Mesh Discretization Research Articles

A new spline method on graded mesh for fourth-order time-dependent PDEs: application to Kuramoto-Sivashinsky and extended Fisher-Kolmogorov equations

In this research, we introduce a two-tier non-polynomial spline approach with graded mesh discretization for addressing fourth-order time-dependent partial differential equations, which find applications in various physical scenarios like the nonlinear Kuramoto-Sivashinsky equation and extended Fisher-Kolmogorov equation. Our method involves considering three spatial points at each time step for scheme development, achieving spatial accuracy of three and temporal accuracy of two. Notably, our approach offers the advantage of directly tackling singular biharmonic problems without the need for discretizing boundary conditions or incorporating fictitious points. It demonstrates unconditional stability when tested on fourth-order problems and outperforms previous methods, as evidenced by comparative analyses. Additionally, we present 2D and 3D plots illustrating numerical solutions for different benchmark scenarios, including the depiction of the chaotic nature of the Kuramoto-Sivashinsky equation under Gaussian initial conditions at various time levels.

A multiscale steel–concrete interface model for structural applications

In this paper, a new multiscale macro-element formulation for the steel–concrete interface modeling is proposed. This element allows for the representation of the behavior of steel and the interface zone surrounding it. It can also model the interfacial bond stresses in between. Compared to conventional interface models of the literature, which employ separate mesh elements for the steel and the interface, utilizing a macro-element to model both components simplifies the creation of a reinforced concrete structure mesh. Moreover, the macro-element equilibrium is solved using a sub-structuring method that aims to reduce the computational cost. At the global level, it is considered as a four-node element linked to two-dimensional and three-dimensional concrete elements. At the local level, an assembly of multiple three-node bar elements with bond stresses is performed. An inner mesh discretization is therefore possible at the local level independently of the global level. The coupling between the two modeling scales is done using a static condensation technique. The formulation of the macro-element is presented in this paper. A selection of numerical examples is provided. The presented applications demonstrate the robustness of the proposed interface model and its capacity to reproduce the experimental behavior of reinforced concrete structural elements.

Modeling of interfacial multi-cracks in dissimilar laminated structures using a nodal-based Lagrange multiplier/cohesive zone approach

Recent decades have witnessed a growing interest in simulating interfacial debonding in laminated structures using cohesive zone (CZ)-based algorithms. In the present work, we develop a nodal-based Lagrange multiplier/cohesive zone (LM/CZ) approach to achieve the end with a special focus on interfacial multi-cracks in dissimilar laminated structures. The main advantages of the developed approach are four-folded: (1) being flexible for the mesh discretization of dissimilar laminated structures and compatible with various element types; (2) being computationally more efficient for dissimilar laminated structures; (3) addressing the so-called artificial compliance issue arising in intrinsic CZ modeling; (4) providing smooth transitions from uncracked to cracking states, and from cracking to contact states. In the presented approach, nodal pairs are constructed with the aid of a searching algorithm capable of dealing with matching and non-matching meshes in a unified manner. For each nodal pair, the displacement continuity is accurately enforced using LMs that are calculated in a predictor–corrector manner with a limited number of iterations. The constraints for nodal pairs are switched from LMs to cohesive forces after a defined criterion is satisfied. The non-penetration condition between interfaces is fulfilled using penalty-based contact forces in the cracking and pure contact states, leading to a smooth transition from cracking to contact states. The accuracy and effectiveness of the presented approach are validated using three benchmark tests. Finally, the capacity of the approach to describe interfacial multi-crack behaviors of dissimilar laminated structures has been exploited and demonstrated.

Optimal design of electro-thermo-mechanical microactuators considering minimum length scale constraints

To ensure the manufacturability of compliant mechanisms, this paper presents a new approach for optimal design of electro-thermo-mechanical microactuators by employing minimum length scale constraints. A sequential coupling method is adopted to carry out finite element analysis for electric-thermal-mechanical coupling multiphysics. The inflection point fields of the solid and void phases are identified by structural indicator functions. The minimum length scales of the solid and void phases are adopted as constraints. The optimization objective is designed to maximize the output displacements of electro-thermo-mechanical microactuators. The validity of the proposed optimal design method is demonstrated through several numerical examples. In the optimal designs obtained by the proposed method, the minimum length scales of the two phases can be properly controlled. The electro-thermo-mechanical microactuator’s output displacements decrease when the allowable value of the minimum length scale is increased. The effects of different mesh discretizations and output spring stiffnesses on the optimized designs are discussed.

