Adaptive Mesh Generation Research Articles

An adaptive mesh generation and higher-order difference approximation for the system of singularly perturbed reaction–diffusion problems

The paper presents a hybrid difference method to solve a coupled system of singularly perturbed reaction–diffusion problems over an adaptive mesh. The solution to the problem manifests a distinctive multi-scale nature, characterized by localized narrow regions where the solution undergoes exponential changes. The mesh-generating procedure relies upon equidistributing a non-negative monitor function. The method utilizes a suitable combination of the cubic and exponential spline difference schemes over a layer adaptive mesh. The proposed method is unconditionally stable and uniformly convergent. The convergence obtained is optimal, being free from logarithmic factors. Moreover, it is free from directional bias. Numerical experiments have been performed and presented for two model problems. The results obtained are in agreement with the theoretical estimates.

Efficient and consistent adaptive mesh generation for geophysical models: A case study over the Gulf of Thailand

Geophysical domains typically exhibit intricate, irregular boundaries characterized by fractal-like geometries, while underlying physical processes operate across a broad spectrum of spatial scales. The challenge lies in generating spatial discretization of these domains that conform to their geographical constraints, utilizing anisotropic, fully adaptive meshes. This problem is compounded by the vast range of scales and a notably heterogeneous parameter space. Current methodologies often rely on ad hoc, model-specific, or application-dependent approaches, which lack comprehensive descriptions. Consequently, the development of new spatial domains is labor-intensive, prone to errors, challenging to replicate, and difficult to maintain consistency due to substantial human involvement. This predicament poses obstacles to the reproducibility of simulations and the establishment of provenance in data handling and model initialization, and it hinders rigorous model intercomparisons. Furthermore, the likelihood of discrepancies in model initialization and forcing parameters increases when employing flexible adaptive meshes. This paper introduces a systematic approach to the automated generation of adaptive meshes for geophysical models. This method is efficient in its generation process and readily reproducible, offering robust and consistent adherence to the source data. The proposed approach facilitates research in complex multi-scale geophysical domains, which would be challenging using existing methods. A simulation of monthly mean currents was carried out as a case study in the Gulf of Thailand. Results revealed that the simulated current circulations agreed with the observation. Examples of its application in various ongoing geophysical modeling endeavors illustrate its effectiveness.

HybridOctree_Hex: Hybrid octree-based adaptive all-hexahedral mesh generation with Jacobian control

We present a new software package, “HybridOctree_Hex,” for adaptive all-hexahedral mesh generation based on hybrid octree and quality improvement with Jacobian control. The proposed HybridOctree_Hex begins by detecting curvatures and narrow regions of the input boundary to identify key surface features and initialize an octree structure. Subsequently, a strongly balanced octree is constructed using the balancing and pairing rules. Inspired by our earlier preliminary hybrid octree-based work, templates are designed to guarantee an all-hexahedral dual mesh generation directly from the strongly balanced octree. With these pre-defined templates, the sophisticated hybrid octree construction step is skipped to achieve an efficient implementation. After that, elements outside and around the boundary are removed to create a core mesh. The boundary points of the core mesh are connected to their corresponding closest points on the surface to fill the buffer zone and build the final mesh. Coupled with smart Laplacian smoothing, HybridOctree_Hex takes advantage of a delicate optimization-based quality improvement method considering geometric fitting, Jacobian and scaled Jacobian, to achieve a minimum scaled Jacobian that is higher than 0.5. We empirically verify the robustness and efficiency of our method by running the HybridOctree_Hex software on dozens of complex 3D models without any manual intervention or parameter adjustment. We provide the HybridOctree_Hex source code, along with comprehensive results encompassing the input and output files and statistical data in the following repository: https://github.com/CMU-CBML/HybridOctree_Hex.

An Implicit Adaptive FDTD Mesh Generation Techniquebased on Tetrahedrons

A novel implicit adaptive FDTD mesh generation method based on tetrahedrons is proposed in this paper. According to the vertex coordinates of tetrahedrons which make up an object, non-uniform grid lines are generated first. These grid lines are constrained by the structure of the object and follow three rules mentioned in the paper. The first rule is to find demarcation points of the object and drop grid lines on these points. The second rule is to make sure all mesh sizes are less than one-tenth of the wavelength by adding more grid lines. The last rule is to densify mesh at the fine structure of the object. Then by comparing the positional relationship between center points of Yee cells and tetrahedrons, the object can be discretized by Yee cells. Finally, numerical examples are given to verify the validity and accuracy of this novel method.

