Adaptive Refinement Technique Research Articles

Adaptive Deep Fourier Residual method via overlapping domain decomposition

The Deep Fourier Residual (DFR) method is a specific type of variational physics-informed neural network (VPINN). It provides a robust neural network-based solution to partial differential equations (PDEs). The DFR strategy is based on minimizing the dual norm of the weak residual of a PDE, which is equivalent to minimizing the energy norm of the error. To compute the dual norm of the weak residual, the DFR method employs an orthonormal spectral basis of the test space, known for rectangles or cuboids for multiple function spaces.In this work, we extend the DFR method with ideas of traditional domain decomposition (DD). This enables two improvements: (a) to solve problems in more general polygonal domains and (b) to develop an adaptive refinement technique in the test space using a Döfler marking algorithm. In the former case, we retain the desirable equivalence between the employed loss function and the H1 error under non-restrictive assumptions, numerically demonstrating adherence to explicit bounds in the case of the L-shaped domain problem. In the latter, we show how refinement strategies lead to potentially significant improvements against a reference, classical DFR implementation with a test function space of significantly lower dimensionality, allowing us to better approximate singular solutions at a more reasonable computational cost.

On adaptive mesh coarsening procedures for the virtual element method for two-dimensional elastic problems

In the context of adaptive remeshing, the virtual element method provides significant advantages over the finite element method. The attractive features of the virtual element method, such as the permission of arbitrary element geometries, and the seamless permission of ‘hanging’ nodes, have inspired many works concerning error estimation and adaptivity. However, these works have primarily focused on adaptive refinement techniques with little attention paid to adaptive coarsening (i.e. de-refinement) techniques that are required for the development of fully adaptive remeshing procedures. In this work novel indicators are proposed for the identification of patches/clusters of elements to be coarsened, along with a novel procedure to perform the coarsening. The indicators are computed over prospective patches of elements rather than on individual elements to identify the most suitable combinations of elements to coarsen. The coarsening procedure is robust and suitable for meshes of structured and unstructured/Voronoi elements. Numerical results demonstrate the high degree of efficacy of the proposed coarsening procedures and sensible mesh evolution during the coarsening process. It is demonstrated that critical mesh geometries, such as non-convex corners and holes, are preserved during coarsening, and that meshes remain fine in regions of interest to engineers, such as near singularities.

A New Definition of the Dual Interpolation Curve for CAD Modeling and Geometry Defeaturing

The present paper provides a new definition of the dual interpolation curve in a geometric-intuitive way based on adaptive curve refinement techniques. The dual interpolation curve is an implementation of the interpolatory subdivision scheme for curve modeling, which comprises polynomial segments of different degrees. Dual interpolation curves maintain various desirable properties of conventional curve modeling methods, such as local adaptive subdivision, high interpolation accuracy and convergence, and continuous and discontinuous boundary representation. In addition, the dual interpolation curve is mainly applied to solve the difficult geometry defeaturing problems for curve modeling in existing computer-aided technology. By adding fictitious and intrinsic nodes inside or at the vertices of interpolation elements, the dual interpolation curve is flexible and convenient for characterizing a set of ordered points or discrete segments. Combined with the Lagrange interpolation polynomial and meshless method, the proposed approach is capable of characterizing the non-smooth boundary for geometry defeaturing. Experimental results are given to verify the validity, robustness, and accuracy of the proposed method.

Adaptive Finite Element Simulation of Double-Diffusive Convection

Double-diffusive convection plays an important role in many physical phenomena of practical importance. However, the numerical simulation of these phenomena is challenging since fine meshes are often required to capture the flow physics. Hence, several different numerical methods have been employed in the past. This work reports the development and application of an adaptive finite element method for the simulation of these phenomena, thereby avoiding the need for the use of very fine meshes over the whole domain. The weak formulation of the conservation equations for mass, momentum, energy and species concentration is used. The Boussinesq approximation relates the density of the fluid to the temperature and/or the species concentration. A second-order backward difference method is used for time discretization and the Galerkin method is employed for spatial discretization. Both adaptive time step and grid refinement techniques are employed, and the code is parallelized using MPI. Three different stabilization methods of the convective-diffusion equations are compared; namely, the streamline upwind Petrov–Galerkin (SUPG) method, and two modified methods aimed at diminishing spurious oscillations that include an artificial diffusion term. This diffusion term may be either isotropic or orthogonal to the streamlines. The addition of artificial isotropic diffusion to the SUPG method provides enhanced stability. The method is applied to double-diffusive finger convection in a sucrose-salt aqueous mixture and a stratified salt solution heated from below. The method accurately reproduces the experimentally observed temporal evolution of the salt fingers in the former case and the location of the interfaces between convective and non-convective zones in the latter.

