Adaptive Grid Generation Research Articles

Adaptive grid generation for discretizing implicit complexes

We present a method for generating a simplicial (e.g., triangular or tetrahedral) grid to enable adaptive discretization of implicit shapes defined by a vector function. Such shapes, which we call implicit complexes, are generalizations of implicit surfaces and useful for representing non-smooth and non-manifold structures. While adaptive grid generation has been extensively studied for polygonizing implicit surfaces, few methods are designed for implicit complexes. Our method can generate adaptive grids for several implicit complexes, including arrangements of implicit surfaces, CSG shapes, material interfaces, and curve networks. Importantly, our method adapts the grid to the geometry of not only the implicit surfaces but also their lower-dimensional intersections. We demonstrate how our method enables efficient and detail-preserving discretization of non-trivial implicit shapes.

A robust adaptive grid method for first-order nonlinear singularly perturbed Fredholm integro-differential equations

<abstract><p>In this paper, a robust adaptive grid method is developed for solving first-order nonlinear singularly perturbed Fredholm integro-differential equations (SPFIDEs). Firstly such SPFIDEs are discretized by the backward Euler formula for differential part and the composite numerical quadrature rule for integral part. Then both a prior and an a posterior error analysis in the maximum norm are derived. Based on the prior error bound and the mesh equidistribution principle, it is proved that there exists a mesh gives optimal first-order convergence which is robust with respect to the perturbation parameter. Finally, the posterior error bound is used to choose a suitable monitor function and design a corresponding adaptive grid generation algorithm. Numerical results are given to illustrate our theoretical result.</p></abstract>

An adaptive grid method for a singularly perturbed convection-diffusion equation with a discontinuous convection coefficient

<abstract><p>In this paper, an adaptive grid method is put forward to solve a singularly perturbed convection-diffusion problem with a discontinuous convection coefficient. First, this problem is discretized by using an upwind finite difference scheme on an arbitrary nonuniform grid except the fixed jump point. Then, a first-order maximum norm a posterior error estimate is derived. Further, based on this a posteriori error estimation and the mesh equidistribution principle, an adaptive grid generation algorithm is constructed. Finally, some numerical experiments are presented that support our theoretical estimate.</p></abstract>

Fast $L^2$ Optimal Mass Transport via Reduced Basis Methods for the Monge--Ampère Equation

Repeatedly solving the parameterized optimal mass transport (pOMT) problem is a frequent task in applications such as image registration and adaptive grid generation. It is thus critical to develop a highly efficient reduced solver that is equally accurate as the full order model. In this paper, we propose such a machine learning--like method for pOMT by adapting a new reduced basis (RB) technique specifically designed for nonlinear equations, the reduced residual reduced over-collocation (R2-ROC) approach, to the parameterized Monge--Ampère equation. It builds on top of a narrow-stencil finite difference method (FDM), a so-called truth solver, which we propose in this paper for the Monge--Ampère equation with the transport boundary. Together with the R2-ROC approach, it allows us to handle the strong and unique nonlinearity pertaining to the Monge--Ampère equation achieving online efficiency without resorting to any direct approximation of the nonlinearity. Several challenging numerical tests demonstrate the accuracy and high efficiency of our reduced solver for solving the parameterized Monge--Ampère equation, effectively transporting the nontrivial boundaries.

An Optimal Adaptive Grid Method Based on L1 Scheme for a Nonlinear Caputo Fractional Differential Equation

A nonlinear fractional differential equation with a Caputo derivative of order α is studied. This problem is discretized by using the L1 scheme on an arbitrary nonuniform mesh. By utilizing the Taylor expansion with integral remainder term, an optimal local truncation error estimation of L1 scheme is proved. Based on this truncation error estimation and the mesh equidistribution principle, a new monitor function is constructed to construct an adaptive grid generation algorithm. Numerical experiments are performed to confirm the accuracy of our new adaptive grid algorithm.

A Second-Order Adaptive Grid Method for a Singularly Perturbed Volterra Integrodifferential Equation

In this paper, an adaptive grid method for a singularly perturbed Volterra integro-differential equation is studied. Firstly, this problem is discretized by a new second-order finite difference scheme, for which a truncation error analysis is conducted. Then, based on this truncation error bound and the mesh equidistribution principle, we show that there is a mesh that provides an optimal error bound of O(N−2), which is robust with respect to the perturbation parameter. Finally, based on an approximation monitor function, an adaptive grid generation algorithm is constructed and some numerical results are given to support our theoretical results.