3D magnetic flux density measurement with reduced sampling and high accuracy using visual localization and adaptive mesh generation

3D magnetic flux density measurement with reduced sampling and high accuracy using visual localization and adaptive mesh generation

Adaptive Isogeometric Analysis using optimal transport and their fast solvers

In this work, we devise fast solvers and adaptive mesh generation procedures based on the Monge–Ampère Equation using B-Splines Finite Elements, within the Isogeometric Analysis framework. Our approach ensures that the constructed mapping is a bijection, which is a major challenge in Isogeometric Analysis. First, we use standard B-Splines Finite Elements to solve the Monge–Ampère Equation. An analysis of this approach shows serious limitations when dealing with high variations near the boundary. In order to solve this problem, a new formulation is derived using compatible B-Splines discretization based on a discrete DeRham sequence. A new fast solver is devised in this case using the Fast Diagonalization method. Different tests are provided and show the performance of our new approach.

A long short-term memory neural network-based error estimator for three-dimensional dynamically adaptive mesh generation

Adaptive meshes are pivotal in numerical modeling and simulation, offering a means to efficiently, precisely, and flexibly represent intricate physical phenomena, particularly when grappling with their intricacies and varying scales. However, the transition from two dimensions (2D) to three dimensions (3D) poses a substantial challenge, as the computational demands of dynamically adaptive mesh techniques increase exponentially. Addressing this challenge effectively, we turn to the cutting-edge realm of artificial intelligence and neural networks. In our study, we harness the innovative power of a long short-term memory (LSTM) neural network as an error estimator for adapting unstructured meshes in both 2D and 3D scenarios. This LSTM network predicts the evolution of the adaptive grid based on specified variables, presenting itself as an artificial intelligence-driven architecture to optimize the adaptive criterion for the target variable. This is achieved by establishing a direct correspondence between the Riemann metric and these variables. To demonstrate the practical applicability of our approach, we seamlessly integrate the LSTM error estimator into the 3D adaptive atmospheric model Fluidity-Atmosphere (Fluidity-Atmos), thereby enabling real-time mesh adaptation during numerical simulations. We assess the effectiveness of this method in terms of simulation precision and computational efficiency through a series of experiments in both 2D and 3D settings. Our results not only reveal that the mesh patterns generated by the LSTM error estimator within Fluidity-Atmos closely resemble those produced by traditional error estimators but also underscore its superior performance in enhancing simulation accuracy. Notably, as the number of nodes increases, the LSTM mesh generator substantially reduces CPU time requirements by up to 50% in 3D cases compared to the conventional mesh generator within Fluidity-Atmos, highlighting its remarkable computational efficiency.

3D Airborne EM Forward Modeling Based on Finite-Element Method with Goal-Oriented Adaptive Octree Mesh

The finite-element (FE) method for three-dimensional (3D) airborne electromagnetic (AEM) modeling can flexibly simulate complex geological structures at high accuracy. However, it has low efficiency and high computational requirements. To solve these problems, one needs to generate meshes more reasonably. In view of this, we develop an adaptive octree meshing scheme for frequency-domain AEM modeling. The octree meshes have the characteristics of regularity and flexibility, while the adaptive algorithm can effectively refine the mesh locally. In our adaptive mesh generation, the posterior errors and weighted coefficients are used to construct the final weighted posterior errors. We verify the accuracy of our method by comparing its results with semi-analytical solutions for a half-space model. Furthermore, we use the spectral-element (SE) method and our method to calculate EM responses for an abnormal block model and compare their computational costs. The results show that our adaptive scheme has obviously technical advantages over SE method for AEM modeling with multiple frequencies and multiple survey stations. Finally, we calculate a model with complex geological structures to verify the feasibility of our algorithm in complex geological circumstances.

