Adaptive Mesh Research Articles

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

Large Deformation Analysis of Hyperelastic Continuum with Hexahedral Adaptive Finite Elements

The use of hyperelastic materials capable of large deformations, such as elastomeric bearings used to reduce seismic effects, is quite common in civil engineering. Such environments are, in most cases, addressed by numerical solution techniques such as the finite element method. In case of large deformations, nonlinear analysis is used in the solution. In the study presented here, large deformations of a hyperelastic continuum expressed by the Mooney-Rivlin material model are calculated using hexahedral adaptive finite elements. A code was written in MATLAB using the total Lagrangian formulation for the nonlinear adaptive finite element solution. Comparisons were made with Abaqus software to check the consistency of the results obtained from this program. It has been observed that local refinements in the adaptive element mesh occur in the regions where they are needed. Considering the variation of maximum displacement and maximum stress with the number of elements, it has been observed that mesh refinement creates a convergent solution.

Quantifying 3D crack propagation in nodular graphite cast iron using advanced digital volume correlation and X-ray computed tomography

Advanced digital volume correlation (DVC) combined with X-ray computed tomography (XCT) was utilized to investigate the 3D crack propagation in nodular graphite cast iron under tensile loading. The objective of this work was to quantify crack growth at the voxel-scale and extract fracture mechanics parameters such as the 3D crack front, crack opening displacements (CODs), and stress intensity factors (SIFs). By employing a crack growth updating strategy and an adaptive multiscale mesh, the 3D crack shape and propagation were accurately captured throughout the entire process. Von Mises strain and COD fields were characterized to provide insights into plastic zones and variations in crack opening behavior. Estimation of SIFs revealed a mode I dominant regime and limited influence from modes II and III. This study provides comprehensive insights into crack propagation and crack opening behavior, providing valuable information to fracture mechanics.

Multi-objective topology optimization of heat transfer surface using level-set method and adaptive mesh refinement in OpenFOAM

The present study proposes a new efficient and robust algorithm for multi-objectives topology optimization of heat transfer surfaces to achieve heat transfer enhancement with a less pressure drop penalty based on a continuous adjoint approach. It is achieved with a customized OpenFOAM solver, which is based on a volume penalization method for solving a steady and laminar flow around iso-thermal solid objects with arbitrary geometries. The fluid-solid interface is captured by a level-set function combined with a newly proposed robust reinitialization scheme ensuring that the interface diffusion is always kept within a single local grid spacing. Adaptive mesh refinement is applied in near-wall regions automatically detected by the level-set function to keep high resolution locally, thereby reduces the overall computational cost for the forward and adjoint analyses. The developed solver is first validated in a drag reduction problem of a flow around a two-dimensional cylinder at the Reynolds numbers of 10 and 40 by comparing reference data. Then, the proposed scheme is extended to heat transfer problems in a two-dimensional flow at the Prandtl number of 0.7 and 6.9. Finally, three-dimensional topology optimization for multi-objective problems is considered for cost functionals with different weights for the total drag and heat transfer. Among various solutions obtained on the Pareto front, 4.0% of heat transfer enhancement with 12.6% drag reduction is achieved at the Reynolds number of 10 and the Prandtl number of 6.9. Moreover, the optimization of a staggered pin-fin array demonstrates that the optimal shapes and arrangement of the fins strongly depend on the number of rows from the inlet. Specifically, the pin-fins in the first and third rows extended in the upstream direction further enhance heat transfer, while the fins in the second row vanish to reduce pressure loss.