Development of Dispersion Calculator SAFEDC of Guided Wave with Complex Material and Waveguide

The propagation characteristics of guided waves (GW) are related to both the material, loading and geometry parameters. Correct understanding of GW is the key step towards further GW-based structural health monitoring. A dispersion calculator of GW propagating in the complex geometry with viscoelastic and anisotropic material parameter in the open waveguide is developed based on the semi-analytical finite element (SAFE) method and perfectly matched layer (PML). By assuming the simple harmonic motion along the propagation direction and adopting the mesh discretization in the cross section, SAFE features the capability of GW calculation in the complex waveguide at only a cost of limited calculation resources. Hence, besides the capabilities of the most off-the-shelf tools focusing on the analytical solution of GW in simple plate or pipeline, the developed algorithm further offers the solution of GW in open waveguide with laminates of arbitrary ply stacking sequences and in arbitrarily complex cross section. Quadratic one-dimensional and triangular elements are adopted to automatically discretize the plate structure and complex cross section, respectively. The obtained results of GW in plates are in a good agreement with the results obtained with the off-the shelf-tools that adopt analytical derivations, and show stronger robustness of mode searching and sorting by avoiding the complex root-finding procedure required in the analytical derivation. The most GW features, including phase velocity, group velocity, attenuation, wave structure in terms of displacement, stress, and strain, and animation of wave propagation are directly calculated. All these functions are integrated and updated in the previously developed freeware SAFEDC, whose 1st version SAFEDC v1.0 was published in September 2023.

Numerical approximation of parabolic singularly perturbed problems with large spatial delay and turning point

PurposeSingular perturbation turning point problems (SP-TPPs) involving parabolic convection–diffusion Partial Differential Equations (PDEs) with large spatial delay are studied in this paper. These type of equations are important in various fields of mathematics and sciences such as computational neuroscience and require specialized techniques for their numerical analysis.Design/methodology/approachWe design a numerical method comprising a hybrid finite difference scheme on a layer-adapted mesh for the spatial discretization and an implicit-Euler scheme on a uniform mesh in the temporal variable. A combination of the central difference scheme and the simple upwind scheme is used as the hybrid scheme.FindingsConsistency, stability and convergence are investigated for the proposed scheme. It is established that the present approach has parameter-uniform convergence of OΔτ+K−2(lnK)2, where Δτ and K denote the step size in the time direction and number of mesh-intervals in the space direction.Originality/valueParabolic SP-TPPs exhibiting twin boundary layers with large spatial delay have not been studied earlier in the literature. The presence of delay portrays an interior layer in the considered problem’s solution in addition to twin boundary layers. Numerical illustrations are provided to demonstrate the theoretical estimates.

Homogenized moduli and local multiphysics fields of unidirectional piezoelectric nanocomposites with energetic surfaces

The study extends the locally-exact homogenization theory (LEHT) to determine the effective parameters and localized multiphysics fields of unidirectional piezoelectric nanocomposites with energetic surfaces. The interface is simulated using the generalized Gurtin-Murdoch model for piezoelectric nanocomposites, taking into account the discontinuities between surface stresses and electrical displacement. By characterizing hexagonal and square repeating unit cells (RUC), the interactions across a fiber's or pore's interface in piezoelectric nanocomposites with energetic surfaces have been analyzed. The extended LEHT possesses the advantages of fast convergence, excellent stability, and computational efficiency due to its avoidance of mesh discretization and pointwise satisfaction of interfacial conditions. To verify the validity of the present theory, this paper also derives the Eshelby solution for piezoelectric nanocomposite with energetic surfaces. The reliability and accuracy of the present theory are demonstrated by comparing the present results from the Eshelby solution and the finite element method (FEM) with generally good agreement. On this basis, the influence of microstructural effects, such as phase volume fractions and geometrical arrangement, on the effective and local responses of piezoelectric nanocomposites with energetic surfaces are investigated. The results show that effective piezoelectric/dielectric moduli and electric displacement field for piezoelectric nanocomposites are significantly affected by the size effect, along with several other parameters such as unit cell arrays and fiber/pore volume fractions. Those results highlight the significance of neighboring pore or fiber interactions for piezoelectric nanocomposites, which are often overlooked in traditional micromechanics models and are challenging to be captured using numerical methods.