Adaptive Mesh Generation Technique for Efficient Electromagnetic Computation in RFIC Designs

A novel adaptive mesh generation technique for efficient electromagnetic simulation of radio-frequency integrated circuits (RFICs) is herein presented. By exploring the geometrical and physical characteristics of RFICs, some adaptive mesh treatments, such as mesh projection, edge refinement, via polymerization, etc., are utilized to improve the accuracy and efficiency of electromagnetic computations. For strong coupling structures, such as two conductors in close proximity for a relatively large area, a projection-based mesh scheme is introduced to improve the accuracy of numerical integration. Moreover, the current most likely concentrates near the edges of conductors due to the edge effect. To better model the edge effect, an edge refinement scheme is applied. For via arrays that appear common in RFICs, an automatic via aggregation approach is adopted to improve computational efficiency yet still keep good computational accuracy. Finally, some numerical examples are given to validate the computational accuracy and efficiency of the novel adaptive mesh generation technique.

Shape Parameterization Optimization of Thermocouples Used in Aeroengines

Aiming at the problem that thermocouples used in different parts of aeroengines need a lot of repeated design work in application and the high precision requirements as special test sensors, a parameterized optimization method for a thermocouple shape combined with a numerical simulation method is proposed. The performance of a dual-screen thermocouple (DST), single-screen thermocouple (SST), and no-screen thermocouple (NST) is tested by a numerical simulation method, and is represented by the velocity error σV and the restitution coefficient r. The dual-screen thermocouple (DST) is the best one, and it is selected as the object to parametric optimization, and the parametric optimization methods based on it, geometrically modeled parametrically, adaptive mesh generation and parametric numerical simulation, are proposed. For a dual-screen thermocouple (DST), eight design structural parameters and four environment parameters are suggested for geometrically modeled parametrically and parametric numerical simulation, respectively. The dichotomy method is used to find the optimal length of the screen L, which is considered the most relevant parameter for thermocouple performance. It can be found that the length of the screen L corresponding to the optimal restitution coefficient r ranges from 56.25 to 62.5 mm.

Construction of Biomimetic Tissues with Anisotropic Structures via Stepwise Algorithm-Assisted Bioprinting.

Tissue-specific natural anisotropic microstructures play an important role in the normal functioning of tissues, yet they remain difficult to construct by current printing techniques. Herein, a stepwise algorithm-assisted bioprinting technology for the construction of biomimetic tissues with a customizable anisotropic microstructure by combining the Adaptive Mesh Generation algorithm and the Greedy Search algorithm is developed. Based on the mechanical topology optimization design mechanism, the Adaptive Mesh Generation algorithm can generate controllable anisotropic mesh patterns with the minimum free energy in plane models according to tissue-specific requirements. Subsequently, the Greedy Search algorithm can program the generated pattern data into optimized printing paths, effectively avoiding structural deformations caused by the multiple stacking of materials and reducing the printing time. The developed bioprinting technique is suitable for various types of bioinks including polymers, hydrogels, and organic/inorganic complexes. After combining with a calcium phosphorus bioink, the compound algorithm-assisted bioprinting technique successfully customizes femurs with biomimetic chemical compositions, anisotropic microstructures, and biological properties, demonstrating its effectiveness. Additionally, algorithm-assisted bioprinting is generally suitable for most commercial extrusion bioprinters that function in the geometric code (G-code) drive mode. Therefore, the algorithm-assisted extrusion bioprinting technology offers an intelligent manufacturing strategy for the customization of anisotropic microstructures in biomimetic tissues.

Singularity-free theory and adaptive finite element computations of arbitrarily-shaped dislocation loop dynamics in 3D heterogeneous material structures

The long-standing problem of arbitrarily-shaped discrete dislocation loops in three-dimensional heterogeneous material structures is addressed by introducing novel singularity-free elastic field solutions as well as developing adaptive finite element computations for dislocation dynamics simulations. The first framework uses the Stroh formalism in combination with the biperiodic Fourier-transform and dual variable and position techniques to determine the finite-valued Peach–Koehler force acting on curved dislocation loops. On the other hand, the second versatile mixed-element method proposes to capture the driving forces through dissipative energy considerations with domain integrals by means of the virtual extension principle of the surfacial discontinuities. Excellent agreement between theoretical and numerical analyses is illustrated from simple circular shear dislocation loops to prismatic dislocations with complicated simply-connected contours in linear homogeneous isotropic solids and anisotropic elastic multimaterials, which also serves as improved benchmarks for dealing with more realistic boundary-value problems with evolving dislocations. Examples of sophisticated dislocation applications include the short-range core reaction between intersecting dislocation loops in interaction with a spherical cavity, as well as the Orowan dislocation-precipitate bypass mechanism in a compressed micropillar of polycrystalline copper. The latter multiscale investigation spans three orders of magnitude in size scale, and is thus enabled by computationally efficient and robust adaptive mesh generation procedures for explicit dislocation propagation, interaction, and coalescence in three-dimensional materials and material structures.

