Meshes For Problems Research Articles

Theory and Application of Surface Generated by Curvilinear Cutter Edge with Two Degrees of Freedom

In order to study the mesh problem in case that the surface is generated by curvilinear cutter with two degrees of freedom a theory of surface generated by curvilinear cutter edge with two degrees of freedom is put forward by using the conjugate conditions of two degrees of freedom motion. The theory is used to calculate the helical surface of the toroidal worm generated by linear cutter edge whose relative motion has two degrees of freedom. This kind of worm is manufactured on a hob machine and its helical surface of the worm is measured on a CMM. The application prospect of this kind of worm gears is studied. The conclusions are: by using the theory the generate curved surface can be deduced when the curvilinear cutter edge and the varying regularity of the two motion parameters are given or the varying regularity of the two motion parameters can be derived when the generated curved surface and the curvilinear cutter edge are given;the calculation of the curved surface generated by curvilinear cutter edge with two degrees of freedom can be simplified if the varying regularity of the two motion parameters is restricted properly; the "modification" curve of the worm generated by linear cutter edge with two degrees of freedom is close to that of the hindley worm which is modified in parabolic curve; the defect of the hindley worm gears can be avoided by using this kind of worm gears without needing modification and there is a good application prospect.

Balanced restoration of geological volumes with relaxed meshing constraints

Balanced restoration consists in removing the effects of tectonic deformation in order to recover the depositional state of sedimentary layers. Restoration thus helps in the understanding of a geodynamic scenario, reduces structural uncertainties by testing the consistency of the structural model, and, under mechanical behavior assumptions, evaluates retro-deformation. We show how an elastic finite element model can be used to solve restoration problems, by setting displacement boundary conditions on the top horizon and contact boundary conditions on the fault cut-offs. This method is generally applied on a tetrahedral mesh, which raises significant meshing problems in complex structural settings, where restoration is particularly useful. Indeed, the mesh has to be conformable to both faults and horizons, including unconformities and onlap surfaces, which may drastically increase the number of elements and decrease the mesh quality. As an alternative, we propose to represent unfaulted horizons as a property of the tetrahedral model, and to transfer the associated boundary conditions onto the neighboring nodes of the mesh, using an “implicit” approach. The proposed methods are demonstrated on a typical example and results show good agreement between both approaches. While the computational time is equivalent in both cases, the time needed for model building is significantly reduced in the implicit case. In addition, the implicit method provides a convenient way to handle unconformities in restoration, both for eroded surfaces, and on onlap layer geometries. In such cases, our method provides a flexible way to specify the amount of eroded material, and generates less mesh elements than the conforming mesh, thereby reducing computational time.

Electromagnetic Field Computation in Magnetic and Conductive Thin Sheets

The finite element method (FEM) is one of the most important techniques of computer modeling because it has a high geometric adaptation capability that makes possible the computation of a great variety of configurations. Nevertheless, the study of special structures that have a geometric dimension smaller than others, creates meshing difficulties. In this paper, the use of shell elements for modeling magnetic and/or conductive thin regions in finite element method is presented. This approach avoids the meshing problems of thin regions. The weak formulation is based on a∗ for (2D) problem and t for (3D) problem. The results are compared with analytic exact solutions.

Conservative interpolation between unstructured meshes via supermesh construction

Mesh adaptivity on unstructured meshes is a proven and popular tool for reducing the computational cost of numerical simulations. Unstructured meshes are often preferred in mesh adaptivity as they allow for greater geometric flexibility and arbitrary anisotropy in resolving simulation features. However, such mesh adaptivity suffers from a significant drawback: the interpolation errors caused by interpolating from the old mesh to the new mesh typically destroys conservation of quantities important to the physical accuracy of the simulation (e.g., density, volume fraction, tracer concentration, etc.). This work presents several globally conservative interpolation operators between general unstructured meshes via the construction of an intermediate supermesh. The construction of the supermesh is performed by transforming the problem to the input to a constrained meshing problem. The performance of the conservative interpolation operators are compared against interpolation using the underlying basis functions.

Parametric Models in Quasi-Static Electromagnetics

A procedure that speeds up parametric modelling is presented. With the help of manifolds and differential geometry it is possible to use a single mesh for problems corresponding to different parameter values.