A reinforcement learning strategy for p-adaptation in high order solvers

Reinforcement learning (RL) has emerged as a promising approach to automating decision processes. This paper explores the application of RL techniques to optimise the polynomial order in the computational mesh when using high-order solvers. Mesh adaptation plays a crucial role in improving the efficiency of numerical simulations by increasing accuracy while reducing the cost. Here, actor-critic RL models based on Proximal Policy Optimization offer an approach for agents to learn optimal mesh modifications based on evolving conditions.The paper provides a strategy for p-adaptation in high-order solvers and includes insights into the main aspects of RL-based mesh adaptation, including the formulation of appropriate reward structures and the interaction between the RL agent and the simulation environment. The proposed strategy does not require a high-fidelity solution during the training process and the formulation is general for any computational mesh and partial differential equation (PDE), solved in a discontinuous Galerkin solver. We discuss the impact of RL-based mesh p-adaptation on computational efficiency and accuracy. We apply the RL p-adaptation strategy to a one-dimensional inviscid Burgers' equation, focusing our analysis on smooth solutions of the equation to showcase its effectiveness. The RL strategy reduces the computational cost and improves accuracy over uniform adaptation, while minimising human intervention.

Optimal surrogate boundary selection and scalability studies for the shifted boundary method on octree meshes

The accurate and efficient simulation of Partial Differential Equations (PDEs) in and around arbitrarily defined geometries is critical for many application domains. Immersed boundary methods (IBMs) alleviate the usually laborious and time-consuming process of creating body-fitted meshes around complex geometry models (described by CAD or other representations, e.g., STL, point clouds), especially when high levels of mesh adaptivity are required. In this work, we advance the field of IBM in the context of the recently developed Shifted Boundary Method (SBM). In the SBM, the location where boundary conditions are enforced is shifted from the actual boundary of the immersed object to a nearby surrogate boundary, and boundary conditions are corrected utilizing Taylor expansions. This approach allows choosing surrogate boundaries that conform to a Cartesian mesh without losing accuracy or stability. Our contributions in this work are as follows: (a) we show that the SBM numerical error can be greatly reduced by an optimal choice of the surrogate boundary, (b) we mathematically prove the optimal convergence of the SBM for this optimal choice of the surrogate boundary, (c) we deploy the SBM on massively parallel octree meshes, including algorithmic advances to handle incomplete octrees, and (d) we showcase the applicability of these approaches with a wide variety of simulations involving complex shapes, sharp corners, and different topologies. Specific emphasis is given to Poisson’s equation and the linear elasticity equations.

A quadtree-based adaptive moment-of-fluid method for interface reconstruction with filaments

Implementation of quadtree adaptive mesh refinement (AMR) to the moment-of-fluid (MOF) method is presented in the context of an interface capturing method. Filaments, thinner than a cell size, are resolved using a computationally efficient technique on an unconstrained quadtree structure. The centroid defect relative to its cell size is used as the refinement criterion, together with an enhanced refinement calculation and subsequently its volume conservation. In addition, different approaches are proposed to ensure mass conservation during the computation. This MOF-AMR framework is validated for a range of benchmark problems which are studied widely in the literature. There is no restriction on the choice of CFL number for the purely Lagrangian advection method considered here and this has advantages when combined with AMR. The current quadtree MOF-AMR method leads to much improved computational efficiency and accuracy relative to its grid size compared with a uniform grid. Higher levels of refinement can be costly, therefore the efficiency of mesh resolution is further discussed in relation to the choice of time step and number of AMR levels.

Numerical and experimental assessment of optimal modeling of laminated glass post-breakage response

Modeling laminated glass response to extreme loading scenarios is a computationally demanding process in designing and analyzing protective structures. Therefore, an optimal modeling scheme requires a delicate trade-off between accuracy and computational demand. This article investigates the failure modeling of laminated glass layups of thin and thick panels (three and 11 layers) under blast loading. This is done by implementing various simulation techniques: the finite element method with element erosion/deletion, the mesh-free method of smoothed particle hydrodynamics, and the hybrid implementation of mesh-based and mesh-free simulation through element conversions into particles. This Article examines the feasibility and limitations of each method, considering both the aspects of accuracy and computational cost in light of experimental testing results obtained from both arena and shock tube scenarios. Mesh sensitivity and the significance of adaptive meshing in capturing the fracture patterns are evaluated. The present paper suggests that using hybrid techniques leads to optimal modeling results. Furthermore, the stability of the modeling results under diverse blast conditions is verified.