Physics-Informed Neural Network Integrating PointNet-Based Adaptive Refinement for Investigating Crack Propagation in Industrial Applications

Crack is one of the critical factors that degrade the performance of machinery manufacturing equipment. Recently, physics-informed neural networks (PINNs) have received attention due to their strong potential in solving physical problems. For fracture problems, PINNs have been used to predict crack paths by minimizing the variational energy of discrete domains where refined meshes are necessary. To obtain refined meshes, posteriori adaptive refinement techniques are commonly used to perform local refinement of the mesh based on errors in the intermediate calculation process; thus, they require pretest calculations. However, it is computationally expensive to precalculate complex problems, especially crack propagation. To solve this problem, we propose a PointNet-based adaptive refinement method to avoid precalculation when constructing the discrete domain. The proposed method is applied to simulate crack propagation using a PINN. Results show that the proposed method can be used to obtain reliable results efficiently when using the PINN framework.

A 3D fully‐nonlinear potential‐flow solver for efficient simulations of large‐scale free‐surface waves

AbstractIn the quest for a numerical method for surface waves and wave‐induced effects applicable when linear or weakly nonlinear methods are insufficient, a three‐dimensional numerical wave tank assuming fully‐nonlinear potential‐flow theory is proposed. When viscous‐flow effects, breaking waves or other violent flow‐phenomena are not of primary importance, potential‐flow methods may have similar capability in capturing the involved physics as Navier–Stokes solvers while being potentially more accurate in handling wave‐propagation mechanism and more computationally efficient. If made sufficiently accurate, efficient and numerically robust, fully‐nonlinear potential flow models can therefore represent a powerful tool in the study of ocean waves and their interaction with marine structures, which is the main motivation behind the present work. The governing Laplace equation for the velocity potential is solved using the harmonic polynomial cell method, which is a field method giving high‐order accuracy provided that the cells used to describe the water domain have no stretching or distortion. This can only be achieved in a grid with cubic cells, which leads to poor numerical efficiency unless measures are introduced to refine the grid locally. Here, to improve the efficiency using strictly cubic cells, an adaptive grid refinement technique is introduced. It is shown that this has the ability to improve the computational speed with a factor of up to 20 without sacrificing accuracy. Numerical results are shown to be in good agreement with highly accurate nonlinear reference solutions for regular and irregular waves of various steepness up to the limit of theoretical wave breaking. For long‐crested irregular waves, significant discrepancies with a second‐order theory for the crest‐height distribution are identified, while the second‐order theory appears to provide a better description of the crest height for the single short‐crested irregular sea state simulated. Having demonstrated that the proposed numerical method accurately models nonlinear wave phenomena up to the limit of wave breaking, future work should seek to implement wave‐body interaction capabilities. The adaptive grid refinement technique, which refines the grid dynamically depending on the position of boundaries of interest, is developed with this application in mind. Except from providing a robust way of dealing with wave‐body intersection points, extending the method to account for wave‐body interactions should therefore involve limited difficulty.

Transfer Learning-Based Artificial Intelligence-Integrated Physical Modeling to Enable Failure Analysis for 3 Nanometer and Smaller Silicon-Based CMOS Transistors

Integral to the success of the semiconductor industry in keeping up with Moore’s law is the importance of failure analysis (FA). Accurate and fast FA is vital in ensuring yield, reliability, and rapid production in the semiconductor industry. However, locating defects among tens of billions of transistors packed in the tiny modern microchip is not a trivial task. Not only the process technology has to achieve such high integration of devices evolved to become astoundingly sophisticated but also debugging for defects in these chips has become remarkably complex. With electrical nanoprobing, we show how artificial intelligence-integrated physical modeling can be effective in finding difficult proverbial needle-in-a-haystack defects based on the electrical responses of the devices. Moreover, the information learned on current devices can be transferred to the latest transistor technologies, enabling machine learning-based defect sleuths of the future. Notably, we achieved a defect region classification accuracy of 99.5% on well-studied fin nanoscale field-effect transistors (FET) using a defect dataset based on an experimentally calibrated TCAD digital twin model and an adaptive boundary refinement technique. With transfer learning, a defect region classification accuracy of 99.58% is also achieved on the next-generation gate-all-around FETs, overcoming the lack of crucial datasets and optimized labels to guide and accelerate the production process of emerging devices. The proposed technique is expected to be the next level of defect identification, an important stepping stone to accelerate the production process for advanced technology nodes beyond 3 nm.

