack damage distribution conditions. The proposed method can be extended to crack damage detection of 2D plate and shell structures and three-dimensional structures with cracks.

A Chebyshev multidomain adaptive mesh method for reaction-diffusion equations

Reaction-diffusion equations can present solutions in the form of traveling waves with stiff wave fronts. Such solutions evolve in different spatial and temporal scales, and it is desired to construct numerical methods that can adopt spatial refinement at locations where the solution becomes stiff. In this work, we develop a high-order adaptive mesh method based on Chebyshev spectral methods with a multidomain approach for the traveling wave solutions of reaction-diffusion equations. The proposed method uses the non-conforming and non-overlapping spectral multidomain method with the temporal adaptation of the computational mesh. The spectral multidomain methods have been used for solving PDEs including the reaction-diffusion equations. However, the non-conformal, non-overlapping and adaptive mesh spectral methods have not been used in the reaction-diffusion community. The proposed multidomain spectral method solves the given reaction-diffusion equations in each subdomain locally first and the boundary and interface conditions are enforced in a global manner. In this way, the method can be parallelizable and is efficient for large reaction-diffusion systems. We show that the proposed method is stable and accurate for solving reaction-diffusion equations with stiff traveling wave solutions. We provide both one- and two-dimensional numerical results that show the efficacy of the proposed method. The application of the adaptive multidomain spectral methods to the reaction-diffusion equations, that yield stiff traveling wave solutions, is new and needs further investigation.

A numerical study on the interaction of droplet collisions and air flow impact in cross-impinging spray

The spray impingement method has been widely used to effectively achieve better atomization of liquid fuel. The factors influencing the formation of impinged spray are analysed using computational fluid dynamics (CFD) modelling. Free spray and impinging sprays with three impinging distances are simulated using the Eulerian-Lagrangian multiphase method. The results are validated with experimental data. Meanwhile, some advanced methods are employed to obtain better results of simulation such as the AMR technique and adaptive collision mesh method. The spray widths in different impingement distances are compared to analyse the distribution of spray droplets. Upon the binary collision theory, the distribution of discrete droplets is attributed to the collisions of huge amounts of particles. In the process of spray impingement, the impact of air flow, generated by gas entrainment of spray, also influences the spray. As a supplement to the collisions of discrete droplets, the impingement of continuous air flow and their interaction is mainly discussed. The impact waves observed in the flow field demonstrate that the impingement of entrained air flow is similar to jet impingement.

An Anisotropic hp-mesh Adaptation Method for Time-Dependent Problems Based on Interpolation Error Control

We propose an efficient mesh adaptive method for the numerical solution of time-dependent partial differential equations considered in the fixed space-time cylinder varOmega times (0,T). We employ the space-time discontinuous Galerkin method which enables us to use different meshes at different time levels in a natural way. The mesh adaptive algorithm is based on control of the interpolation error in the L^infty (0,T; L^q(varOmega ))-norm. The goal is to construct a sequence of conforming triangular meshes in such a way that the interpolation error bound is under a given tolerance and the number of degrees of freedom is minimal. The resulting grids consist of anisotropic mesh elements with varying polynomial approximation degrees with respect to space. We present a theoretical framework of this approach as well as several numerical examples demonstrating the accuracy, efficiency, and applicability of the method.

Adaptive mesh refinement for finite element analysis of elastic buckling disturbance of circularly curved beams due to multiple micro-cracks damage

PurposeThis study aimed to overcome the challenging issues involved in providing high-precision eigensolutions. The accurate prediction of the buckling load bearing capacity under different crack damage locations, sizes and numbers, and analysing the influence mechanism of crack damage on buckling instability have become the needs of theoretical research and engineering practice. Accordingly, a finite element method was developed and applied to solve the elastic buckling load and buckling mode of curved beams with crack damage. However, the accuracy of the solution depends on the quality of mesh, and the solution inevitably introduces errors due to mesh. Therefore, the adaptive mesh refinement method can effectively optimise the mesh distribution and obtain high-precision solutions.Design/methodology/approachFor the elastic buckling of circular curved beams with cracks, the section damage defect analogy scheme of a circular arc curved beam crack was established to simulate the crack size (depth), position and number. The h-version finite element mesh adaptive analysis method of the variable section Euler–Bernoulli beam was introduced to solve the elastic buckling problem of circular arc curved beams with crack damage. The optimised mesh and high-precision buckling load and buckling mode solutions satisfying the preset error tolerance were obtained.FindingsThe results of testing typical examples show that (1) the established section damage defect analogy scheme of circular arc curved beam crack can effectively realise the simulation of crack size (depth), position and number. The solution strictly satisfies the preset error tolerance; (2) the non-uniform mesh refinement in the algorithm can be adapted to solve the arbitrary order frequencies and modes of cracked cylindrical shells under the conditions of different ring wave numbers, crack positions and crack depths; and (3) the change in the buckling mode caused by crack damage is applicable to the study of elastic buckling under various curved beam angles and crack damage distribution conditions.Originality/valueThis study can provide a novel strategy for the adaptive mesh refinement for finite element analysis of elastic buckling of circular arc curved beams with crack damage. The adaptive mesh refinement method established in this study is fundamentally different from the conventional finite element method which employs the user experience to densify the meshes near the crack. It can automatically and flexibly generate a set of optimised local meshes by iteratively dividing the fine mesh near the crack, which can ensure the high accuracy of the buckling loads and modes. The micro-crack in curved beams is also characterised by weakening the cross-sectional stiffness to realise the characterisation of locations, depths and distributions of multiple crack damage, which can effectively analyse the disturbance behaviour of different forms of micro-cracks on the dynamic behaviour of beams.

An arbitrary Lagrangian–Eulerian discontinuous Galerkin method for two‐dimensional compressible flows on adaptive quadrilateral meshes

AbstractIn this article, a discontinuous Galerkin (DG) scheme on the adaptive quadrilateral meshes is proposed to simulate two‐dimensional compressible flows in the direct arbitrary Lagrangian–Eulerian (ALE) framework. In our scheme, the Euler equations are discretized in the reference element with the help of a bilinear map. A kind of Taylor expansion basis functions in the reference element is used to construct the interpolation polynomials of variables. We describe the property that the material derivatives of the basis functions used in the DG discretization are equal to zero, with which the scheme is simplified. Furthermore, the mesh velocity in our ALE framework is obtained by implementing the approach of mesh movement based on the variational principle from [Adaptive mesh methods for one‐ and two‐dimensional hyperbolic conservation laws. SIAM J Numer Anal. 2003;41:487–515]. This approach of mesh movement automatically concentrates the mesh nodes near the regions with large gradient values of the variables and can greatly improve the resolution of the solution near these regions. In addition, a WENO (weighted essentially non‐oscillatory) reconstruction helps our scheme remove the numerical oscillations. Some numerical examples are presented to demonstrate the accuracy, high resolution, and robustness of our scheme.