Mesh Convergence Study Research Articles

Drag, added mass and radiation damping of oscillating vertical cylindrical bodies in heave and surge in still water

Forced heave and surge motion of axisymmetric vertical cylindrical bodies with flat and rounded bases are simulated using the advanced CFD software STAR-CCM+ where the overset method is used so that the mesh local to the body moves within a stationary outer mesh. Viscous effects generating drag, and also influencing added mass and radiation damping, are determined. The Reynolds-Averaged Navier–Stokes (RANS) equations are adopted with different turbulence closure models and the water surface is captured by the volume of fluid (VOF) method. These results are of basic interest but the main motive is to assess appropriate drag coefficients for use with linear diffraction models of wave energy converters (WEC) and we have particular interest in the multi-body WEC M4. A basic dynamical model is set up so that drag, added mass and radiation damping coefficients may be obtained from the CFD results. Added mass and radiation damping coefficients were also obtained from the potential flow solver WAMIT for comparison. Mesh convergence studies were undertaken and while mesh independence was achieved for total force it was not possible for the very small shear force. In the laboratory a free heave decay test was undertaken without mechanical contact for bodies with rounded bases and the inferred drag and added mass coefficients were very close to those from CFD. Some general observations are possible for motion in heave. For the hemispherical base the drag coefficient Cd is very low for small amplitudes but this increases as amplitude is increased. For the rounded base with a flat central area the Cd is larger and for the wholly flat base it is larger again but with values less than 0.35. For a larger geometric scale (times 32) for the hemisphere and round base cases the Cd are generally somewhat reduced. For surge motion the Cd show less variation and are always greater than heave values by at least a factor of 2 which is indicative of effects due to separation and wake generation.

Mesh convergence study and estimation of discretization error of hub in clutch disc with integration of ANSYS

Structural analysis is an essential tool for design engineers. Mesh generation is the basic step in any simulation. In practice of finite-element stress analysis, the engineer first needs to know if key stresses are converging, and second if they have converged to a reasonable level of accuracy. In order to achieve results that are reliable when using the finite element method, it must be ensured that an acceptable mesh is used with respect to the shape and size of the elements. Mesh quality and mesh density values are directly linked with the solution accuracy. This paper presents a mesh convergence study on the hub which is the most challenging part in the automotive clutch disc assembly. Finite element analysis convergence defines the relationship between the number of elements or DOF and the analysis accuracy. Discretization error is key concept for the study of mesh convergence study, also discretization is function of number of degree of freedom of model. A detailed comparison of different results is carried out using convergence test based on displacement and energy norms. Error between exact and ANSYS results are compared in this study.

Multi-scale modelling of granular materials: numerical framework and study on micro-structural features

A multi-scale model for the analysis of granular systems is proposed, which combines the principles of a coupled FEM–DEM approach with a novel servo-control methodology for the implementation of appropriate micro-scale boundary conditions. A mesh convergence study is performed, whereby the results of a quasi-static biaxial compression test are compared with those obtained by direct numerical simulations. The comparison demonstrates the capability of the multi-scale method to realistically capture the macro-scale response, even for macroscopic domains characterized by a relatively coarse mesh; this makes it possible to accurately analyse large-scale granular systems in a computationally efficient manner. The multi-scale framework is applied to study in a systematic manner the role of individual micro-structural characteristics on the effective macro-scale response. The effect of particle contact friction, particle rotation, and initial fabric anisotropy on the overall response is considered, as measured in terms of the evolution of the effective stress, the volumetric deformation, the average coordination number and the induced anisotropy. The trends observed are in accordance with notions from physics, and observations from experiments and other DEM simulations presented in the literature. Hence, it is concluded that the present framework provides an adequate tool for exploring the effect of micro-structural characteristics on the macroscopic response of large-scale granular structures.

A Numerical Analysis Method of Impurity Precipitation on Mesh Wire of Cold Trap in Fast Breeder Reactor