Measurement of Torsionally Induced Shear Stresses in Shrink-Fit Assemblies

Shear stresses along the shaft/hub interface in shrink-fit components, generated by torsional loads, can drive premature failure through fretting mechanisms. It is difficult to numerically predict these shear stresses, and the associated circumferential slip along the shaft/hub interface, due to uncertainties in frictional behaviour and the presence of steep stress gradients which can cause meshing problems. This paper attempts to provide validation of a numerical modelling methodology, based on finite element analysis, so the procedure may be used with confidence in fitness-for-purpose cases. Very few experimental techniques offer the potential to make measurements of stress and residual stress interior to metallic components. Even fewer techniques provide the possibility of measuring shear stresses. This paper reports the results of neutron diffraction measurements of shear stress and residual shear stress in a bespoke test specimen containing a shrink-fit. One set of measurements was made with a torsional load ‘locked-in’. A second set of measurements was made to determine the residual shear stress when the torsional load had been applied and removed. Overall, measurement results were consistent with numerical models, but the necessity for a small test specimen to allow penetration of the neutron beam to the measurement locations meant the magnitude of shear stresses was at the limits of what could be measured experimentally.

Finite Element Analysis of Periodic Structures Without Constrained Meshes

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> A finite element analysis is detailed of periodic structures without constrained meshes. The infinite periodic analysis domain is reduced to the unit cell analysis by means of the Floquet-Bloch decomposition, where an admittance-type matrix approach is carried out. Special emphasis is placed on the admittance-type matrix generation in such a way that the constrained meshing problem is no longer required. As a result, periodic ports are permitted in the unit cell, and the Floquet-Bloch periodic conditions are easily applied in the admittance-type matrix by means of a connection-like handling. Several infinite periodic antennas are shown to be analyzed appropriately. </para>

Mesh generation technology for nuclear reactor simulation; barriers and opportunities

Mesh generation in support of nuclear reactor simulation has much in common with the requirements of other application areas, such as computational fluid dynamics (CFD). Indeed, fluid dynamics analysis of the coolant behavior inside the reactor core is an internal flow problem that requires the resolution of spatial and temporal variations in the flow caused by complex component configurations, fluids/structure interaction, turbulence, and thermal heating of the coolant. Typical concerns of meshing complex geometries; the use of hexahedral vs. tetrahedral elements, element geometric quality, mesh smoothness, use of anisotropic elements in the thermal boundary layer, etc., are all considerations important to the reactor meshing problem. Reactor meshing begins to become more specialized as the need to employ reactor simulation as a predictive design and safety analysis capability grows in importance. First, a predictive capability will require more precise physical models to be included, and these models will need to be supported by a computational science framework that will allow them to be accurately approximated both spatially and temporally during the reactor core analysis. Both the multiphysical nature of the composite reactor model and details of the physics algorithms themselves will place new requirements on the meshing process needed to support multidimensional reactor simulation. This article discusses the current state of meshing technology applied to reactor simulation and examines a set of issues that are important in the generation of high-quality reactor meshes today and in the future.

A direct simulation method for flows with suspended paramagnetic particles

A direct numerical simulation method based on the Maxwell stress tensor and a fictitious domain method has been developed to solve flows with suspended paramagnetic particles. The numerical scheme enables us to take into account both hydrodynamic and magnetic interactions between particles in a fully coupled manner. Particles are assumed to be non-Brownian with negligible inertia. Rigid body motions of particles in 2D are described by a rigid-ring description implemented by Lagrange multipliers. The magnetic force, acting on the particles due to magnetic fields, is represented by the divergence of the Maxwell stress tensor, which acts as a body force added to the momentum balance equation. Focusing on two-dimensional problems, we solve a single-particle problem for verification. With the magnetic force working on the particle, the proper number of collocation points is found to be two points per element. The convergence with mesh refinement is verified by comparing results from regular mesh problems with those from a boundary-fitted mesh problem as references. We apply the developed method to two application problems: two-particle interaction in a uniform magnetic field and the motion of a magnetic chain in a rotating field, demonstrating the capability of the method to tackle general problems. In the motion of a magnetic chain, especially, the deformation pattern at break-up is similar to the experimentally observed one. The present formulation can be extended to three-dimensional and viscoelastic flow problems.