Effects of fretting wear on rolling contact fatigue

In this study, a three-dimensional (3D) continuum damage mechanics (CDM) finite element (FE) model was developed to investigate the effects of fretting wear on rolling contact fatigue (RCF) of bearing steels. In order to determine the fretting scar geometry, a 3D arbitrary Lagrangian-Eulerian (ALE) adaptive mesh (AM) FE model was developed to simulate fretting wear between two elastic bodies for different initially pristine fretting pressures (0.5, 0.75 and 1 GPa) and friction coefficients (0.15, 0.175 and 0.25) resulting in stick zone to contact width ratios, c/a = 0.35, 0.55 and 0.75. The resulting wear profiles were subjected to various initially pristine RCF pressures (1, 2.2 and 3.4 GPa). The pressure profiles for RCF were determined by moving the contact over the fretted wear profiles in 21 steps. These pressure profiles were then used in the CDM-FE model to predict the RCF life of fretted surfaces. The CDM-FE model uses Fatemi-Socie (FS) criteria for damage evolution and accounts for scatter in RCF life by implementing material topological variation through Voronoi tessellation. The results from CDM-FE RCF model indicate that fretting scar generated at 1 GPa with c/a = 0.35 has the most detrimental effect on RCF life at 1 GPa, reducing life by as much as 99.8% from pristine condition. However, it is to be noted that the remaining life expectancy at this RCF pressure (1 GPa) is significant and more than 17 billion cycles. As the RCF pressure increases (PRCF ≥ 2.2 GPa), the effect of fretting on RCF life decreases for all fretting pressures and c/a values, indicating that life is primarily governed by the RCF pressure. The results from CDM-FE model were also used to develop a life equation for evaluating the L10 life of fretted M-50 bearing steel for the range of tested conditions.

An R-adaptive algorithm based on self-organizing maps for solving incompressible flows with high-order discontinuous Galerkin methods

Mesh quality is critical for the numerical accuracy of CFD (Computational Fluid Dynamics). Although various techniques have been developed to improve mesh applicability to complex flows, r-adaptive methods have received far too little attention. This paper introduces an r-adaptive algorithm based on the self-organizing maps (SOM) of Kohonen and applies it to unsteady CFD applications using the Discontinuous Galerkin (DG) method. Inspired by the properties of the DG method, the numerical discontinuity over the element interface is quantified as an adaptation indicator. The method requires only one adaptation, followed by solution projection onto the new mesh to save computational costs in unsteady cases. A significant advantage of this method is its higher efficiency without solving complex equations during the adaptation. The performance of the developed mesh adaptation algorithm is tested on the steady laminar viscous flow past a circular cylinder at Re=40 and the unsteady laminar viscous flow past a circular cylinder at Re=100 and two circular cylinders in a side-by-side arrangement at Re=200. Significant improvements in flow contours and coefficients are obtained after adaptation.

On the potential of transfer entropy in turbulent dynamical systems

Information theory (IT) provides tools to estimate causality between events, in various scientific domains. Here, we explore the potential of IT-based causality estimation in turbulent (i.e. chaotic) dynamical systems and investigate the impact of various hyperparameters on the outcomes. The influence of Markovian orders, i.e. the time lags, on the computation of the transfer entropy (TE) has been mostly overlooked in the literature. We show that the history effect remarkably affects the TE estimation, especially for turbulent signals. In a turbulent channel flow, we compare the TE with standard measures such as auto- and cross-correlation, showing that the TE has a dominant direction, i.e. from the walls towards the core of the flow. In addition, we found that, in generic low-order vector auto-regressive models (VAR), the causality time scale is determined from the order of the VAR, rather than the integral time scale. Eventually, we propose a novel application of TE as a sensitivity measure for controlling computational errors in numerical simulations with adaptive mesh refinement. The introduced indicator is fully data-driven, no solution of adjoint equations is required, with an improved convergence to the accurate function of interest. In summary, we demonstrate the potential of TE for turbulence, where other measures may only provide partial information.