Abstract A fast breeder reactor (FBR) is considered as the promising technology in terms of load reduction on the environment, because the FBR has capability to improve usage efficiency of uranium resources and can reduce high-level radioactive waste which needs to be managed for millions of years. A cold trap is one of the important components in the FBR to control the impurity concentration of the liquid sodium. For accurate evaluation of the cold trap performance, we have been proposing the three-dimensional (3D) numerical analysis method of the cold trap. In this method, the evaluation of the impurity precipitation phenomena on the surface of the mesh wire of the cold trap is the key. For this, the numerical analysis method which is based on the lattice kinetic scheme (LKS) has been proposed. In order to apply the LKS to the impurity precipitation simulation of the cold trap, two models (the low Reynolds number model and the impurity precipitation model) have been developed. In this paper, we focused on the validation of these models. To confirm the validity of the low Reynolds number model, the Chapman–Enskog analysis was applied to the low Reynolds number model. As a result, it has been theoretically confirmed that the low Reynolds number model can recover the correct macroscopic equations (incompressible Navier–Stokes equations) with small error. The low Reynolds number model was also validated by the numerical simulation of two-dimensional (2D) channel flow problem with the low Reynolds number conditions which correspond to the actual cold trap conditions. These results have confirmed that the error of the low Reynolds number model is ten times smaller than that of the original LKS. The validity of the impurity precipitation model was investigated by the comparison to the precipitation experiments. In this comparison, the mesh convergence study was also conducted. These results have confirmed that the proposed impurity precipitation model can simulate the impurity precipitation phenomena on the surface of the mesh wire. It has been also confirmed that the proposed impurity precipitation model can simulate the impurity precipitation phenomenon regardless of the cell size which were tested in this investigation.

Verification and validation of residual stresses in composite structures

Process-induced residual stresses occur in composite structures composed of dissimilar materials. As these residual stresses could result in fracture, their consideration when designing composite parts is necessary. However, the experimental determination of residual stresses in prototype parts can be time and cost prohibitive. Alternatively, it is possible for computational tools to predict potential residual stresses. Therefore, the objectives of the presented work are to demonstrate an efficient method for simulating residual stresses in composite parts, as well as the potential value of statistical methods during analyses for which material properties are unknown. Specifically, a simplified residual stress modeling approach is implemented within Sandia National Laboratories’ SIERRA/SolidMechanics code. Concurrent with the model development, bi-material composite structures are designed and manufactured to exhibit significant residual stresses. Then, the presented modeling approach is rigorously verified and validated through simulations of the bi-material composite structures’ manufacturing processes, including a mesh convergence study, sensitivity analysis, and uncertainty quantification. The simulations’ final results show adequate agreement with the experimental measurements, indicating the validity of a simple modeling approach, as well as a necessity for the inclusion of material parameter uncertainty in the final residual stress predictions.

An oscillation-free flow solver based on flux reconstruction

In this paper, a segregated algorithm is proposed to suppress high-frequency oscillations in the velocity field for incompressible flows. In this context, a new velocity formula based on a reconstruction of face fluxes is defined eliminating high-frequency errors. In analogy to the Rhie–Chow interpolation, this approach is equivalent to including a flux-based pressure gradient with a velocity diffusion in the momentum equation. In order to guarantee second-order accuracy of the numerical solver, a set of conditions are defined for the reconstruction operator. To arrive at the final formulation, an outlook over the state of the art regarding velocity reconstruction procedures is presented comparing them through an error analysis. A new operator is then obtained by means of a flux difference minimization satisfying the required spatial accuracy. The accuracy of the new algorithm is analyzed by performing mesh convergence studies for unsteady Navier–Stokes problems with analytical solutions. The stabilization properties of the solver are then tested in a problem where spurious numerical oscillations arise for the velocity field. The results show a remarkable performance of the proposed technique eliminating high-frequency errors without losing accuracy.

Numerical Studies on Low-Velocity Impact Failure Response of Al 1100 under Blunt and Hemispherical Impactors

Numerical simulation is performed to study the deformation and failure modes of Al 1100 plate of 2.4 mm thickness, subjected to low-velocity impact. Blunt and hemispherical nose shaped impactors are used in this study. The quasi-static tensile test is performed at a strain rate of 0.01/s to obtain the Johnson-Cook material parameters which are used in numerical simulation software, ABAQUS/CAE to perform impact analysis. Mesh convergence study is carried out to decide the appropriate number of elements for numerical analysis. The impact behavior of Al 1100 plate for each impactor shapes are studied at 22 J impact energy. Result indicate that increased in the nose radius of impactor will increase the amount of deformation energy for aluminium plate.