Enhanced strategy for adaptive Cartesian grid generation with arbitrarily complex 3D geometry

This article presents an enhanced strategy for generating Cartesian grids for arbitrary complex three-dimensional (3D) geometry. It addresses the major challenges, including data structures managing surface triangles and Cartesian cells, background Cartesian generation, intersection cell determination, and adaptive mesh refinement. To overcome the problem of inefficiency, we developed a robust method for determining intersecting cells based on the k-dimensional tree data structure combined with the fast 3D-Cartesian box-triangle overlap testing algorithm. Furthermore, the minimum distance query algorithm based on the k-dimensional tree was enhanced to improve the query efficiency by reducing the number of backtrackings. The correctness and efficiency of the proposed strategy were evaluated and verified by 3D tests on different configurations. The results show that in terms of mesh generation, the CPUtime∕cell was nearly 5.33–5.54×10−6 s; in terms of the minimum distance query, the query efficiency of the improved backtracking method improved by up to 870.9 times compared with the traversal method. Meshing experiments of several complicated configurations demonstrated that the proposed algorithms were accurate, robust, and efficient. At last, an example was examined, and the result demonstrates that the mesh generation strategy developed in this study meets the actual flow field calculation requirements.

An Adaptive Irregular Grid Generation Method for Scattering Simulation of Electrically Large Terrains

This communication provides a novel adaptive irregular grid model (AIGM) to improve the scattering simulation efficiency of electrically large terrains. This model distinguishes the steep and flat regions of the measured topographic point cloud models by a threshold. To determine the optimal threshold, we present the height gradient as the criterion to describe the topographic undulation. The optimal threshold is located at the inflection point of the curve describing topographic similarity in the range of height gradient. This generated AIGM can reduce the grid quantity and maintain consistency with the original point cloud model. It is also valuable in the popular coherent and incoherent (C&I) scattering model of global navigation satellite system reflectometry (GNSS-R). In this communication, the coherent component can be expressed by a simple product of the electric field of the conductor grid, the characteristic function of roughness, and the Fresnel reflection coefficient. The incoherent component comes from the diffuse scattering simulated by the integral equation method (IEM). Compared with the traditional solution to C&I scattering by using the Kirchhoff approximation (KA), the modified C&I model has a more extensive application. The precision of the AIGM-based-C&I model is verified by traditional physical optics (PO). The excellent simulation efficiency of the AIGM-based-C&I model is also demonstrated.

Simulation of viscoelastic squeeze flows for adhesive joining applications

A backwards-tracking Lagrangian–Eulerian method is used to simulate planar viscoelastic squeeze flow. The momentum and continuity equations are discretized with the finite volume method and implicit immersed boundary conditions are used to describe objects in the domain. The viscoelastic squeeze flow, which involves moving solid geometry as well as free surface flow, is chosen for its relevance in industrial applications, such as adhesive parts assembly and hemming. The main objectives are to validate the numerical method for such flows and to outline the grid resolution dependence of important flow quantities. The main part of the study is performed with the Oldroyd-B model, for which the grid dependence is assessed over a wide range of Weissenberg numbers. An important conclusion is that the load exerted on the solids can be predicted with reasonable accuracy using a relatively coarse grid. Furthermore, the results are found to be in excellent agreement with theoretical predictions as well as in qualitative resemblance with numerical results from the literature. The effects of different viscoelastic properties are further investigated using the PTT model, revealing a strong influence of shear-thinning for moderate Weissenberg numbers. Finally, a reverse squeeze flow is simulated, highlighting important aspects in the context of adhesive joining applications. • Lagrangian–Eulerian used to simulate viscoelastic two-fluid flow. • Viscoelastic squeeze flow and extensional flow with applications for adhesive joining processes. • Viscoelastic flow with moving geometry treated with immersed boundaries and automatic adaptive grid generation. • Wide range of Weissenberg numbers for the Oldroyd-B model as well as the linear and exponential form PTT models.

