Coarse Mesh Research Articles

An adaptive continuous–discontinuous approach for the analysis of phase field fracture using mesh refinement and coarsening schemes and octree-based trimmed hexahedral meshes

An adaptive continuous–discontinuous approach for the analysis of phase field fracture using mesh refinement and coarsening schemes and octree-based trimmed hexahedral meshes

Effect of Mesh Coarseness on Slope Stability Analysis Using 2D and 3D Finite Element Method

A comprehensive understanding of slope stability is essential for ensuring the safety and durability of structures built on or near slopes and mitigating the risks associated with landslides and slope failures. Slope stability is typically evaluated using the factor of safety (FOS) based on the critical slip surfaces. The calculation of FOS is commonly executed using Limit Equilibrium Method (LEM) by dividing the slope into several vertical slices. However, the stability analyses using Finite Element Method (FEM) have gained significant attention in geotechnical engineering due to their ability to simulate slope behaviour and predict stability accurately by employing mathematical models and computational algorithms. Hence, this paper aims to analyse the FOS of the unreinforced slope using 2D FEM and 3D FEM conducted through computer software while examining the influence of different mesh coarseness. Besides that, the formation of critical slip surfaces and the displacement behaviour of the slope are also presented. A slope geometry model was analysed using PLAXIS 2D and PLAXIS 3D with different mesh coarseness. The findings were compared and discussed. The findings reveal that the values of FOS generated by 3D FEM are slightly larger compared to 2D FEM analysis, ranging from 1.27% to 2.56%. On the other hand, the effect of mesh coarseness indicates that coarser mesh sizes yield higher FOS values compared to finer mesh sizes. The shape, location and depth of the critical slip surfaces are consistent for each analysis in both methods. However, the maximum displacement values differ for each mesh coarseness, as the locations of maximum total displacement are identified at different nodes due to varying numbers of elements but still within the same potential failure zone. Overall, this comparative study is crucial in ensuring the validity of the performed analyses. Understanding the capabilities and limitations of 2D and 3D numerical analyses to achieve reliable and accurate results is important to balance mesh coarseness and computational efficiency.

Directional enrichment functions for finite element solutions of transient anisotropic diffusion

The present study proposes a novel approach for efficiently solving an anisotropic transient diffusion problem using an enriched finite element method. We develop directional enrichment for the finite elements in the spatial discretization and a fully implicit scheme for the temporal discretization of the governing equations. Within this comprehensive framework, the proposed class of exponential functions as enrichment enhance the approximation of the finite element method by capturing the directional based behaviour of the solution. The incorporation of these enrichment functions leverages a priori knowledge about the anisotropic problem using the partition of unity technique, resulting in significantly improved approximation efficiency while retaining all the advantages of the standard finite element method. Consequently, the proposed approach yields accurate numerical solutions even on coarse meshes and with significantly fewer degrees of freedom compared to the standard finite element methods. Moreover, the choice of mesh coarseness remains independent of the anisotropy in the problem, enabling the use of the same mesh regardless of changes in the anisotropy. Using extensive numerical experiments, we consistently demonstrate the efficiency of the proposed method in attaining the desired levels of accuracy. Our approach not only provides reliable and precise solutions but also extends the capabilities of the finite element method to effectively address aspects of the heterogeneous anisotropic transient diffusion problems that were previously considered ineffective when using this method.

Seismic collaborative reliability analysis for a slope considering spatial variability base on optimized subset simulation