On the volume conservation of emulsion drops in boundary integral simulations

Boundary Integral simulations are very common to study the microhydrodynamics of viscous drops and predict the rheology of emulsions. The standard boundary integral formulation for the velocity field at the drop surface is given by surface integrals of singular Green’s functions in space and depends on geometric quantities of the mesh, as the local mean curvature and the unitary exterior normal vector. Typically, the drops deform and rotate under the flow action, which leads to a great distortion of the computational mesh and, therefore, might produce several numerical errors in the surface velocity calculation. In this work, we investigate the numerical errors in the calculation of the surface velocity in boundary integral simulations of a viscous drop in simple shear flows. Assuming that the flow is incompressible, such that the total volume of the drops must remain constant, we use the net flow rate across the drop surface as a measure of the numerical errors in the simulations. For both small and moderate regimes of drop surface deformation, we show that the net mass flow rate across the drop surface is positive, and strongly depends on the drop-to-basis fluid viscosity ratio, capillary number of the flow, and mesh refinement. These results indicate that the volume of the drops increases continuously during the simulations. To remove the spurious mass flow rate from the simulations, we present a mesh convergence study based on an extrapolation procedure to an almost continuous mesh, so that the results become independent of mesh refinement and recover the theoretical prediction of a null net flow rate and constant drop volume in incompressible flows. The extrapolation method proposed here can be straightforwardly implemented to improve the accuracy of BI simulations used to study drop microhydrodynamics and emulsion rheology. As a matter of example, we applied the method to predict the surface deformation of a high-viscosity emulsion drop, and the result is compared in very good agreement with asymptotic theories available in the related literature.

Prediction of mining-induced surface subsidence and ground movements at a Canadian diamond mine using an elastoplastic finite element model

A full three dimensional elastoplastic finite element model of Diavik Diamond Mine is developed to predict surface-induced ground movement due to underground blasthole stoping activities. A mesh convergence study has been conducted to reach to an optimum meshing scenario when further mesh refinements produce a negligible change in the results. This way the calculation cost and time has been managed successfully. Furthermore, sequences of the excavation/backfilling in the model are defined according to the mine production plan; however, some simplifications in the shape of the stopes have been made. Consequently, the developed finite element model simulates the complete stress-strain path through the entire excavation and backfilling simulation steps in full three dimensional space. The model is calibrated using two extensometers installed on the back of two secondary undercut drifts in one of the Kimberlite pipes. The results of the calibrated model are validated using pit surface prism monitoring system data. The average relative error between the measured data and FE model predictions is 7.95%. It is shown that the numerical predictions of the mining-induced surface subsidence, due to the blasthole stoping mining method, matched well with the Gaussian distribution. A significant increase, approximately by 44%, to the amount of the induced settlements on the surface occurred as the mining activities reaches near surface ground levels. Finally, the comparison between the predicted results of the finite element model and monitoring data showed that the predictive capacity of the numerical model is a valuable tool for stability and design analysis of underground mines.

Extended Hertz Theory of Contact Mechanics for Case-Hardened Steels With Implications for Bearing Fatigue Life

The analytical expressions currently available for Hertzian contact stresses are applicable only for homogeneous materials and not for case-hardened bearing steels, which have inhomogeneous microstructure and graded elastic properties in the subsurface region. Therefore, this article attempts to determine subsurface stress fields in ball bearings for graded materials with different ball and raceway geometries in contact. Finite element models were developed to simulate ball-on-raceway elliptical contact and ball-on-plate axisymmetric contact, to study the effects of elastic modulus variation with depth due to case hardening. Ball bearings with low, moderate, and heavy load conditions are considered. The peak contact pressure for case-hardened steel is always more than that of through-hardened steel under identical geometry and loading conditions. Using equivalent contact pressure approach, effective elastic modulus is determined for case-carburized steels, which will enable the use of Hertz equations for different gradations in elastic modulus of raceway material. Nonlinear regression tools are used to predict effective elastic modulus as a weighted sum of surface and core elastic moduli of raceway material and design parameters of ball–raceway contact area. Mesh convergence study and validation of equivalent contact pressure approach are also provided. Implications of subsurface stress variation due to case hardening on bearing fatigue life are discussed.

Numerical hull resistance calculation of a catamarán using OpenFOAM

In the present study, numerical resistance predictions using OpenFOAM were performed considering the Cormorant Evolution Catamaran, which provides travel services in the Galapagos Islands. These predictions were compared with experimental data published by Chávez and Lucín [1] and with systematic series [2].Simulations were made at model scale of 2 [m] in two load conditions, considering demi and twin hull (s=0.56 [m]) configurations. A mesh convergence study was performed with 3 different meshes for V=1.05 [m/s] at Light Condition (T=0.086 [m]). The converged mesh, with 1 million of cells approximately, has the lower standard deviation and a 5% error when compared to its experimental value of 1.79 [N]. The errors between the experimental data and the numerical simulations for demi hull configuration were 43% and 36% for Light and Full conditions, respectively. Besides, for twin hull configuration the errors were around 14% and 32% for Light and Full conditions, respectively.