Extended finite element method with cell-based smoothing for modeling frictional contact crack-induced acoustic nonlinearity involving distorted mesh

This study proposes a 2D cell-based smoothed extended finite element method (CS-XFEM) for accurate and efficient simulation of nonlinear ultrasonic wave propagation in solid structures, specifically addressing the effects of frictional contact in cracks. Traditional mesh discretization methods for cracks often suffer from mesh distortion and computational instability owing to their high aspect ratios. To overcome this, CS-XFEM integrates a cell-based smoothing technique into XFEM to model the frictional contact of a crack. A comprehensive numerical example demonstrates the advantages of CS-XFEM. The results show that CS-XFEM exhibits a higher convergence rate and enables a larger critical time increment than XFEM. Specifically, the critical time increment of CS-XFEM was found to be twice that of XFEM, leading to a 50% reduction in the total computational time. These findings confirm that CS-XFEM is an efficient, accurate, and robust numerical method for studying the acoustic nonlinearity induced by crack-induced frictional contact.

Mixed convection characteristics in a long horizontal lid‐driven channel with periodically distributed local flow modulators

AbstractThis study explores the effectiveness of periodically placed rotating blades in enhancing heat transfer in a channel. The channel consists of a cold top plate moving at a constant speed and a fixed hot plate at the bottom. Thin rotating blades are placed periodically along the channel's centerline, with the spacing between their axes equal to the channel's height. This paper analyzes a transient, two‐dimensional, laminar flow problem using energy, momentum, and continuity equations. To address the challenges posed by moving blades, the Galerkin finite element method is implemented within an arbitrary Lagrangian–Eulerian framework, employing a triangular mesh discretization scheme. This study comprehensively explores thermal and hydrodynamic characteristics, including overall heat transfer, thermal frequency, and power consumption of the rotating blade for heat transfer in mixed convection scenarios with Richardson numbers (Ri) ranging from 0.1 to 10 at varying rotational frequency of the blade. Outcomes demonstrate that the inclusion of a rotating blade increases heat transfer up to 50% at lower Ri, after which the impact of the rotating blade diminishes and heat transfer reduces up to 20% at higher Ri. In addition, heat transfer enhances with increasing blade frequency up to Ri = 6.5, beyond which the effect of the frequency overturns. Examining thermal and hydrodynamic characteristics reveals that the blade achieves optimal performance when operating at f = 1 and Ri = 3. The study's insights into mixed convection heat transfer offer versatile applications, benefiting industries and equipment such as electronic cooling, chemical reactors, food processing, material fabrication, solar collectors, and nuclear reactor systems. Moreover, the findings are instrumental in the thermal ventilation of buildings and the development of micro‐electromechanical systems.

Validation of a Eulerian–Lagrangian numerical algorithm for simulating ultra-coarse particles transported in horizontal and vertical hydraulic pipes

Investigation on the characteristics of coarse and ultra-coarse particles transported in hydraulic conveying pipe is essential to design hydraulic conveying system with high efficiency and reliability. However, the modeling of the particles with a finite size larger than the mesh for fluid discretization is intractable, because the unresolved point-particle methods are difficult to converge in the calculation and the particle-resolved methods cannot afford the computational cost for engineering problem. A numerical algorithm based on the Eulerian–Lagrangian framework is developed to simulate such two-phase flows. The solvers of the computational fluid dynamics (CFD) and the discrete element method (DEM) are coupled through an in-house developed interface code. In the code, a diffusion-based averaging method is employed to smooth the contributions of large particle in the particle volume fraction, the interphase forces and the interactions between turbulence and particles. The interpolation of fluid velocity is also improved so that the calculations of the interphase forces are more accurate. The authenticity of the numerical algorithm is validated against available experimental results. The validating scenarios are the millimeter- and centimeter-sized particles conveyed in both horizontal and vertical pipes under various conveying speeds and particle concentrations. The results show the numerical algorithm can simulate large particles in the fine CFD meshes. More importantly, some distinct features in the particle distribution and the characteristics of turbulence in the experiments are well reproduced by the simulations.