Adaptive phase field method using novel physics based refinement criteria

Amongst the available methods to model fracture processes, the phase field approach has proved to be efficient and has received extensive attention. However, the approach is computationally demanding as it requires a very high resolution, both in space and time to resolve the fracture characteristics. In this paper, a novel adaptive phase field method is proposed for modelling quasi-static crack propagation in brittle materials. The adaptive refinement is based on the stability analysis which is combined with the quadtree decomposition. The stability analysis is based on a linear perturbation method that is used herein to determine the onset of fracture initiation on the fly. Three standard benchmark problems are solved to show the efficiency and the robustness of the proposed method. It is shown that the proposed framework does not require any post processing technique for adaptive refinement and yet is effective.

Multilevel correction adaptive finite element method for solving nonsymmetric eigenvalue problems

Large-scale nonsymmetric eigenvalue problems are common in various fields of science and engineering computing. However, their efficient handling is challenging, and research on their solution algorithms is limited. In this study, a new multilevel correction adaptive finite element method is designed for solving nonsymmetric eigenvalue problems based on the adaptive refinement technique and multilevel correction scheme. Different from the classical adaptive finite element method, which requires solving a nonsymmetric eigenvalue problem in each adaptive refinement space, our approach requires solving a symmetric linear boundary value problem in the current refined space and a small-scale nonsymmetric eigenvalue problem in an enriched correction space. Since it is time-consuming to solve a large-scale nonsymmetric eigenvalue problem directly in adaptive spaces, the proposed method can achieve nearly the same efficiency as the classical adaptive algorithm when solving the symmetric linear boundary value problem. In addition, the corresponding convergence and optimal complexity are verified theoretically and demonstrated numerically.

Spurious interface and boundary behaviour beyond physical solutions in lattice Boltzmann schemes

The scalar field and the non-equilibrium solutions of the linear advection-diffusion d2Q9 Lattice Boltzmann (LBM) two-relaxation-times (TRT) scheme are constructed analytically. The scheme copes with an infinite number of suitable, second-order accurate, equilibrium weights. Here, the simplest, translation-invariant geometry with an implicitly located, straight or diagonal, grid-aligned interface (boundary) is addressed. We show that these two interface (boundary) orientations are accommodated with the help of two distinctive, anisotropic, discrete-exponential algebraic solution components, referred to as the A-layer and the B-layer. Being unpredicted by the perturbative analysis, such as the Chapman-Enskog, asymptotic or truncation, their solution is derived symbolically from the TRT recurrence equations, subject to the local mass conservation solvability and effective closure conditions. When the interface (boundary) is “diagonal”, the A-layer perturbs the simplest physical solutions, like the piece-wise linear, polynomial or exponential scalar field, rendering the macroscopic solution weight-dependent and delaying its convergence to the first order; the A-layer base depends upon the weights, free relaxation parameter Λ and physical numbers. In contrast, the B-layer, invisible to the scalar field, typically accommodates the non-equilibrium discrepancy between the normal and diagonal directions on the “straight” interface (boundary); the B-layer base is fixed by Λ alone. The A-layer and B-layer may coexist and degrade the physical solution gradient and its convergence. Only the D2Q5 model is free from all these effects in the straight and diagonal orientations, while the diagonally-rotated D2Q5 model is unsuitable because of the “checkerboard” effect. These spurious corrections are not the Knudsen layers, but they present the LBM response for any-order bulk mismatch with the implicit or explicit interface (boundary) treatment; the A-layer and B-layer bring them in evidence and provide excellent benchmarks for their attenuation through interface-conjugate or adaptive refinement techniques. Our approach extends to any lattice, linear collision, source term, heterogeneity and LBM problem class.

Adaptive analysis using scaled boundary finite element method in 3D

In this paper, an adaptive refinement technique using the scaled boundary finite element method (SBFEM) is proposed. The salient feature of this technique is that it is not required to regenerate the mesh for the whole model during the iterations. To this end, a local mesh refinement strategy is implemented based on a polytree algorithm in three dimensions, which can be applied to polyhedral elements with arbitrary number of nodes, edges and faces. These elements constructed by the SBFEM can be used in analysis with their boundaries discretized only, which reduce the difficulty to connect elements with different sizes. An explicit residual based error indicator is developed using the discontinuity of the stress field to guide the adaptive mesh refinement. The accuracy and efficiency of the proposed method are demonstrated using five numerical examples, including complex geometry and stress singularity.

