Coarse Mesh Research Articles

Mesh size effect on finite source inversion with 3-D finite-element modelling

SUMMARY Three-dimensional finite-element models, which can handle the stress perturbations caused by subsurface mechanical heterogeneities and fault interactions, have been combined with the finite source inversion to estimate the coseismic slip distribution over the fault plane. However, the mesh grid for discretizing the governing equations in the finite-element model significantly affects the numerical accuracy. In this study, we performed kinematic finite source inversion with idealized (regular observation point array; M1A–M1D) and regional (GEONET, GPS Earthquake Observation Network System stations in Japan; M2A–M2H) models with different mesh sizes to quantitatively analyse the effect of the mesh grid size around the fault plane on the inverted fault slip distribution. Synthetic observation data vectors obtained from the finest models (M1A and M2A) are compared with those from the coarser models (M1B–M1D and M2B–M2H), which were adopted to construct Green's function matrix. We found that the coarser mesh models derived a smaller surface displacement, leading to a decrease in the norm of Green's function matrix, which in turn influences the fault slip magnitude from the finite source inversion. Finally, we performed the source inversion for the fault slip distribution of the 2011 Mw 9.0 Tohoku–Oki earthquake using the coseismic surface displacements recorded at the GEONET and seafloor stations and finite-element modelling. By reducing the mesh size on the fault, we confirmed that the estimated magnitude of fault slip converged to approximately 80 m, which is consistent with the range of fault slip amounts from previous studies based on the Okada model. At least 0.88 million total domain elements and a 6.7 km2 mesh size on the fault plane with an area of 240 × 720 km2 are required for the convergence of the fault slip. Furthermore, we found that the location of the maximum fault slip is less sensitive to the mesh size, implying that source inversion based on a coarse mesh model (i.e. less than 0.5 million elements and > ∼60 km2 mesh size) can quickly provide the rough fault slip distribution.

Adaptive ANCF method described by arbitrary Lagrange-Euler formulation with application in variable-length underwater tethered systems moving in limited spaces

The nonlinear effect of the variable-length cable may dominate the design of Remotely Operated Vehicles (ROVs) due to the interaction between the flexible cable and surroundings, especially when moving in limited spaces. This paper develops a novel adaptive Absolute Nodal Coordinate Formulation (ANCF) described by Arbitrary Lagrange-Euler (ALE) formulation to investigate the nonlinear dynamics of variable-length underwater tethered systems going across obstacles. The adaptive element techniques are adopted to efficiently simulate the moving contact. Fine meshes are generated autonomously to capture the detailed contact behaviors in contact areas while coarse meshes emerge in non-contact areas, thus significantly reducing DOFs and contact pairs. Additionally, the precontact searching algorithm based on the axis-aligned bounding box (AABB) is employed for the acceleration of the contact detection, which is crucial when element number increases as the mass flow. Combined with the experimental records and analytical solutions in two numerical examples, the accuracy and efficiency of the proposed methodology are further assessed by a comprehensive comparison among the fine, coarse, and adaptive mesh results. The numerical algorithm provides an efficient means for developing a feed-forward controller to compensate the nonlinear cable effects on ROVs when moving in limited space.

Multi-fidelity reduced-order surrogate modelling

High-fidelity numerical simulations of partial differential equations (PDEs) given a restricted computational budget can significantly limit the number of parameter configurations considered and/or time window evaluated. Multi-fidelity surrogate modelling aims to leverage less accurate, lower-fidelity models that are computationally inexpensive in order to enhance predictive accuracy when high-fidelity data are scarce. However, low-fidelity models, while often displaying the qualitative solution behaviour, fail to accurately capture fine spatio-temporal and dynamic features of high-fidelity models. To address this shortcoming, we present a data-driven strategy that combines dimensionality reduction with multi-fidelity neural network surrogates. The key idea is to generate a spatial basis by applying proper orthogonal decomposition (POD) to high-fidelity solution snapshots, and approximate the dynamics of the reduced states—time-parameter-dependent expansion coefficients of the POD basis—using a multi-fidelity long short-term memory network. By mapping low-fidelity reduced states to their high-fidelity counterpart, the proposed reduced-order surrogate model enables the efficient recovery of full solution fields over time and parameter variations in a non-intrusive manner. The generality of this method is demonstrated by a collection of PDE problems where the low-fidelity model can be defined by coarser meshes and/or time stepping, as well as by misspecified physical features.