Neural Physical Simulation with Multi-Resolution Hash Grid Encoding

We explore the generalization of the implicit representation in the physical simulation task. Traditional time-dependent partial differential equations (PDEs) solvers for physical simulation often adopt the grid or mesh for spatial discretization, which is memory-consuming for high resolution and lack of adaptivity. Many implicit representations like local extreme machine or Siren are proposed but they are still too compact to suffer from limited accuracy in handling local details and a long time of convergence. We contribute a neural simulation framework based on multi-resolution hash grid representation to introduce hierarchical consideration of global and local information, simultaneously. Furthermore, we propose two key strategies: 1) a numerical gradient method for computing high-order derivatives with boundary conditions; 2) a range analysis sample method for fast neural geometry boundary sampling with dynamic topologies. Our method shows much higher accuracy and strong flexibility for various simulation problems: e.g., large elastic deformations, complex fluid dynamics, and multi-scale phenomena which remain challenging for existing neural physical solvers.

An Unstructured Mesh Generation Tool for Efficient High‐Resolution Representation of Spatial Heterogeneity in Land Surface Models

AbstractThis study develops an unstructured mesh generation tool for land surface modeling using a multi‐scale hexagon discrete mesh. The tool can automatically determine the required mesh resolution for different regions based on multi‐objective criteria such as elevation, slope, land cover, and land use. The refined unstructured meshes demonstrate significant enhancement in the representation of spatial heterogeneity. The tool is coupled with the Common Land Model (CoLM) to enable land surface simulations using unstructured meshes. Evaluations focused on runoff, river discharge, and inundation indicate improved model performance compared to traditional structured mesh‐based CoLM simulations under the same computational cost constraints. This tool provides new capabilities for more efficiently capturing localized land surface processes and extreme events.

Approximate inner solvers for block preconditioning of the incompressible Navier–Stokes problems discretized by isogeometric analysis

AbstractWe deal with efficient numerical solution of the steady incompressible Navier–Stokes equations (NSE) using our in‐house solver based on the isogeometric analysis (IgA) approach. We are interested in the solution of the arising saddle‐point linear systems using preconditioned Krylov subspace methods. Based on our comparison of ideal versions of several state‐of‐the‐art block preconditioners for linear systems arising from the IgA discretization of the incompressible NSE, suitable candidates have been selected. In the present paper, we focus on selecting efficient approximate solvers for solving subsystems within these preconditioning methods. We investigate the impact on the convergence of the outer solver and aim to identify an effective combination. For this purpose, we compare convergence properties of the selected solution approaches for problems with different viscosity values, mesh refinement levels and discretization bases.

Multi-Directional Viscous Damping Absorbing Boundary in Numerical Simulation of Elastic Wave Dynamic Response

Numerical seismic wave field simulation is essential for studying the dynamic responses in semi-infinite space, and the absorbing boundary setting is critical for simulation accuracy. This study addresses spherical waves incident from the free boundary by applying dynamic equations and Rayleigh damping. A new multi-directional viscous damping absorbing boundary (MVDB) method is proposed based on regional attenuation. An approximate formula for the damping value is established, which can achieve absorbing the boundary setting by only solving the mass damping coefficients without increasing the absorbing region grid cells or depending on the spatial and temporal walking distance. The validity and stability of the proposed method are proven through numerical calculations with seismic sources incident from different angles. Meanwhile, the key parameters affecting the absorption of the MVDB are analyzed, and the best implementation scheme is provided. In order to meet the requirements of mediums with different elastic parameters for boundary absorption and ensure the high efficiency of numerical calculations, the damping amplitude control coefficients k can be set between 1.02 and 1.12, the thickness of the absorbing region L is set to 2–3 times of the wavelength of the incident transverse wave, and the thickness of the single absorbing layer is set to the size of the discrete mesh of the model Δl.