Assessment of immersed boundary methods for hypersonic flows with gas-surface interactions

The efficacy of immersed boundary (IB) methods with adaptive mesh refinement (AMR) techniques is assessed in the context of atmospheric entry applications, including effects of chemical nonequilibrium (CNE) and gas-surface interactions (GSI). We scrutinize a conservative cut-cell IB method and two non-conservative ghost-cell IB methods, comparing their results with analytical solutions, data from the literature, and results obtained with a reference solver that operates on body-fitted grids. All solvers employ the same external thermochemistry library, ensuring that all observed differences can be attributed solely to differences in the underlying numerical methodologies. We present results for eight benchmark cases. Four verification cases verify the implementation of chemistry, transport properties, catalytic boundary conditions, and shock capturing. Four validation cases encompass blunt geometries with adiabatic and isothermal, as well as inert, catalytic and ablative boundary conditions. Overall, the results obtained with the IB solvers are in very good agreement with the reference data. Discrepancies arise in cases with large temperature or concentration gradients at the wall, and these are linked to conservation errors inherent to ghost-cell methods. Only a strictly conservative cut-cell IB method is on par with body-fitted grid methods.

Studying the Influence of Finite Element Mesh Size on the Accuracy of Ventricular Tachycardia Simulation.

Ventricular tachycardia (VT) is a life-threatening heart condition commonly seen in patients with myocardial infarction (MI). Although personalized computational modeling has been used to understand VT and its treatment noninvasively, this approach can be computationally intensive and time consuming. Therefore, finding a balance between mesh size and computational efficiency is important. This study aimed to find an optimal mesh resolution that minimizes the need for computational resources while maintaining numerical accuracy and to investigate the effect of mesh resolution variation on the simulation results. We constructed ventricular models from contrast-enhanced magnetic resonance imaging data from six patients with MI. We created seven different models for each patient, with average edge lengths ranging from 315 to 645 µm using commercial software, Mimics. Programmed electrical stimulation was used to assess VT inducibility from 19 sites in each heart model. The simulation results in the slab model with adaptive tetrahedral mesh (same as in the patient-specific model) showed that the absolute and relative differences in conduction velocity (CV) were 6.1 cm/s and 7.8% between average mesh sizes of 142 and 600 µm, respectively. However, the simulation results in the six patient-specific models showed that average mesh sizes with 350 µm yielded over 85% accuracy for clinically relevant VT. Although average mesh sizes of 417 and 478 µm could also achieve approximately 80% accuracy for clinically relevant VT, the percentage of incorrectly predicted VTs increases. When conductivity was modified to match the CV in the model with the finest mesh size, the overall ratio of positively predicted VT increased. The proposed personalized heart model could achieve an optimal balance between simulation time and VT prediction accuracy when discretized with adaptive tetrahedral meshes with an average edge length about 350 µm.

A Stable Time-Dependent Mesh Method for Generalized Credit Rating Migration Problem

AbstractThe r-adaptive difference scheme is advanced in this article for solving the generalized credit rating migration model for arbitrary volatility with multiple terminal conditions. The r-adaptive moving mesh method defines the coordinate mapping from the physical to the computational domain and then uses piece-wise polynomials to approximate the physical coordinates. The central implicit semi-discretization scheme is imposed on space, and the backward Euler time marching scheme, coupled with several moving mesh partial differential equations, is used to achieve the numerical solution. The numerical operations are performed with several examples, and the obtained results are sufficiently accurate. The convergence of the proposed scheme is second-order, which is verified with the analysis. The article also investigates the stability and convergence of the adaptive mesh discretization scheme, which are not available in the literature; the convergence of the scheme is second-order in space and first-order in time.

Quokka-based understanding of outflows (QED) – I. Metal loading, phase structure, and convergence testing for solar neighbourhood conditions