Two-grid parallel stabilized finite element method based on overlapping domain decomposition for the Stokes problem

In this paper, one non-iterative two-grid parallel stabilized finite element method based on overlapping domain decomposition is developed and investigated for the Stokes problem. The lowest-order finite element pairs P1−P1 are utilized to approximate the velocity and pressure, respectively. The difference between a consistent and an under-integrated mass matrices is chosen as the stabilized term to circumvent the so-called inf-sup condition. Algorithm distributes legitimately residuals into local domains with a partition of unity. Error estimates are rigorously established. Theoretical results indicate that patches of diameter H|lnH|(|lnH|−ln|lnH|) or H|lnH| are sufficient to preserve optimal convergence rates with a proper configuration between the coarse mesh size H and the fine mesh size h in 2-D or 3-D case, respectively. Finally, some numerical experiments are reported to support our theoretical results.

Numerical simulation of the bentonite-bonded sand mould compaction process using micro-mechanical parameter relationships

ABSTRACT Commercial software packages for FEM analysis have been used to numerically simulate the behaviour of the complex systems of bentonite-bonded sand mould under pressure and subjected to stress distributions and to predict their performance. The Drucker-Prager model and the Mohr-Coulomb model are two well-known mathematical models used to describe the plastic non-linear behaviour of the soil. Conducting direct shear tests on varying densities of sand can provide the individual parameters necessary for the simulation of the moulding process. A new approach is based on making relationships between micro-mechanical parameters and changes in sand density during the compaction process. COMSOL Multiphysics is a popular software tool used to implement FEM simulations. The steps involved drawing geometry, inserting material properties, mesh generation and time-dependent density, and solving the model. The boundary conditions depend on the particular problem being analysed, which defines the external forces and constraints acting on the structure. The use of a coarse mesh and stationary study may be a computationally efficient approach for the evaluation of the compaction process of green sand. The study found that the maximum displacement value is 6.1*10−3 mm, the maximum volumetric strain value is 8.88*10−5, and the von Mises stress is 4.14*103 N/m2.

A two grid partition of unity finite element method for electrically conducting incompressible fluid flows

In this article, a two grid partition of unity finite element method is proposed and investigated for electrically conducting incompressible fluid flows. This algorithm involves solving a much smaller nonlinear problem on a coarse grid, utilizing a partition of unity to decompose reasonably the residual problem into a series of independent subproblems on a fine grid, and carrying out a further coarse correction on the coarse grid. Rigorously theoretical analysis is presented and convergence results indicate that the method could reach the optimal convergence orders with proper configurations between the coarse mesh size H and the fine mesh size h. Finally, some numerical results are reported to verify our theoretical findings.

A probabilistic peridynamic framework with an application to the study of the statistical size effect

The high computational expense of peridynamic models remains a major limitation, hindering ‘outer-loop’ applications that require a large number of simulations, for example, uncertainty quantification. This contribution presents a framework that makes such computations feasible. By employing a Multilevel Monte Carlo framework, where the majority of simulations are performed using a coarse mesh, and performing relatively few simulations using a fine mesh, a significant reduction in computational cost can be realised, and statistics of structural failure can be estimated. The maximum observed speed-up factor is 16 when compared to a standard Monte Carlo estimator, thus enabling the efficient forward propagation of uncertain parameters in a computationally expensive peridynamic model. Furthermore, the multilevel method provides an estimate of both the discretisation error and sampling error, thereby improving confidence in numerical predictions. The performance of the approach is demonstrated through an examination of the statistical size effect in quasi-brittle materials.

Practical considerations for the adoption of Anderson acceleration in nonlinear diffusion accelerated transport

The use of the Anderson Acceleration (AA) method within the context of the coarse mesh finite difference (CMFD)-accelerated neutron transport k-eigenvalue calculation is explored for practical problems. Current application of CMFD only has a linear convergence rate. The AA method is similar to a quasi-Newton method and can improve the convergence rate of fixed-point iterations. We therefore implement AA for the 2D calculation in MPACT and study its performance in pressurized water reactor (PWR) problems. The initial results show that the traditional AA method can improve the convergence rate in a trivial problem and improve the stability of the CMFD method for some problems. However, the success of AA in the practical problems, where simulations with symmetry or domain decomposition, relies on one essential factor–the treatment of iterative angular flux boundary conditions in the AA. Additionally, the efficiency of AA is also affected by the computational cost for solving the optimization problem, and the computational cost of the transport solver and the CMFD solver. To address these issues, we propose a new implementation of AA that considers only the mutligroup scalar flux in the optimization problem, but the angular flux and the scalar flux in the mixing. The new approach achieves better stability and efficiency in practical simulations, and can be explained by the theory of Randomized Numerical Linear Algebra (RandNLA).