Numerical study of a fish swimming in hydrokinetic turbine wake

The environmental effects of hydrokinetic turbines are still under investigation, reflecting the emerging status of this technology. This study investigates the interaction between hydrokinetic rotor wakes and fish swimming, revealing insights into fish biomechanics in complex flows and assessing the environmental implications of marine energy solutions. We conducted numerical simulations with the URANS approach and k−ω−SST turbulence closure model to predict three-dimensional turbulent flow in the OpenFOAM software. The hydrokinetic rotor wake was simulated employing the actuator line method, providing a computationally efficient alternative to full geometry simulations. For accurate replication of the motion of a fish-like tuna (Thunnus atlanticus), dynamic adaptive mesh discretization was employed. The results offer a comparative analysis of fish swimming performance within the wake rotor, particularly when immersed in the tip blade vortex, contrasted with scenarios where fish swim in undisturbed flow conditions. The analysis encompasses three-dimensional wake structures, force generation, efficiency, and equilibrium states (balancing drag and thrust) across varying Swimming numbers (Sw). Key findings include the enhanced attachment of the leading-edge vortex due to the caudal fin’s interaction with the tip blade vortex, resulting in improved auto-propulsive force production; a reduced tail stride frequency observed in fish swimming downstream of the rotor to achieve longitudinal force balance compared to unperturbed flow; and transverse hydrodynamic forces pushing fish radially away from the wake’s influence zone, potentially mitigating the risk of collision with turbine blades.

Iterative dynamics-based mesh discretisation for multi-scale coastal ocean modelling

Flow in coastal waters contains multi-scale flow features that are generated by flow separation, shear-layer instabilities, bottom roughness and topographic form. Depending on the target application, the mesh design used for coastal ocean modelling needs to adequately resolve flow features pertinent to the study objectives. We investigate an iterative mesh design strategy, inspired by hydrokinetic resource assessment, that uses modelled dynamics to refine the mesh across key flow features, and a target number of elements to constrain mesh density. The method is solver-agnostic. Any quantity derived from the model output can be used to set the mesh density constraint. To illustrate and assess the method, we consider the cases of steady and transient flow past the same idealised headland, providing dynamic responses that are pertinent to multi-scale ocean modelling. This study demonstrates the capability of an iterative approach to define a mesh density that concentrates mesh resolution across areas of interest dependent on model forcing, leading to improved predictive skill. Multiple design quantities can be combined to construct the mesh density, refinement can be applied to multiple regions across the model domain, and convergence can be managed through the number of degrees of freedom set by the target number of mesh elements. To apply the method optimally, an understanding of the processes being model is required when selecting and combining the design quantities. We discuss opportunities and challenges for robustly establishing model resolution in multi-scale coastal ocean models.

An isoparametric inclusion model for determining the thermo-elastic fields produced by varying Eigen-temperature gradients

The existence of inclusions or inhomogeneities may disrupt heat flow, resulting in increased stresses and temperature fluctuations at the material interface. Analyzing the thermo-elastic fields surrounding such microstructures is crucial for unveiling the intricate failure mechanisms within materials. Based on the method of Green's function and contour integral, this work presents a complete set of explicit analytical solutions for thermal and elastic fields of an arbitrary polygonal inclusion subjected to linearly varying eigen-temperature gradients or thermal eigenstrains. In contrast to the previous studies on potentials showing analogy with anti-plane elasticity, a mathematical analogy of Green's functions between steady-state heat conduction and thermal inclusion for plane elasticity is accordingly established. By employing the linear interpolation of the nodal eigen-temperature gradients or eigenstrains, the work further extends the traditional isoparametric element idea to the thermos-elastic inclusions, and proposes a novel isoparametric inclusion model to solve arbitrarily shaped inclusions with any distributed eigen-temperature gradients. Unlike the finite element method, which requires meshing for both the matrix and inclusion, the present model exhibits efficiency, flexibility, and versatility with the mesh discretization only conducted inside the inclusion domain. The inclusion based isoparametric method may have potential applications in mechanical engineering and material science for analyzing the thermo-structural behavior of various microstructures.