A locally refined adaptive isogeometric analysis for steady-state heat conduction problems

This paper presents an isogeometric analysis with adaptivity using locally refined B-splines (LR B-splines) for steady-state heat conduction simulations in solids. Within this framework, the LR B-splines, which have an efficient and simple local refinement algorithm, are used to represent the geometry, and are also employed for spatial discretization, thus providing a seamless interaction between the CAD models and the numerical analysis. A Zienkiewicz-Zhu a posteriori error estimator in terms of the temperature gradient recovery is used to identify the regions for local mesh refinement. The accuracy and convergence properties of the proposed framework are demonstrated through several two-dimensional isotropic examples. Numerical results indicate good performance of the present method as the adaptive refinement technique yields more accurate solution compared to the uniform global refinement with an improved convergence rate.

Effects of dynamic contact angle on immiscible two-phase flow displacement in angular pores: A computational fluid dynamics approach

Analyzing multiphase flow in capillaries can provide valuable insight into fluid flow modeling in different branches of engineering, particularly petroleum engineering, carbon dioxide sequestration, polymer processing, and drug delivery. Wall wettability is one of the most critical parameters significantly affecting fluid distribution within a capillary system. The tendency for wettability is usually described using the contact angle. In this research, the impact of dynamic contact angles was investigated, and the results were compared with published experimental data. For this purpose, a three-dimensional numerical scheme using computational fluid dynamics was developed to simulate two-phase immiscible flow through angular pores at micro-scale. In comparison with previously published numerical simulations assuming a static contact angle, a better agreement between the experimental data and numerical simulations was observed in this study. Results of this study demonstrated that using dynamic contact angle models can reduce the numerical error from 150% to 35%. Therefore, it can be concluded that the effects of dynamic contact angles on viscous dominant flows are not negligible. In addition, analyzing multiphase flow within different angular pores demonstrated the importance of dynamic contact angle implementation for angular pores with sharp corners. The accuracy and computational efficiency of the proposed numerical scheme were improved using a dynamic mesh adaptive refinement technique. This method improved the numerical simulations' accuracy while simultaneously reducing computational costs.

Error indicators and adaptive refinement of the discrete thin plate spline smoother

The discrete thin plate spline is a data fitting and smoothing technique for large datasets. Current research only uses uniform grids for this discrete smoother, which may require a fine grid to achieve a certain accuracy. This leads to a large system of equations and high computational costs. Adaptive refinement adapts the precision of the solution to reduce computational costs by refining only in sensitive regions. The error indicator is an essential part of the adaptive refinement as it identifies whether certain regions should be refined. Error indicators are well researched in the finite element method, but they might not work for the discrete smoother as data may be perturbed by noise and not uniformly distributed. Two error indicators are presented: one computes errors by solving an auxiliary problem and the other uses the bounds of the finite element error. Their performances are evaluated and compared with 2D model problems.
 
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Extension of the [formula omitted]-[formula omitted]-SPH model for simulating Vortex-Induced-Vibration problems

In this paper, the Lagrangian particle method δ-Plus-SPH model is used to solve a series of benchmark test-cases of vortex induced vibrations (VIV). An Adaptive Particle Refinement (APR) technique is adopted to resolve correctly the boundary layer regions of the moving bodies, and to de-refine the particles that are transported far away. Furthermore, a switch correction on the pressure forces term is adopted in the momentum equation to completely remove the occurrence of the so-called Tensile Instability that leads to the development of numerical cavitation in negative pressure regions. Because of the Lagrangian nature of the method, difficulties arise when intense vortex wakes, typically developed in VIV problems, cross the outflow boundary. To this purpose, a damping zone is also implemented close to the outlet to improve the numerical stability of the scheme. The fluid–structure coupling technique is based on ghost particles properly generated inside the solid bodies. The validation is performed against test-cases for which reference solutions are available in the literature. Within those test-cases challenging benchmarks involving small mass ratios, large-amplitude body motions and multi-body interactions have been selected. For all the benchmark test-cases the δ-Plus-SPH results are in good agreement with the reference solutions, demonstrating the ability of this particle method in solving complex VIV problems.