Non-dimensionalization and scaling of fracture processes in concrete and rock

Many types of engineering structures may fracture, due to extreme external loads or due to inherent damages arising while in service or during their manufacturing/creation. The fractures may form at various scales, particularly within geological or cementitious materials like rock and concrete. Understanding and simulating accurately the fracturing processes at different scales, from laboratory to field, assists with the optimization of structural design and enhances reliability. This paper proposes novel non-dimensional scaling laws for fracture mechanics problems. They improve the prediction and interpretation of crack initiation and evolution and the subsequent mechanical response of structures. They have direct applicability in advanced numerical models like phase field models, where the governing parameters and mesh sizes can be appropriately scaled prior to conducting the numerical simulation. This leads to a unique dimensionless model output for any given loading configuration. Rescaling the unique non-dimensional output, which may incorporate varying pre-peak and post-peak fracture resistances, allows for the fracture behavior of a structure at any scale to be determined. The non-dimensionalization and rescaling techniques are demonstrated and validated against independent analytical and experimental benchmark problems. The scaling laws enable the adoption of coarse meshes in numerical models, significantly reducing computational cost while maintaining accuracy and capturing the fundamental mechanics. It is shown that the challenges associated with modeling fracture processes across scales can be mitigated as long as the problem is appropriately scaled.

An immersed boundary velocity correction method combined with virtual body-fitted grid for simulation of incompressible flows

In this work, a virtual body-fitted grid is introduced into the velocity correction-based immersed boundary method (IBM) to simulate incompressible flows. The impact of the immersed boundary is indirectly transmitted to the flow field via a virtual body-fitted grid. In this method, the fractional step technique consisting of the predictor and the corrector is adopted. The prediction step is executed on the Eulerian mesh, and the correction step is done on the virtual grid to fulfill the no-slip boundary condition. After the correction step, the corrected velocity field on the virtual grid is then assigned to that on the Eulerian mesh to update the flow field. Being able to adjust the grid spacing flexibly, the virtual body-fitted grid alleviates the shortcomings of the conventional IBM that uses the smooth Dirac delta function to associate Lagrangian points with their surrounding Eulerian points. As a result, the present method is easy to apply to non-uniform Cartesian grids, which is inapplicable to the conventional IBM with the smooth Dirac delta function. Numerical experiments concerning flow past a circular cylinder and a NACA0012 airfoil demonstrate the advantages of the present method, i.e., fewer Lagrangian points are required to avoid the streamline penetration of boundary and the range of “diffuse interface” can be narrowed by reducing the normal grid spacing of the virtual body-fitted grid to improve numerical results on a coarse mesh. In addition, an accuracy assessment on the decaying vortex problem reveals that the present IBM has a second-order accuracy.

In‐Plane Free Vibration Analysis by Using an Efficient Simple Plane Element

This article presents an in‐plane free vibration analysis using an efficient plane element based on strain states. In this element formulation, Taylor’s expansion of the strain field is exploited around the origin of the coordinates. The magnitudes of the strain gradient at the origin are referred to as strain states. The second‐order deformation field is calculated by assuming a linear strain field and using the relationship between displacement and strain. The number of unknowns in the formulation is reduced through the establishment of equilibrium equations. Shape functions are obtained by utilizing a dependency between the strain states and nodal displacement. The mass and stiffness matrices of the element are computed using these shape functions. Several benchmark tests are employed to demonstrate the efficiency of the element in free vibration analysis. The results of well‐established elements by other researchers are available in these tests. In some of these tests, the insensitivity of the elements’ results to distortion is evaluated in free vibration analysis. A comparison of the frequency answers of the elements shows that the element created based on strain state has high accuracy even in coarse distorted meshes. Also, the convergence rate of this element is investigated in the regular and distorted meshes. The accuracy of this element in the coarse mesh is significant. Furthermore, the sensitivity of the element created based on strain state to distortion is less than other elements.