Adaptive stochastic morphology simulation and mesh generation of high-quality 3D particulate composite microstructures with complex surface texture

Particulate composite materials have a broad range of potential applications in engineering and other disciplines. Accurate modeling of their microstructures and fast generation of the finite element meshes play a vital role in investigating many micromechanical phenomena and improving understanding of the underlying failure mechanisms. Due to the exceedingly intricate multiscale internal structures that they possess, the modeling and meshing of their microstructures still remain difficult in general. In this work, we present a computational framework and methodology for the representation, simulation, and mesh generation of 3D stochastic microstructures of particulate composites. Towards this goal, we propose a multi-level multiscale scheme that allows for capturing the multiscale structures of particulate composite materials at both the coarse and fine scales. A briging scale approach based on heat kernel smoothing is also presented to seamlessly link the coarse and fine scales. In addition to the microstructural modeling of particulate composite materials, we also develop an adaptive curvature-based surface and volume mesh generation algorithm for particulate composite microstructures with complex surface texture. Following the implementation of the morphology and mesh generation algorithm, a series of numerical examples are presented to demonstrate the capability and potential of the proposed method.

Tasking framework for adaptive speculative parallel mesh generation

Tasking framework for adaptive speculative parallel mesh generation

Parametric Study of Nonequilibrium Shock Interference Patterns over a Fuselage-and-Wing Conceptual Vehicle

Predicting shock/shock and shock/boundary-layer interactions in gas flows that surround high-speed vehicles is key in aerodynamic design. Under typical hypersonic conditions, these flow structures are influenced by complex nonequilibrium phenomena leading to high-temperature effects. In this work, the conceptual Bedford wing-body vehicle is studied to analyze flow patterns in shock/shock and shock/boundary-layer interactions with thermochemical nonequilibrium. A parametric computational fluid dynamics study is carried out for different hypersonic operating conditions, with respect to the freestream Mach number. Simulations are performed with the SU2-NEMO solver coupled to the Mutation++ library, which provides all the necessary thermodynamic, kinetic, and transport properties of the mixture and chemical species. The Adaptive Mesh Generation library is used for automatic anisotropic mesh adaptation. Numerical results show that increasing the freestream Mach number from 4 to 10 leads to changes in the shock layer, locations of shock impingement, and regions of boundary-layer separation. Despite these changes, the change in freestream Mach number has little impact on the overall shock interaction structures.

Full-Wavefield, Full-Domain Deterministic Modeling of Shallow Low-Magnitude Events for Improving Regional Ground-Motion Predictions

ABSTRACT At the fundamental level, seismic risk analysis relies on good modeling tools for predicting the ground motion resulting from hypothetical earthquake events, which is traditionally approximated using many variations of ground-motion prediction equations (GMPEs). The main benefit of these equations lies in their low computational cost, allowing one to run Monte Carlo simulations in which event probabilities are dictated by regional catalogs comprising historical observations. These equations, however, rely on approximations that are only accurate in a statistical sense. In this study, we consider cases in which regional high-resolution 3D earth models are available from exploration reflection seismology. These high-fidelity velocity models allow us to perform deterministic elastic ground-motion simulations at local distances, given a prescribed synthetic earthquake event, and compare the results with those predicted by GMPEs. This full-wavefield full-domain modeling approach is significantly more costly and particularly challenging due to the slow shear-wave velocity at the near surface, which requires fine spatial and temporal discretizations. With the aid of powerful computational resources, we use an adaptive mesh generator and an efficient wave solver to model the 3D elastic and anelastic wave propagation from the hypocenter all the way to the ground surface. This approach can simultaneously account for 3D subsurface structures, near-surface site effects, topographic relief, and the radiation pattern of the source. In areas where observations are sparse, the modeling results can fill the gap between stations and provide a test bed that can be used for improving the development and accuracy of GMPEs. This approach is well suited for areas where shallow low-magnitude-induced seismic events can occur. Lastly, to demonstrate our approach, we consider an observed seismic event at the Groningen gas field and compare the recorded ground motions with both—those predicted by our approach and those predicted by GMPEs.