Seismic reliability analysis of actual slopes considering the spatial variability of soil materials is crucial. However, for the discretization of large-scale random fields, high-precision finite element analysis and analysis of small failure probability events, the random analysis of slopes under seismic loads is inefficient. To address this situation, this study proposes a collaborative reliability analysis framework based on the modified linear estimation method (MLEM) and optimized subset simulation (OSS). First, the random field of the uncertain parameters of the Jinping-I left bank slope model is efficiently discretized by the MLEM, and a sensitivity analysis is carried out. Then, considering the adoption of different degrees of cross-correlation of the sensitive random parameters, the OSS method is used to perform random finite element analysis on the coarse mesh model. Finally, the fine mesh samples are obtained according to the response conditioning method (RCM). The MLEM is used to ensure the consistency of the two sets of random fields, and the seismic failure probability and reliability index of the slope under different cross-correlation coefficients of uncertain parameters are obtained. The results suggest that the degree of cross-correlation of parameters has a great influence on the seismic reliability of the slope. Considering that the shear strength parameters of geotechnical materials are often negatively correlated, the fine analysis based on a fine model is necessary.

Multiscale Modeling of Flow in Rod Bundles: From Direct Numerical Simulation and Subchannel to Coarse-Mesh CFD

Fast and accurate evaluation of flow and heat transfer phenomena in rod bundles is a problem of long-standing interest in nuclear engineering. Computational fluid dynamics (CFD) can provide accurate but relatively time-intensive estimates, such that simulations of very long transients with high spatial detail are infeasible. On the other hand, subchannel codes require relatively low computational time and can provide pin-level estimates, but have substantial empiricism in evaluating aspects such as crossflows and turbulent mixing coefficients. A multiscale method (SC+) for bridging this accuracy/speed gap, based on the Subchannel CFD (SubChCFD) method of Liu et al. [Nucl. Eng. Design, Vol. 355, paper 110318 (2019)], is demonstrated and developed here with a focus on a 5 × 5 square rod bundle geometry. The method has been newly implemented into the commercial code STAR-CCM+ and benchmarked against Liu et al.’s data. The 5 × 5 geometry was chosen in part due to the availability of direct numerical simulation (DNS) data at a relevant Reynolds number of a similar configuration for comparison. The SC+ results are variously compared against results from DNS, large eddy simulation, wall-resolved Reynolds-averaged Navier-Stokes, and coarse-mesh CFD methods. As an expansion to the original SubChCFD approach, a simple Hi2Lo approach is demonstrated using the DNS data as a correction to the original friction factor correlations employed. This is verified to improve the predictions. Additional test cases with geometric perturbations are pursued, illustrating the flexibility of SC+. The potential of this method for modeling the narrow gap vortex instability, which would represent an advancement over standard subchannel approaches, is also assessed. The method is expanded to include transverse flow losses, which was found to improve the results for modeling the gap instability. Initial extensions of SC+ for hexagonal rod bundles are also presented; some inaccuracies for the coarsest meshes prompted a detailed investigation of the mesh convergence behavior of the method. Geometric correction factors were devised that provided substantial improvement on these very coarse meshes, improving the prospects of SC+ for wider usage. Future work plans are to expand the methodology to wire-wrapped rod bundles and to implement the method into Pronghorn, with a unified pipeline via the Cardinal wrapper between the codes NekRS, Pronghorn, and BISON, to solve fuel performance problems of direct interest to industry.

Fully higher-order coupling of finite element and level set methods for two-phase flow with a new explicit projection method

This work presents the development of a higher-order finite element method for modeling unsteady, incompressible and inviscid two-phase flows using the level set method. A new explicit projection method is introduced to solve the incompressible Euler equation, where the pressure and the velocity fields are solved separately. To implement high order temporal schemes, the Hodge decomposition in this projection method is not applied to the velocity but to its time derivative. A discontinuous Galerkin discretization is used to solve both the level set and the nonlinear advection terms present in the Euler equations, while a Galerkin discretization is used to solve the pressure Poisson equation. The present method is a fully high order numerical approach, both in space and temporal dimensions. Numerical results provided by several 2D benchmark test cases in naval hydrodynamics (i.e hydrostatic equilibrium, sloshing and dam break) validate the computational code, illustrating the power of the proposed method. Compared to the CFD codes based on low order VOF/finite-volume methods, the present approach demonstrates greater accuracy on coarse meshes, leading to reduced CPU time.