Performance Analysis of the Vortex Cuttings Cleaner: Turbine Hydraulic Drive and Cuttings Transport in Wellbore Annulus

Summary In the process of directional and horizontal well drilling, cuttings tend to settle and form a bed at the low side of the annulus due to gravity, which decreases the drilling rate and even causes accidents in severe cases. This paper analyzes the performance of a new tool, the vortex cuttings cleaner, which can be effective without rotation of the drillpipe. Based on the computational fluid dynamics (CFD) approach, together with the discrete phase, Euler, and dynamic mesh models, the vortex cuttings cleaner is investigated with respect to the turbine torque, turbine velocity, pressure drop, and cuttings transport in the annulus. The working mechanism of the vortex cuttings cleaner is clarified. Finally, field tests are conducted on the tool to evaluate its application in terms of service life, wellbore friction, and rate of penetration (ROP). The results show that the turbine can rotate continuously under hydraulic drive. The turbine torque/velocity and the tool’s pressure drop increase with increasing displacement. The cuttings transport in the annulus is jointly affected by factors such as turbine velocity, fluid velocity, and particle size. A too low or high turbine velocity is unfavorable for cuttings transport. Through the analysis of the number of particles and particle concentration, the optimal velocity is determined to be 125 rev/min. The swirling flow intensity in the annulus flow field increases with the increase in turbine velocity. Field applications suggest a service life longer than 200 hours, a notable decrease in wellbore friction, and an average increase in ROP by more than 20%. This study provides a theoretical basis for the research on wellbore cleaning tools.

A fast computational technique to solve fourth-order parabolic equations: application to good Boussinesq, Euler-Bernoulli and Benjamin-Ono equations

In this article, we introduce a novel cubic spline method for the numerical solution of a class of fourth-order time-dependent parabolic partial differential equations. The method employs uniform mesh discretization, achieving a spatial accuracy of four and a temporal accuracy of two. Here, the key contribution is that this method can be directly applied to singular parabolic problems due to the consideration of half-step points in spatial direction. The model linear fourth-order partial differential equation is used to discuss the method's stability. The method we propose has multiple advantages, including high accuracy and seamless point-to-point interpolation capabilities. It is a versatile technique that can easily adapt to solve various nonlinear problems without the need to change or linearize the nonlinear terms to produce numerical solutions. Benchmark linear and nonlinear problems like single and double-soliton of the good Boussinesq equation, the Euler-Bernoulli beam model and the Benjamin-Ono equation, are solved.

Effects of Water-to-Air Mass Ratio on Long-Term Washing Efficiency and Erosion Risk in an Axial Compressor Under Online Washing Conditions

Abstract One of the main reasons for gas turbines’ performance losses is the deposition of dirt on the compressor blades. Dirt deposit has to be periodically removed to keep the engine performance as high as possible. This is the reason motivating the presence of online water washing systems in most compressor gas turbines. Such systems aim at cleaning the compressor blades to recover efficiency; thus, the larger the water flow, the better it is assumed the compressor is cleaned (fixing all the other conditions). In the present work, we simulate the long-term behavior of a real axial compressor, from the inlet to the first-stage rotor, subject to online water washing with different water flowrates. The frozen rotor approach is adopted to solve the flow field in the rotor region. Simulations are performed by using the unsteady k-ɛ realizable model coupled with a Lagrangian tracking of the injected liquid phase. Water droplet erosion is handled by using a semi-empirical model developed by the authors. In each simulation, 504,000 parcels have been tracked, providing statistically reliable predictions. To simulate the long-term evolution of the washing process, a discrete mesh morphing technique coupled with the use of specific scale factors is adopted. Each of the tested configurations is composed of three successive erosive steps up to the blade compressor end-of-life. By varying the water-to-air mass fraction (WAMF*), six different injection configurations are assessed in terms of long-time average washing efficiency and erosion risk. The results predicted show the dependence of the considered washing indices on water mass flowrate and set the stage for the development of a washing optimization tool, which can help the design and management processes. In scenarios where washing indices are given minimal importance and the objective is to reduce the risk of erosion, the optimal injection configuration was shown to correspond to a WAMF* value of 0.250. Conversely, when washing efficiency is prioritized, the optimal injection configuration has been shown to correspond to the case where WAMF* = 0.750.