Application of high-order numerical schemes and Newton-Krylov method to two-phase drift-flux model

This paper concerns the application and solver robustness of the Newton-Krylov method in solving two-phase flow drift-flux model problems using high-order numerical schemes. In our previous studies, the Newton-Krylov method has been proven as a promising solver for two-phase flow drift-flux model problems. However, these studies were limited to use first-order numerical schemes only. Moreover, the previous approach to treating the drift-flux closure correlations was later revealed to cause deteriorated solver convergence performance, when the mesh was highly refined, and also when higher-order numerical schemes were employed. In this study, a second-order spatial discretization scheme that has been tested with two-fluid two-phase flow model was extended to solve drift-flux model problems. In order to improve solver robustness, and therefore efficiency, a new approach was proposed to treating the mean drift velocity of the gas phase as a primary nonlinear variable to the equation system. With this new approach, significant improvement in solver robustness was achieved. With highly refined mesh, the proposed treatment along with the Newton-Krylov solver were extensively tested with two-phase flow problems that cover a wide range of thermal-hydraulics conditions. Satisfactory convergence performances were observed for all test cases. Numerical verification was then performed in the form of mesh convergence studies, from which expected orders of accuracy were obtained for both the first-order and the second-order spatial discretization schemes. At last, the drift-flux model, along with numerical methods presented, were validated with three sets of flow boiling experiments that cover different flow channel geometries (round tube, rectangular tube, and rod bundle), and a wide range of test conditions (pressure, mass flux, wall heat flux, inlet subcooling and outlet void fraction).

A numerical stabilization framework for viscoelastic fluid flow using the finite volume method on general unstructured meshes

SummaryA robust finite volume method for viscoelastic flow analysis on general unstructured meshes is developed. It is built upon a general‐purpose stabilization framework for high Weissenberg number flows. The numerical framework provides full combinatorial flexibility between different kinds of rheological models on the one hand, and effective stabilization methods on the other hand. A special emphasis is put on the velocity‐stress‐coupling on colocated computational grids. Using special face interpolation techniques, a semi‐implicit stress interpolation correction is proposed to correct the cell‐face interpolation of the stress in the divergence operator of the momentum balance. Investigating the entry‐flow problem of the 4:1 contraction benchmark, we demonstrate that the numerical methods are robust over a wide range of Weissenberg numbers and significantly alleviate the high Weissenberg number problem. The accuracy of the results is evaluated in a detailed mesh convergence study.

A Finite Element Study of an Elasto-Plastic Disk or Cylindrical Contact Against a Rigid Flat in Plane Stress with Bilinear Hardening

This work presents a finite element study of elasto-plastic cylindrical contact. The geometry could also be described as a vertically aligned disk whose axis of symmetry is parallel to the contact surface. The cylinder is considered to be in the plane stress state. The material of the cylinder is modeled as elasto-plastic with bilinear hardening (also known as linear hardening). Simulations for a range of material properties and deflections typical to engineering applications are carried out. A mesh convergence study has also been performed. By employing symmetry, the cylinder has been modeled as a quarter circle and a straight line is used to model the opposing rigid flat surface. The finite element results for the elastic and fully plastic cylindrical contact cases are compared to other existing models such as Hertz contact and spherical elasto-plastic models. Since the case considered is plane stress, the stress distribution is significantly different from elasto-plastic spherical contacts, which would be closer to a plane strain case. An empirical relationship is fit to the results to allow for prediction of the contact width as a function of displacement and force.

Size-dependant behaviour of functionally graded microplates based on the modified strain gradient elasticity theory and isogeometric analysis

This paper presents a robust numerical model, which takes into account both size-dependent and shear deformation effects, for the bending, buckling and free vibration analyses of functionally graded microplates. The size-dependent effect is captured by using the modified strain gradient elasticity theory with three length scale parameters, whilst the shear deformation effect is accounted by using the third-order shear deformation theory. The rule of mixture is employed to describe the distributions of material phrases through the plate thickness. By using Hamilton’s principle, the governing equations are derived and then discretized by employing an Isogeometric Analysis (IGA) approach, where the Non-Uniform Rational B-Splines (NURBS) basis functions are adopted to meet the C2-continuity requirement. Physical mesh convergence and verification studies are performed to prove the accuracy and reliability of the present model. In addition, parametric studies are also carried out to investigate the size effect in conjunction with the influences of gradient index, shear deformation effect and boundary conditions on the responses of microplates.