Exact domain truncation for the Morse-ingard equations

Morse and Ingard [1] give a coupled system of time-harmonic equations for the temperature and pressure of an excited gas. These equations form a critical aspect of modeling trace gas sensors. Like other wave propagation problems, the computational problem must be closed with suitable far-field boundary conditions. Working in a scattered-field formulation, we adapt a nonlocal boundary condition proposed in [2] for the Helmholtz equation to this coupled system. This boundary condition uses a Green's formula for the true solution on the boundary, giving rise to a nonlocal perturbation of standard transmission boundary conditions. However, the boundary condition is exact and so Galerkin discretization of the resulting problem converges to the restriction of the exact solution to the computational domain. Numerical results demonstrate that accuracy can be obtained on relatively coarse meshes on small computational domains, and the resulting algebraic systems may be solved by GMRES using the local part of the operator as an effective preconditioner. These numerical results taken together combine several advanced techniques, including higher-order finite elements, geometric multigrid in curvilinear geometry, native use of complex arithmetic, and incorporation of nonlocal operators. These are tied together in a high-level simulation using the Firedrake library.

Influence of vapor core design on thermal and hydraulic performance of mesh-type ultra-thin vapor chamber

The mesh-type ultra-thin vapor chamber(UTVC), with an internal mesh porous structure, is a cost-effective and efficient heat transfer element for compact spaces. The design of the vapor core plays an important part in enhancing the heat transfer performance of the mesh-type UTVC within the confined space requirements. The present study proposes to design cavities in the supporting coarse mesh inside the vapor core to form a multi-scale combination of vapor channels with the mesh channels. Utilizing a numerical model that considers microporous evaporation as well as the Marangoni effect, the effects of different flow channel designs and designed channel widths on the performance of the UTVC were analyzed. The study determined the flow and heat transfer characteristics of the working medium within the mesh-type UTVC, and found that the most significant performance enhancement was achieved when the designed flow channel passed through part of the UTVC evaporation and condensation sections, reducing the thermal resistance by 17.45%. Whereas the design flow channel width has less influence on the performance of mesh-type UTVC, it satisfies the changing trend of vapor flow pressure drop and thermal resistance decreasing firstly and subsequently increasing with the expansion of the designed channel width.

Robust domain decomposition methods for high-contrast multiscale problems on irregular domains with virtual element discretizations

Our research focuses on the development of domain decomposition preconditioners tailored for second-order elliptic partial differential equations. Our approach addresses two major challenges simultaneously: i) effectively handling coefficients with high-contrast and multiscale properties, and ii) accommodating irregular domains in the original problem, the coarse mesh, and the subdomain partition. The robustness of our preconditioners is crucial for real-world applications, such as the efficient and accurate modeling of subsurface flow in porous media and other important domains.The core of our method lies in the construction of a suitable partition of unity functions and coarse spaces utilizing local spectral information. Leveraging these components, we implement a two-level additive Schwarz preconditioner. We demonstrate that the condition number of the preconditioned systems is bounded with a bound that is independent of the contrast. Our claims are further substantiated through selected numerical experiments, which confirm the robustness of our preconditioners.

Improved well-balanced AWENO schemes with hydrostatic reconstruction for the Euler equations under gravitational fields

The Euler equations under gravitational fields allow the hydrostatic equilibrium states, which requires that the numerical scheme of the system should also have this characteristic. In our previous work, a well-balanced finite difference conservative AWENO scheme has been constructed to preserve the isothermal equilibrium state accurately (Fu et al., Appl. Numer. Math., 180:1-15, 2022). However, the scheme fails to maintain more complicated hydrostatic equilibrium states, such as the isentropic equilibrium state. In this study, the improved well-balanced AWENO schemes with hydrostatic reconstruction are designed to maintain the general equilibrium states numerically. The key technologies to achieve the well-balanced property are rewriting the source term into an appropriate form based on the equilibrium equation, acting on the conserved variables with an affine-invariant WENO or linearized WENO interpolation operator, employing hydrostatic reconstruction methods to create auxiliary conserved variables, and modifying the numerical flux with auxiliary conserved variables and higher-order conserved terms based on pressure. The proposed well-balanced schemes maintain the equilibrium solution near the machine precision and capture small perturbations of the equilibrium state without spurious oscillations even at the coarse mesh. Furthermore, they can be applicable to the general hydrostatic equilibrium more than the isothermal equilibrium state. Several numerical experiments are tested to show that the developed well-balanced schemes perform well in maintaining equilibrium states and capturing small perturbations.