A simplified fracture mechanics method for fatigue life analysis of weld roots

Volvo CE currently uses the effective notch stress method for assessing fatigue life in welded structures. For large structures such as frames, a sub modelling technique with a very fine mesh is necessary using millions of DOF. Stress tensors from unit load cases, estimated with FEA, are combined with time history loading from Multi Body Simulations of the complete machine operating on customer representative digitized test tracks. A critical plane approach (the Modified Wöhler Curve Method) is used to estimate the fatigue life. The usage of the effective notch method implies a very time-consuming design process. An LEFM approach for weld roots would be even more time-consuming but would be a more natural approach since the weld root consists of crack like defects. To overcome the drawbacks with the current design process that involves usage of the effective notch method, a hybrid method is defined using pre-calculated geometry factors for different crack sizes and weld geometries. The geometry factors as a function of crack size, and the weld geometry, are stored in a database. From the unit load cases the structural stress using a coarse mesh in the weld is extracted. Together with the geometry factors, the stress intensity factors may be estimated and used with Paris law to calculate the fatigue life of welds. A comparison between the effective notch method, LEFM and the new hybrid method is carried out for typical fillet welds is presented in this paper.

Hybrid Resolution Method for Efficient Depletion Analyses using Explicit Geometry Multigroup Monte Carlo

The hybrid resolution (HR) method is a recently developed calculational algorithm tailored for efficient nuclear reactor analysis by leveraging multigroup (MG) Monte Carlo (MC) coupled with continuous energy MC calculations. While the HR method has demonstrated promising results in various scenarios, an anticipated challenge arises from inconsistencies of MG MC results caused by spatial homogenization when relatively coarse mesh is adopted, particularly in heterogeneous problems characterized by a strong flux gradient. To address this issue, the HR method incorporates MG MC calculations on the same explicit geometry used in continuous energy MC calculations. In other words, this approach preserves accuracy in geometry handling while solely simplifying energy treatment. In our investigation, a 3-dimensional BWR-type fuel assembly was focused on for a comparative study regarding the different geometry treatments in MG MC calculations. Various reactor parameters such as k-effective, power, flux, and atomic densities were estimated, and numerical performance was also compared in terms of the computing time and figure-of-merits. Our evaluation revealed that the explicit geometry-based HR method yielded more consistent and reliable parameters compared to the simplified coarse mesh-based scheme, reducing errors in power and flux distributions by about 10%.

Efficient Mesh Space Mapping Optimization for Tunable Filters Incorporating Structurally Simplified Coarse Mesh Model Without Tunable Elements

Efficient Mesh Space Mapping Optimization for Tunable Filters Incorporating Structurally Simplified Coarse Mesh Model Without Tunable Elements

2D/3D GPU-accelerated deterministic neutronics solver based on planar method of characteristics and hybrid finite element method

The newly developed 2D/3D method operating over the unstructured mesh geometry is introduced. Implemented in the NuDEAL code, the method alternates the planar method of characterization (MOC) and the diffusion-based hybrid finite element (HFEM), with planar MOC solving the 2D decoupled fine mesh and HFEM providing 3D coarse mesh flux distributions based on the MOC solution. The consistency between the planar MOC and the diffusion HFEM is made with the devised consistency factors operating on volumetric and surface flux distributions. Furthermore, the consistent HFEM is fully accelerated using GPUs, which results in a 50-speedup in the red-black sweep compared to the runtime of the single CPU core. As a result, the accelerated HFEM takes less than 30 seconds to solve the multigroup benchmark problem, which is 20 times faster with a single gaming GPU than with a single CPU core.

Impact of Weirs on Decomposition Rates of Dombeya goetzenii (K. Schum) Leaf Litter in the River Njoro, Kenya

Water storage impoundments have deleterious effects on the structure and functioning of running water ecosystems. Many studies have been undertaken on the impacts of large impoundments on river ecosystems. However, comparatively few researches have determined the impact of small weirs on the functioning and structure of running water ecosystems. The main objective of the current study was to determine the impact of weirs on the decomposition rates of Dombeya goetzenii leaf litter in the Njoro River. The study was undertaken at five study sites located along the middle reaches of the Njoro River, near Egerton University, between November 2021 and January 2022. Four study sites were situated at upstream and downstream areas (i.e., approximately 10 m) of two weirs (i.e., Treetops and Canning weirs). A reference site was situated approximately 100 m upstream from the impoundment of the first weir. Physicochemical factors such as temperature, conductivity and nutrients (e.g., nitrates, ammonium) were determined at each study site. Additionally, decomposition rates of one of the most common tree species (i.e., Dombeya goetzenii) found growing along the Njoro River were determined. Leaf litter decomposition was determined using fine (100 µm) and coarse (500 µm) mesh litter bags and the negative exponential decay model was used to estimate decomposition rates (-k per day). The mean leaf litter decomposition rate was highest at the reference site (-k/day = 0.04) and lowest (-k/day = 0.02) at the upstream side of the first weir (i.e., Treetops weir).Leaf litter decomposition rates differed significantly (p = 0.002) between the aforementioned sites. Weirs had a significant local effect on the decomposition rates of Dombeya goetzenii leaf litter. The study provides crucial information to managers of riverine ecosystems about the impact of water storage weirs on an ecosystem function. The concerned government agencies should take action to prevent negative effects of weirs on riverine ecosystems.