Effect of Grade on Thermal–Mechanical Behavior of Steel During Initial Solidification

Thermal–mechanical analysis of solidification is important to understand crack formation, shape problems, and other aspects of casting processes. This work investigates the effect of grade on thermal–mechanical behavior during initial solidification of steels during continuous casting of a wide strand. The employed finite element model includes non-linear temperature-, phase-, and carbon content-dependent elastic–viscoplastic constitutive equations. The model is verified using an analytical solution, and a mesh convergence study is performed. Four steel grades are simulated for 30 seconds of casting without friction: ultra-low-carbon, low-carbon, peritectic, and high-carbon steel. All grades show the same general behavior. Initially, rapid cooling causes tensile stress and inelastic strain near the surface of the shell, with slight complementary compression beneath the surface, especially with lower carbon content. As the cooling rate decreases with time, the surface quickly reverses into compression, with a tensile region developing toward the solidification front. Higher stress and inelastic strain are generated in the high-carbon steel, because it contains more high-strength austenite. Stress in the δ-ferrite phase near the solidification front is always very small, owing to the low strength of this phase. This modeling methodology is a step toward designing better mold taper profiles for continuous casting of different steels.

Galerkin finite element method for cancer invasion mathematical model

A finite element scheme for the solution of a cancer invasion model is proposed. The cancer dynamics model consists of three coupled partial differential equations which describe the evolution of cancer cell density, extra cellular matrix and the matrix degrading enzymes. The model incorporates proliferation and haptotaxis effect of cancer cells, their interaction with extracellular matrix, the production of matrix degrading enzymes and consequent degradation of the extracellular matrix. The coupled partial differential equations are discretized in space with the standard Galerkin finite elements and in time with the Crank–Nicolson method. Moreover, the nonlinear terms in the coupled equations are treated semi-implicitly in the finite element scheme. The numerical scheme is validated with numerical results taken from the literature. In addition to the mesh convergence study, the effects of haptotactic rate, proliferation rate and remodelling rate of matrix components of the considered mathematical model are investigated.

Modeling and optimization of acoustic black hole vibration absorbers

Recent studies have shown that the acoustic black hole (ABH) effect can be used to provide vibration absorption and improved structural-acoustic response. In this talk, several aspects of the design and implementation of ABH vibration absorbers will be discussed. First, to address the competing nature of the best ABH taper for vibration reduction and the underlying theoretical assumptions, a multi-objective approach is used to find the best ABH parameters where both criteria are sufficiently met. Next, the modeling challenges associated with one- and two-dimensional ABH design will be discussed and a mesh convergence study will be presented. Finally, the use of multiobjective optimization for ABH design will be discussed for aerospace and marine applications.

Adaptive unified continuum FEM modeling of a 3D FSI benchmark problem.

In this paper, we address a 3D fluid-structure interaction benchmark problem that represents important characteristics of biomedical modeling. We present a goal-oriented adaptive finite element methodology for incompressible fluid-structure interaction based on a streamline diffusion-type stabilization of the balance equations for mass and momentum for the entire continuum in the domain, which is implemented in the Unicorn/FEniCS software framework. A phase marker function and its corresponding transport equation are introduced to select the constitutive law, where the mesh tracks the discontinuous fluid-structure interface. This results in a unified simulation method for fluids and structures. We present detailed results for the benchmark problem compared with experiments, together with a mesh convergence study.

Numerical modelling of the mechanical response of cellular solids made from sintered Titanium powders

Two algorithms are developed to generate virtual microstructures of Titanium foams of medium relative density, in the range 0.37 – 0.48; one of the algorithms captures the effect of pore shape and size distribution, while the other algorithm only accounts for the relative density of the material. Foams are modelled as two-phase composites consisting of a solid material and voids; the elasto-plastic and damage response of the parent material is deduced from its measured mechanical responses in tension and compression, while the material occupying the voids is assigned a plastically compressible constitutive response, to mimic the effects of pore collapse and subsequent self-contact. Finite Element (FE) simulations are conducted to predict the measured macroscopic material response in uniaxial tension and compression, as well as in pure shear; a mesh convergence study is performed and the minimum RVE size is determined. The simulations are found in good agreement with measurements and can predict accurately and effectively the material's mechanical response up to large strains, as well as the scaling of mechanical properties with relative density.