An SPIM-FEM adapting coupling approach for the analysis of quasi-brittle media

This paper presents an adaptive coupling approach between meshless Smoothed Point Interpolation Methods (SPIMs) and the Finite Element Method (FEM) for the physically nonlinear analysis of quasi-brittle media. The nonlinear behaviour is represented by scalar damage and smeared-crack models. In the proposed adaptive coupling approach, the domain is initially discretised with a relatively coarse FEM mesh. During the solution process, FEM patches are replaced by meshless discretisations according to certain selection criteria. Refinement is also considered.

A two-level finite element method with grad-div stabilizations for the incompressible Navier–Stokes equations

This article presents and studies a two-level grad-div stabilized finite element discretization method for solving numerically the steady incompressible Navier–Stokes equations. The method consists of two steps. In the first step, we compute a rough solution by solving a nonlinear Navier–Stokes system on a coarse grid. And then, in the second step, we pass the coarse grid solution to a fine grid to linearize the nonlinear term, update the solution by solving a linearized problem based on Newton iterations. In both steps, a grad-div stabilization term is incorporated into the system to reduce the influence of pressure on the approximate velocity. We analyze stability and asymptotic convergence of the approximate solutions, derive explicit dependence of the solution errors on the grad-div stabilization parameter and viscosity. We perform also some numerical tests to validate the theoretical analysis and illustrate the efficiency of the proposed method. Compared with the standard two-level method without stabilizations, the grad-div stabilization term added in present method improves the accuracy of the approximate velocity, accelerates the convergence of the nonlinear iterations for the coarse mesh nonlinear system, and reduces the computational time.

An interface tracking, finite volume code for modeling axisymmetric implosion of a rotating liquid metal liner with free surface

We present Integrated System Model-hydrodynamics (ISM-hydro)—an interface tracking, finite volume code for modeling a shaped implosion of a rotating, initially cylindrical, fluid shell (liner) with a free surface. The code is a novel implementation of the mixed Lagrangian–Eulerian approach, applied to a compressible fluid in an axisymmetric geometry described by cylindrical coordinates (r, ϕ, z). In ISM-hydro, a structured quadrilateral mesh follows fluid elements in the r-direction (radially Lagrangian) and is fixed in the z-direction (axially Eulerian). This approach accurately captures the motion of the liner's free surface, making it an interface tracking method. Using this mesh, we derive a finite volume discretization of the axisymmetric Euler equations for a rotating compressible fluid that has an exact balance of kinetic energy. An extensive comparison between ISM-hydro and the open-source software OpenFOAM is presented; results for different test cases show very good agreement in simulated implosion trajectories and flow fields. ISM-hydro is the purely hydrodynamic component of the Integrated System Model (ISM), a framework developed at General Fusion (GF) for comprehensive predictive modeling of GF's magnetized target fusion (MTF) scheme, where an imploding rotating liquid metal liner compresses a magnetized plasma target to fusion conditions. Among advantages of the code is its speed: a full implosion simulation with a coarse mesh takes on the order of one minute on a single core while preserving high accuracy. This makes ISM-hydro a valuable tool for the design optimization of GF's MTF machines.