ABSTRACT Multiphase galactic outflows, generated by supernova (SN) feedback, are likely to be more metal rich than the interstellar media from which they are driven due to incomplete mixing between SN ejecta and the ambient interstellar medium. This enrichment is important for shaping galactic metallicities and metallicity gradients, but measuring it quantitatively from simulations requires resolution high enough to resolve mass, momentum and energy exchanges between the different phases of the outflows. In this context, we present QED, which are simulations of outflows, driven by SN feedback, conducted using Quokka, a new GPU-optimized adaptive mesh refinement radiation-hydrodynamics code. This code allows us to reach combinations of resolution, simulation volume, and simulation duration larger than those that have previously been possible, and to resolve all gas phases from cold neutral medium, T ∼ 100 K, to hot ionized gas, T ≳ 107 K. In this, a first of a series of papers exploring generation and evolution of multiphase outflows from a wide range of galactic environments and star formation rates, we quantify the extent of selective metal loading in solar neighbourhood-like environments. We explain the selective metal loading, we find as a result of the transport of metals within and between phases, a phenomenon we can study owing to the parsec-scale resolution that our simulations achieve. We also quantify the sensitivity of metal loading studies to numerical resolution, and present convergence criteria for future studies.

The flow around a stepped cylinder with turbulent wake and stable shear layer

The turbulent external flow around a three-dimensional stepped cylinder is studied by means of direct numerical simulations with the adaptive mesh refinement technique. We give a broad perspective of the flow regimes from laminar to turbulent wake at $Re_D=5000$ , which is the highest ever considered for this flow case. In particular, we focus on the intermediate Reynolds number $Re_D=1000$ that reveals a turbulent wake coupled with a stable cylinder shear layer (subcritical regime). This flow shows a junction dynamics similar to the laminar $Re_D=150$ , where no hairpin vortex appears around the edges, and just two horseshoe vortices are visible. A new stable vortex in the form of a ring, which coils around the rear area, is also identified. In the turbulent wake, the presence of three wake cells is pointed out: the large and small cylinder cells together with the modulation region. However, the modulation dynamics varies between the subcritical and turbulent regimes. A time-averaged, three-dimensional set of statistics is computed, and spatially coherent structures are extracted via proper orthogonal decomposition (POD). The POD identifies the (long-debated) connection between the N-cell and the downwash behind the junction. Furthermore, as the Reynolds number increases, the downwash phenomenon becomes less prominent. Eventually, a reduced-order reconstruction with the most energetically relevant modes is defined to explain the wake vortex interactions. This also serves as a valuable starting point for simulating the stepped cylinder wake behaviour within complex frameworks, e.g. fluid–structure interaction.

SuperAdjoint: Super-resolution neural networks in adjoint-based error estimation

Numerical simulations and optimisation methods, such as mesh adaptation, rely on the accurate and inexpensive use of error estimation methods. Adjoint-based error estimation is the most accurate method, and generally the most costly. A strong contributor to this cost is the need to compute a higher resolution adjoint solution. Here, it is proposed to use super-resolution neural networks to super-resolve a fine adjoint solution from a lower-cost coarse adjoint solution: a superAdjoint. The method is compared to reference error estimators on an unsteady Burgers’ equation using the method of manufactured solutions. Two forms of the superAdjoint were implemented, a twice and a four times refining super-resolution neural network. These were used to demonstrate both the computational cost reduction and the potential for the reduction of the storage footprint of the primal problem. The first, referred to as 2×CNN, was able to reconstruct the spatially enriched adjoint solution, thus providing a robust and inexpensive local output error. The second, the 4×CNN, was able to demonstrate the reconstruction ability of super-resolution neural networks for higher upscaling factors. This was leveraged in order to subsample the primal solution in space, thus reducing substantially the storage footprint of the discrete primal solution. Both superAdjoints could achieve the desired level of accuracy when compared to the reconstruction of the refined adjoint-based error estimate. Moreover, superAdjoint was shown to be able to generalise to a new QoI that was untrained for. This gives great confidence in the use of super-resolution neural networks for the reduction of both computational cost and storage requirements of adjoint-based error estimation, and goal-oriented mesh adaptation.