Numerical simulation of a silicon-based latent heat thermal energy storage system operating at ultra-high temperatures

This study investigates numerically a silicon-based latent heat storage system operating at ultra-high temperatures (∼1410–2000 °C). Owing to the silicon’s high latent heat (1230 kWh·m−3), storage densities of almost an order of magnitude higher than the state-of-the-art molten salt-based systems can be achieved. Prior to fabricating this system, there is a necessity to scrutinize the complex heat transfer mechanisms occurring during silicon phase change and indicate a vessel design that enables quick charge rates. A validated transient computational fluid dynamics model, combining the multiphase volume of fluid method with the enthalpy porosity approach and an adaptive local grid refinement technique is applied. This model simultaneously takes into account the phase-change material (PCM) volumetric change and the effect of dendritic formations and buoyancy-driven natural convection on the melting process. Numerical results reveal that the system charge rate is highly dependent on the PCM permeability, vessel design and operating conditions. Between five shapes investigated, i.e. cube, truncated cone, sphere, cut-off sphere and cylinder of volume V = 3.75E−03 m3, the truncated cone is the most preferable considering both design flexibility- its height can be easily altered, without increasing its volume- and melting rates. Actually, the PCM melting time achieved with the cone is 21% quicker compared to sphere and 6% slower compared to cube. Concerning the vessel size, the melting time increases by almost 28%, when the vessel volume increases to V′ = 2V. Finally, reduction in the silicon melting time, almost by 31%, is achieved by increasing the Stefan number by 35%.

Prediction of burning velocity and quenching distance of hydrogen flames

Atmospheric, laminar, stoichiometric, premixed hydrogen-air flames in a diverging channel are investigated by means of Computational Fluid Dynamics. This configuration has been recently used in a series of experimental investigations to determine the burning velocities and quenching distances for premixed flames of different fuels. The purpose of the present investigation is the validation of the prediction procedures for the burning speeds and quenching distances for hydrogen flames by comparing them with these measurements. Global and detailed reaction mechanisms are applied to describe the combustion process. For assuring an adequately fine resolution of the flame fronts, adaptive grid refinement techniques are applied. A reasonable agreement is observed with the experiments, where the detailed and global mechanisms are slightly overpredicting and underpredicting the quenching distance, respectively.

Angular adaptive discrete ordinates method for deuterium–tritium neutron streaming experiments simulating narrow gaps

Shielding analyses for fusion reactors pose significant computational challenges, including strong anisotropic scattering and a combination of deep-penetration shielding and complex channeling. The discrete ordinates method is one of the most widely used techniques to solve the linear Boltzmann equation. We propose a regional angular adaptive algorithm to reduce angular discretization errors for shielding calculations. Angular adaptive refinement techniques are based on estimated local errors. The local errors are weighted by the importance distribution toward the computational goal, providing an appropriate goal-oriented adaptivity. The accuracy of this algorithm with FENDL/MG-3.0 libraries was investigated through analyses of streaming experiments with deuterium–tritium neutron sources at JAERI/FNS. From comparing reaction rates of measurements and calculations, the angular adaptive algorithm suffices in predicting the fast neutron field inside the gap. The solutions show that the algorithm can improve the accuracy of calculation results.

A novel error indicator and an adaptive refinement technique using the scaled boundary finite element method

In this paper, an adaptive refinement strategy based on the scaled boundary finite element method on quadtree meshes for linear elasticity problems is discussed. Within this framework, the elements with hanging nodes are treated as polygonal elements and thus does not require special treatment. The adaptive refinement is supplemented with a novel error indicator. The local error is estimated directly from the solution of the scaled boundary governing equations. The salient feature is that it does not require any stress recovery techniques. The efficacy and the robustness of the proposed approach are demonstrated with a few numerical examples.

Sharp interface approaches and deep learning techniques for multiphase flows

We present a review on numerical methods for simulating multiphase and free surface flows. We focus in particular on numerical methods that seek to preserve the discontinuous nature of the solutions across the interface between phases. We provide a discussion on the Ghost-Fluid and Voronoi Interface methods, on the treatment of surface tension forces that avoid stringent time step restrictions, on adaptive grid refinement techniques for improved efficiency and on parallel computing approaches. We present the results of some simulations obtained with these treatments in two and three spatial dimensions. We also provide a discussion of Machine Learning and Deep Learning techniques in the context of multiphase flows and propose several future potential research thrusts for using deep learning to enhance the study and simulation of multiphase flows.