An adaptive moving grid finite difference method

The grid generation is very crucial for the accuracy of the numerical solution of PDEs, especially for problems with very rapid variations or sharp layers, such as shock waves, wing leading and trailing edges, regions of separation, and boundary layers. The adaptive grid generation is an iterative approach to accommodate these complex structures. In this paper, we introduce a deformation based adaptive grid generation method, in which a differentiable and invertible transformation from computational domain to physical domain is constructed such that the cell volume (Jacobian determinant) of the new grid is equal to a prescribed monitor function. A vector field is obtained by solving the div-curl system and can be used to move the grids to the desired locations. By computing the inverse of Jacobian, any deformed grids can also be transformed back to the uniform grid. Several numerical results in two dimensions are presented. Some applications in image registration are discussed.

An efficient adaptive grid method for a system of singularly perturbed convection-diffusion problems with Robin boundary conditions

A system of singularly perturbed convection-diffusion equations with Robin boundary conditions is considered on the interval [0,1]. It is shown that any solution of such a problem can be expressed to a system of first-order singularly perturbed initial value problem, which is discretized by the backward Euler formula on an arbitrary nonuniform mesh. An a posteriori error estimation in maximum norm is derived to design an adaptive grid generation algorithm. Besides, in order to establish the initial values of the original problems, we construct a nonlinear optimization problem, which is solved by the Nelder–Mead simplex method. Numerical results are given to demonstrate the performance of the presented method.

奇异摄动反应扩散方程的后验误差估计及自适应算法

An adaptive grid method for singularly perturbed semilinear reactiondiffusion equations was studied. The equation was discretized by means of the upwind finite difference scheme on an arbitrary nonuniform mesh. A posteriori error estimation of the presented numerical scheme was built. In turn, the posteriori error bound was derived, and the adaptive grid generation algorithm was designed. Numerical experiments prove the effectiveness of the proposed adaptive grid method.

Analysis of a finite difference scheme for a nonlinear Caputo fractional differential equation on an adaptive grid

<abstract> A nonlinear initial value problem whose differential operator is a Caputo derivative of order $ \alpha $ with $ 0 &lt; \alpha &lt; 1 $ is studied. By using the Riemann-Liouville fractional integral transformation, this problem is reformulated as a Volterra integral equation, which is discretized by using the right rectangle formula. Both a priori and an a posteriori error analysis are conducted. Based on the a priori error bound and mesh equidistribution principle, we show that there exists a nonuniform grid that gives first-order convergent result, which is robust with respect to $ \alpha $. Then an a posteriori error estimation is derived and used to design an adaptive grid generation algorithm. Numerical results complement the theoretical findings. </abstract>

A second-order adaptive grid method for a nonlinear singularly perturbed problem with an integral boundary condition

In this paper, a nonlinear singularly perturbed problem with an integral boundary condition is studied. A hybrid finite difference method based on the midpoint difference scheme and the upwind difference scheme is constructed. An adaptive grid generation algorithm is generated by equidistributing a monitor function. The convergence analysis of the hybrid difference method on the adaptive grid is derived. It is proved by theory and experiments that the scheme is second-order uniformly convergent, which improves previous results.

A posteriori error estimation in maximum norm for a system of singularly perturbed Volterra integro-differential equations

In this paper, a system of singularly perturbed Volterra integro-differential equations is considered. The backward Euler formula is used to discretize the differential part and the right-hand rectangle rule is applied to approximate the integral term. The stabilities of the continuous and discrete solutions are carried out using the Gronwall’s inequality, respectively. The a posterior error bounds are given to design an adaptive grid generation algorithm. Numerical results complement the theoretical results.

Maximum Norm a Posteriori Error Estimation for a System of Singularly Perturbed Semilinear Reaction-Diffusion Equations

In this paper, a system of singularly perturbed semilinear reaction-diffusion equations is discretized on an arbitrary nonuniform mesh using an upwind finite difference scheme. Based on the quadratic polynomial interpolation function, a second-order maximum norm a posteriori error estimation is derived. Then, this bound is used to design an adaptive grid generation algorithm. Numerical results are given to verify the effectiveness of the presented adaptive grid method.<br />

Numerical Investigation of Interaction of Counter Flow Jet and Hypersonic Capsule Flow Via Modified DsmcFoam