Investigation of the ultrasound-induced collapse of air bubbles near soft materials

A numerical investigation into the ultrasound-induced collapse of air bubbles near soft materials, utilizing a novel multi-material diffuse interface method (DIM) model with block-structured adaptive mesh refinement is presented. The present work expands from a previous five-equation DIM by incorporating Eulerian hyperelasticity. The model is applicable to any arbitrary number of interacting fluid and solid material. A single conservation law for the elastic stretch tensor enables tracking the deformations for all the solid materials. A series of benchmark cases are conducted, and the solution is found to be in excellent agreement against theoretical data. Subsequently, the ultrasound-induced bubble-tissue flow interactions are examined. The bubble radius was found to play a crucial role in dictating the stresses experienced by the tissue, underscoring its significance in medical applications. The results reveal that soft tissues primarily experience tensile forces during these interactions, suggesting potential tensile-driven injuries that may occur in relevant treatments. Moreover, regions of maximal tensile forces align with tissue elongation areas. It is documented that while early bubble dynamics remain relatively unaffected by changes in shear modulus, at later stages of the penetration processes and the deformation shapes, exhibit notable variations. Lastly, it is demonstrated that decreasing standoff distances enhances the interaction between bubbles and tissue, thereby increasing the stress levels in the tissue, although the behavior of the bubble dynamics remains largely unchanged.

Tidal turbine array modelling using goal-oriented mesh adaptation

To examine the accuracy and sensitivity of tidal array performance assessment by numerical techniques applying goal-oriented mesh adaptation. The goal-oriented framework is designed to give rise to adaptive meshes upon which a given diagnostic quantity of interest (QoI) can be accurately captured, whilst maintaining a low overall computational cost. We seek to improve the accuracy of the discontinuous Galerkin method applied to a depth-averaged shallow water model of a tidal energy farm, where turbines are represented using a drag parametrisation and the energy output is specified as the QoI. Two goal-oriented adaptation strategies are considered, which give rise to meshes with isotropic and anisotropic elements. We present both fixed mesh and goal-oriented adaptive mesh simulations for an established test case involving an idealised tidal turbine array positioned in a channel. With both the fixed meshes and the goal-oriented methodologies, we reproduce results from the literature which demonstrate how a staggered array configuration extracts more energy than an aligned array. We also make detailed qualitative and quantitative comparisons between the fixed mesh and adaptive outputs. The proposed goal-oriented mesh adaptation strategies are validated for the purposes of tidal energy resource assessment. Using only a tenth of the number of degrees of freedom as a high-resolution fixed mesh benchmark and lower overall runtime, they are shown to enable energy output differences smaller than 2% for a tidal array test case with aligned rows of turbines and less than 10% for a staggered array configuration.

An immersed selective discontinuous Galerkin method in particle-in-cell simulation with adaptive Cartesian mesh and polynomial preserving recovery

In this paper, a selective discontinuous Galerkin (SDG) method and a polynomial preserving recovery (PPR) method are developed and integrated with the immersed-finite-element particle-in-cell (IFE-PIC) method, in order to carry out the plasma-material interaction simulation on adaptive Cartesian meshes (ACM). To significantly save the computational cost in practice, the PIC simulation often needs to use Cartesian meshes for the whole domain and the ACM for some focused regions in the plasma-material interaction problems. Therefore, we utilize the SDG method with immersed finite elements for the field solver of the IFE-PIC method, as the key technique to allow the local Cartesian mesh refinement which will generate hanging nodes. To minimize the degree of freedom of the SDG on the ACM and reduce the computational cost, a new selective technique of the discontinuous global basis functions is proposed for the SDG. Various types of nodes in the ACM are discussed for the implementation of this selective technique, including the hanging nodes. Meanwhile, the gathering and scattering steps of the PIC simulation also need to be appropriately performed on the ACM. The PPR method is a generic gradient recovery technique to be incorporated into the IFE-PIC method for more accurate force deposition on the ACM. In addition, two other algorithmic structures of the traditional IFE-PIC method have been modified to accommodate the ACM in the simulation, including a new mapping array structure for the particle positioning and a new charge deposition with some special particle positions in the ACM. As a result, the integrated immersed-selective-discontinuous-Galerkin particle-in-cell (ISDG-PIC) method can efficiently perform the plasma-material interactions simulation on the Cartesian meshes with local mesh refinement independent of the interface. Several numerical experiments are performed to demonstrate the convergence, efficiency, and applicability of the proposed ISDG-PIC method.