Prediction of Mean Heat Transfer Characteristics of Multiple Impinging Jets with Steady RANS Simulation Using a Coarse Mesh

The capability of the standard SST k-ω turbulence model for the prediction of jet impingement cooling characteristics using a coarse mesh is investigated. The discussion is based on a sensitivity study with five computational grids, differing from each other in topology and resolution. The analysis considers a hexagonal configuration of turbulent jets at the inlet Reynolds number equal to 20,000, with the distance between the nozzle and target plates equal to four nozzle diameters. The results of steady RANS simulations are validated against the time-averaged LES results and data from experiments. The mean heat transfer characteristics of turbulent impinging jets have been successfully reproduced with all tested grids, which indicates that for a rather accurate mean heat transfer prediction, it is not necessary to resolve all the small-scale flow features of impinging jets above the target plate.

Intelligent active flow control of long-span bridge deck using deep reinforcement learning integrated transfer learning

Aerodynamic forces of the Great Belt Bridge are mitigated using deep reinforcement learning (DRL) based active flow control (AFC) techniques at a Reynolds number of 5×105 in this study. The flow control method involves placing a suction slit at the bottom of the trailing edge of the bridge. The DRL agent as a controller is able to optimize the velocity of the suction to reduce the fluctuating coefficients of bending moment, drag, and lift by 99.1%, 73.7%, and 95.8% respectively. To reduce the high computational demands associated with DRL-based AFC training, this study integrates transfer learning technique into DRL, which calls transfer learning based deep reinforcement learning (TL-DRL) method. Specifically, the transfer learning method based on a DRL pre-trained model trained with a coarse mesh scheme is implemented. Results indicate that with TL-DRL method, the training cost can be reduced by 53%, while achieving the same control strategy and control effect as the DRL training from scratch. This study shows that the TL-DRL based flow control method is highly effective in reducing aerodynamic forces on long-span bridges. Furthermore, TL-DRL based flow control approach can adapt flexibly to different flow field environments, ultimately enhancing energy utilization efficiency.

The deterministic neutron transport calculation for the HTR-PM by the three-dimensional method of characteristics code ARCHER

Due to the strong geometric adaptability, the Three-Dimensional (3-D) Method Of Characteristics (MOC) is a promising candidate for 3-D whole-core high-fidelity neutron transport calculations. However, the 3-D MOC can hardly be applied to large-scale full core problems on account of the huge computational costs in memory and time. In the past, the 3-D MOC was mainly used in traditional Light Water Reactor (LWR) calculations, but rarely used for more complex core calculations. For pebble-bed High Temperature gas-cooled Reactors (HTRs), most 3-D MOC codes are out of ability to construct complex pebble beds, let alone efficient acceleration methods. The 3-D MOC code ARCHER can efficiently simulate the large-scale pebble-bed HTRs through using the Linear Source Approximation (LSA), the Coarse Mesh Finite Difference (CMFD) and the hybrid MPI-OpenMP parallel. In this work, the two-level CMFD acceleration and efficient preconditioners on Krylov subspace linear solvers are implemented in the ARCHER code in order to achieve better performance. In addition, the practical HTR-PM criticality problem is calculated by ARCHER, which is the world’s first high-fidelity neutron transport solution of large-scale pebble-bed HTRs by deterministic numerical method. It also shows that the 3-D MOC has great application potential in other complex reactors.

Collapsing Embedded Cell Complexes for Safer Hexahedral Meshing

We present a set of operators to perform modifications, in particular collapses and splits, in volumetric cell complexes which are discretely embedded in a background mesh. Topological integrity and geometric embedding validity are carefully maintained. We apply these operators strategically to volumetric block decompositions, so-called T-meshes or base complexes, in the context of hexahedral mesh generation. This allows circumventing the expensive and unreliable global volumetric remapping step in the versatile meshing pipeline based on 3D integer-grid maps. In essence, we reduce this step to simpler local cube mapping problems, for which reliable solutions are available. As a consequence, the robustness of the mesh generation process is increased, especially when targeting coarse or block-structured hexahedral meshes. We furthermore extend this pipeline to support feature alignment constraints, and systematically respect these throughout, enabling the generation of meshes that align to points, curves, and surfaces of special interest, whether on the boundary or in the interior of the domain.