Abstract In this work, numerical investigation of interaction of counter flow jet and hypersonic re-entry capsule flow has been carried out through modification of DsmcFoam solver. The DsmcFoam modification includes implementation of variable soft sphere (VSS) collision model for more accurate collision model, adjustment of nonuniform initial condition for faster convergence and nonuniform boundary condition, calculation of local Knudsen number in postprocessing for adaptive grid generation, and implementation of different gas species for multigas flow interaction simulations. Therefore, the modified DsmcFoam can be used for investigation of the effect of counter flow jet on the vehicle aerodynamics and aerothermodynamics. New validation test cases from Von Karman gas dynamics facility (VKF) tunnel data of Apollo and blunt-cone re-entry geometries are studied via DsmcFoam in which a suitable agreement of results is observed compared to experimental data and also MONACO code computations. Also, the influence of counter flow jet has been presented, i.e., changing of bow shock configuration and its distance from the vehicle. Consequently, it is observed that by increasing counter flow jet velocity or density, reduction of the drag coefficient and heat flux on the vehicle will occur. Furthermore, variation of the velocity or density of counter flow jet leads to different jet-flow interaction patterns which are presented evidently.

Rapid Aerodynamic Shape Optimization With Payload Size Constraints for Hypersonic Vehicle

Nowadays, aerodynamic shape optimization of the hypersonic vehicle is mostly optimized for local details such as the wing. In the overall shape optimization, the conceptual design of the vehicle is more common, in which the shape is described in fewer parameters and optimized roughly. Meanwhile, in the optimization, the capacity is considered only by the volumetric ratio, and it can hardly meet the actual payload size constraints, which is difficult to apply to engineer practice. In this paper, a rapid aerodynamic shape optimization method with payload size constraints is proposed. The analytic method combined with a clamped cubic spline curve is used to establish the parametric model with the payload size constraints. In order to ensure the accuracy and improve the optimization efficiency, the aerodynamic characteristics of the vehicle are quickly obtained based on the adaptive Cartesian grid generation method and viscosity modified Euler equations. The lift-to-drag ratio is used as the objective, and a sequence iteration method with multi-point selecting strategy is adopted to make convergence faster. In order to compare and verify the effectiveness of the proposed method, the optimal solution is compared with a reference shape obtained through a conventional multi-objective aerodynamic shape optimization with lift-to-drag ratio and the volumetric ratio as objectives. Through comparative analysis, the method proposed in this paper can find the optimal shape that satisfies the payload size constraints quickly, which can effectively improve the aerodynamic performance of the hypersonic vehicle and can be better applied to practical engineering optimization problems.

Development of information technologies and algorithms for generation of block-structured adaptive grids in complex domains

The paper covers the area of block-structured adaptive grid generation for finite-difference, finite-element and finite-volume methods. It deals with a problem of development of information technologies and algorithms for generating of such grids in complex geometries. Its key idea is to bring together the different existing techniques for computer-aided geometric design of domains, grid generation, and subdivision. The methods used for a construction of domain geometries and working with block-structured quasi-continuous grids are discussed. The results in the generation of three-dimensional adaptive grids for the domain of external flow near a hemisphere, flow around a surface of the high-speed aircraft of the Falcon HTV-2 type and for the domain of the gap between a wheel and a body of an aircraft are analyzed. The presented methods and algorithms for grid generation have the following advantages: 1. The grids are structured and mesh cells are hexahedra in the entire domain. 2. The grids can be applied for finite-difference methods. 3. The grid lines are adjusting with the flow in the whole domain.

Development of adaptive coarse grid generation methods for fast fluid dynamics in simulating indoor airflow

The control of indoor air quality and thermal comfort requires faster-than-real-time simulations of air distribution. One approach to achieve such simulations is to use coarse grid. This study developed two adaptive coarse grid generation methods by analysing the grid-related truncation errors in the solution of the Navier–Stokes equations by fast fluid dynamics (FFD). Method 1 generates a grid that is coarser than that for a grid-independent solution but can still provide accurate FFD predictions. Method 2 generates a very coarse grid that provides acceptable FFD predictions. This study assessed the performance of the developed methods by using them to simulate three typical indoor airflows. The assessment showed that the coarse grid generated by Method 1 accelerated the FFD simulation by at least two to five times, with similar computational accuracy. The coarse grid generated by Method 2 provided faster-than-real-time FFD simulations for the cases tested in this study.