A thermodynamically consistent strain difference-based nonlocal damage mechanics approach for failure analysis of quasi-brittle materials

The nonlocal integral approaches are established methods known for their effectiveness in dealing with the challenges of ill-posed behaviour and mesh sensitivity during the strain softening process, especially within classical continuum theories. The present work extends the strain difference-based nonlocal approach to continuum damage mechanics (CDM) theories for predicting the failure behaviour of quasi-brittle materials. The strain difference approach is adopted for two reasons: (i) it maintains physical realism and indirectly incorporates nonlocal effects while computing both the damage variable and the damage energy release rate, and (ii) the constitutive model maintains a symmetry nature in constructing the nonlocal part of the stiffness matrix, reducing the computational cost in strain-softening problems. The state and damage evolution equations for the present nonlocal approach are formulated within the thermodynamic framework. Numerical examples involving 2D and 3D cases are solved: (i) to study the influence of nonlocal parameter ‘α’ on the damage distribution while capturing the post-peak behaviours, (ii) to demonstrate the effectiveness of the present methodology by obtaining mesh-independent results and accurately predicting damage propagation paths, and (iii) to compare the experimental results and numerical structural response for coarse mesh, highlighting both computational efficiency and reduction of mesh-bias dependency. The results obtained by the strain difference-based nonlocal damage model agree well with the experimental results in the literature.

Advanced Mesh Space Mapping Approach With Fast Coarse Mesh Models Comprising Sharpening Structural Processing and Mesh Deformation

Advanced Mesh Space Mapping Approach With Fast Coarse Mesh Models Comprising Sharpening Structural Processing and Mesh Deformation

Application of a hierarchical quadrature element method to interface crack analysis in bi-material systems

This paper presents an extension of the hierarchical quadrature element method (HQEM) with p-convergence to evaluating fracture mechanical parameters of bi-material interface cracks. The numerical implementation of HQEM in combination with the virtual crack closure method (VCCM) is comprehensively illustrated, and the formulas for calculating the strain energy release rate (SERR) are rederived. The relationship between components of SERR and the complex stress intensity factors (SIFs) is established. In addition, the effectiveness of an auxiliary parameter related to the dual energy release rates in measuring the accuracy of the present method is examined.The feasibility of applying HQEM to the interface crack problems is demonstrated by investigating the benchmark problem of an infinite bi-material plate with a central interface crack. Two loading cases are considered, including pure tensile loading and mixed tensile-shear loading. The influence of the crack tip element size Δa on the convergence characteristics and computational accuracy of fracture parameters is quantitatively investigated to provide guidance for choosing the most reasonable value of Δa. Numerical results show that even with very coarse meshes, together with a virtual crack extension consisting of only one element, HQEM yields highly accurate results for the total SERR and the complex SIFs, making it suitable for complex, large-scale interface crack problems in dissimilar media.

Aircraft Simulations Using the New CFD Software from ONERA, DLR, and Airbus

This paper presents a thorough comparison between RANS simulations performed with the computational fluid dynamics by ONERA, DLR, and Airbus (CODA) new-generation flow solver and reference legacy codes TAU (DLR) and elsA (ONERA) for high-speed cruising NASA Common Research Model (CRM) configurations that were considered in the context of the 5th, 6th, and 7th Drag Prediction Workshops. The solver’s accuracy is assessed with several meshing strategies, including block-structured, hybrid-structured/unstructured, and fully unstructured tetrahedral meshes. This solver features both a cell-centered finite-volume (FV) scheme suited to arbitrary meshes as well as a modern high-order discontinuous Galerkin (DG) scheme. We show that for all cases considered, the FV component recovers an equivalent accuracy compared to elsA (cell-centered FV) on block-structured meshes and TAU (node-centered FV) on hybrid and unstructured meshes. The high-order DG scheme (third-order accurate) is found to significantly enhance the drag prediction on coarse meshes compared to legacy FV methods, both for structured and unstructured meshes.