A graded mesh refinement approach for boundary layer originated singularly perturbed time‐delayed parabolic convection diffusion problems

In this work, we consider a graded mesh refinement algorithm for solving time‐delayed parabolic partial differential equations with a small diffusion parameter. The presence of this parameter leads to boundary layer phenomena. These problems are also known as singularly perturbed problems. For these problems, it is well‐known that one cannot achieve a convergent solution to maintain the boundary layer dynamics, on a fixed number of uniform meshes irrespective of the arbitrary magnitude of perturbation parameter. Here, we consider an adaptive graded mesh generation algorithm, which is based on an entropy function in conjunction with the classical difference schemes, to resolve the layer behavior. The advantage of the present algorithm is that it does not require to have any information about the location of the layer. Several examples are presented to show the high performance of the proposed algorithm.

Performance Analysis of FEM Solvers on Practical Electromagnetic Problems

The paper presents a comparative analysis of different commercial and academic software. The comparison aims to examine how the integrated adaptive grid refinement methodologies can deal with challenging, electromagnetic-field related problems. For this comparison, two bench-mark problems were examined in the paper. The first example is a solution of an L-shape domain like test problem, which has a singularity at a certain point in the geometry. The second problem is an induction heated aluminum rod, which accurate solution needs to solve non-linear, coupled physical fields. The accurate solution of this problem requires applying adaptive mesh generation strategies or applying a very fine mesh in the electromagnetic domain, which can significantly increase the computational complexity. The results show that the fully-hp adaptive meshing strategies, which are integrated into Agros Suite, can significantly reduce the task's computational complexity compared to the automatic h-adaptivity, which is part of the examined, popular commercial solvers.

Simulation of Quasi-Static Crack Propagation by Adaptive Finite Element Method

The finite element method (FEM) is a widely used technique in research, including but not restricted to the growth of cracks in engineering applications. However, failure to use fine meshes poses problems in modeling the singular stress field around the crack tip in the singular element region. This work aims at using the original source code program by Visual FORTRAN language to predict the crack propagation and fatigue lifetime using the adaptive dens mesh finite element method. This developed program involves the adaptive mesh generator according to the advancing front method as well as both the pre-processing and post-processing for the crack growth simulation under linear elastic fracture mechanics theory. The stress state at a crack tip is characterized by the stress intensity factor associated with the rate of crack growth. The quarter-point singular elements are constructed around the crack tip to accurately represent the singularity of this region. Under linear elastic fracture mechanics (LEFM) with an assumption in various configurations, the Paris law model was employed to evaluate mixed-mode fatigue life for two specimens under constant amplitude loading. The framework includes a progressive analysis of the stress intensity factors (SIFs), the direction of crack growth, and the estimation of fatigue life. The results of the analysis are consistent with other experimental and numerical studies in the literature for the prediction of the fatigue crack growth trajectories as well as the calculation of stress intensity factors.

A moving mesh refinement based optimal accurate uniformly convergent computational method for a parabolic system of boundary layer originated reaction–diffusion problems with arbitrary small diffusion terms

In this paper, a system of time dependent boundary layer originated reaction dominated problems with diffusion parameters of different magnitudes, is considered for numerical analysis. The presence of these parameters lead to the boundary layer phenomena. Here, an optimal order uniformly accurate boundary layer adaptive method moving mesh method is proposed. This method is able to capture the layer phenomena without using a priori information of the solution. The problem is discretized by a modified implicit-Euler scheme in time direction. For the present system, adaptive mesh generation is required in space due to the singularly perturbed nature of the problem. For this purpose, a positive error monitor function is used whose equidistribution will move the mesh points towards the boundary layers. Parameter uniform error estimates are derived to show that the convergence rate is optimal with respect to the problem discretization. Numerical experiments strongly verify the theoretical findings and confirm the efficiency and accuracy of the proposed method.