Effects of Soil Fauna on the Home-Field Advantage of Litter Total Phenol and Condensed Tannin Decomposition

Soil fauna play a vital role in contributing to the home-field advantage (HFA: litter decomposes faster in its natural habitat than elsewhere) during litter decomposition. Whether the presence of soil fauna affects the HFA of the decomposition of total phenols and condensed tannins, which are important components of litter, has rarely been investigated. In this study, litterbags with different mesh sizes were transplanted reciprocally, 0.04 mm (basically excluding soil fauna) and 3 mm (basically allowing all soil fauna to enter), in Lindera megaphylla and Cryptomeria fortunei forests. The results illustrated that the loss rates of total phenols and condensed tannins reached 64.07% to 84.49% and 69.67% to 88.37%, respectively, after 2 months of decomposition. Moreover, soil fauna positively contributed to the decomposition of condensed tannins in high-quality litter. After 2 months of decomposition, a significantly positive HFA (HFA index: 10.32) was found for total phenol decomposition in the coarse mesh, while a significantly negative HFA (HFA index: −1.81) was observed for condensed tannin decomposition in the fine mesh after 10 months of decomposition. Polyphenol oxidase (PPO) and peroxidase (POD) activities were significantly influenced by litter types. The loss rates of total phenols and condensed tannins were significantly negatively correlated with the initial N content, P content, N/P ratio, and POD activity and were positively related to the initial C content, total phenol content, condensed tannin content, C/P ratio, and C/N ratio. Only the loss of condensed tannins was negatively correlated with PPO activity (after 2 months’ decomposition). However, none of these correlations were observed after 10 months of decomposition. Our study illustrated that (1) soil fauna contributed to the decomposition of total phenols and condensed tannins but were influenced by litter type for condensed tannins. (2) The soil fauna had inconsistent effects on the HFA of total phenols and condensed tannins, possibly due to the combined regulatory effects of environmental context, litter quality, and rapid decomposition rates. In sum, the results indicated that soil fauna played an important role in the decomposition of condensed tannins and total phenols in litter, and additional studies on the effects of soil faunal abundance and class on HFA of condensed tannins and total phenols are needed.

The phase-field fracture model enriched by interpolation cover functions for brittle fracture problems

In this study, interpolation cover functions were introduced into the phase-field fracture model (PFM) to improve its convergence and efficiency. The traditional PFM has proven to be successful in simulating brittle fracture problems. However, one disadvantage is that it requires a relatively dense mesh. The PFM model enriched by interpolation cover functions (EPFM) was used for numerical fracture analysis under a relatively coarse mesh. The rank deficiency problem caused by the introduction of interpolation cover functions was resolved using the rank deficiency repair technique. Higher gradients were captured by introducing interpolation cover functions without changing the mesh structure. The numerical examples showed the improvement in convergence. The high efficiency of the proposed EPFM was demonstrated through a series of numerical examples. The numerical results of the EPFM and PFM under different length-scale parameters and meshes were also compared. The results showed that the proposed EPFM is an excellent alternative for evaluating engineering fracture problems.

Free vibration analysis employing strain gradient notation four-node plane elements and considering the effects of parasitic shear

In this work, plane stress free vibration is performed using four-node elements formulated using strain gradient notation. Accurate results are sought, and the effects of parasitic shear are investigated. By applying strain gradient notation, which is a physically interpretable notation, parasitic shear terms are identified a priori in the shear strain expressions of the four-node plane element. Parasitic shear is the source of shear locking in that element, and the a priori removal of the parasitic shear terms corrects the element for locking. Several studies in the literature show that this modeling error causes improper behavior of the element in bending, which affects the computed results of stress and displacements of the structure. Based on this, this work applies strain gradient notation to examine the effects of parasitic shear on the natural frequencies and corresponding mode shapes of plane stress structures. The results are compared with literature results, and with the isoparametric FEM formulation modeled by ANSYS software. It is observed that parasitic shear has a higher preponderance in coarse meshes of flexural vibration modes. Furthermore, the deleterious effects are emphasized as bending deformation becomes more important in the model. Strain gradient notation shows advantages in the numerical aspect and successfully eliminates the parasitic shear terms from the four-node element. The problems studied here display accurate results.