A Two-Level Discretization Method for the Smagorinsky Model

A two-level method for discretizing the Smagorinsky model for the numerical simulation of turbulent flows is proposed and analyzed. In the two-level algorithm, the solution to the fully nonlinear coarse mesh problem is utilized in a single-step linear fine mesh problem. When modeling parameters are chosen appropriately, the error in the two-level algorithm is comparable to the error in solving the fully nonlinear problem on the fine mesh. We demonstrate numerically that, for an appropriate choice of grids, the two-level algorithm is significantly more efficient than the standard one-level algorithm. We also provide a rigorous numerical analysis of the two-level method, which yields appropriate scalings between the coarse and fine mesh-sizes (H and h, respectively), and the radius of the spatial filter used in the Smagorinsky model ($\delta$). Our analysis provides a mathematical answer to the large eddy simulation mystery of finding the scaling between the filter radius and numerical mesh-size.

Numerical approximation of the Smagorinsky model by a two‐level method

AbstractA two‐level method for discretizing the Smagorinsky model for the numerical simulation of turbulent flows is proposed. In the two‐level algorithm, the solution to the fully nonlinear coarse mesh problem is utilized in a single‐step linear fine mesh problem. When modeling parameters are chosen appropriately, the error in the two‐level algorithm is comparable to the error in solving the fully nonlinear problem on the fine mesh. We provide an a priori error estimate for the two‐level method, which yields appropriate scalings between the coarse and fine mesh‐sizes (H and h, respectively), and the radius of the spatial filter used in the Smagorinsky model (δ). In addition, we provide an algorithm in which a coarse mesh correction is performed to further enhance the accuracy. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Parallel adaptive simulations on unstructured meshes

This paper discusses methods being developed by the ITAPS center to support the execution of parallel adaptive simulations on unstructured meshes. The paper first outlines the ITAPS approach to the development of interoperable mesh, geometry and field services to support the needs of SciDAC application in these areas. The paper then demonstrates the ability of unstructured adaptive meshing methods built on such interoperable services to effectively solve important physics problems. Attention is then focused on ITAPs' developing ability to solve adaptive unstructured mesh problems on massively parallel computers.

A 3D adaptive mesh moving scheme

AbstractThis paper presents an adaptive mesh moving technique for three‐dimensional (3D) fluid flow problems that involve moving fluid boundaries and fluid–solid interfaces. Such mesh moving techniques are an essential ingredient of fluid–structure interaction methods that typically employ arbitrary Lagrangian–Eulerian (ALE) frameworks. In the ALE frame, the velocity field representing motion of the underlying continuum is integrated in the fluid flow equations. In the discretized setting, the velocity field of the underlying continuum gives rise to the mesh displacement field that needs to be solved for in addition to the flow equations and the structural equations. Emphasis in the present work is on the motion and deformation of 3D grids that are composed of linear tetrahedral and hexahedral elements in structured and unstructured configurations. The proposed method can easily be extended to higher‐order elements in 3D. A variety of moving mesh problems from different fields of engineering are presented that show the range of applicability of the proposed method and the class of problems that can be addressed with it. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Boundary recovery after 3D Delaunay tetrahedralization without adding extra nodes

AbstractIn this paper, we investigate boundary recovery, the problem that has troubled researchers ever since Delaunay‐based methods were applied to generate mesh. There are a number of algorithms for boundary recovery already and most of them depend heavily on adding extra nodes. In this paper, we make an effort to seek a method to recover boundaries without using extra nodes.It was noted that some previous algorithms imposed artificial boundary constraints on a meshing problem at the recovering stage; we first try to discard these artificial constraints and thus make things easier. Then a new method is proposed by which the boundaries can be recovered by means of two operations: (1) creating a segment in the mesh and (2) removing a segment from the mesh. Both operations are special cases of a general local transformation called small polyhedron reconnection operation. The method works well when coupled with the sphere‐packing method proposed by the first author. If the mesh sizing function is suitable, a good configuration of nodes will be created accordingly by the sphere‐packing method and the boundary can be recovered by the local transformation presented here without inserting extra nodes. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Efficient solution of stochastic systems: Application to the embankment dam problem

The embankment dam problem of the benchmark study is treated using the newly developed Stochastic Model Reduction for Polynomial Chaos Representations method. The elastic and shear moduli of the material, in the present problem, are modeled as two stochastic processes that are explicit functions of the same process possessing a relatively low correlation length. The state of the system can thus be viewed as a function defined on a high dimensional space, associated with the fluctuations of the underlying process. In such a setting, the spectral stochastic finite element method (SSFEM) for the specified spatial discretization is computationally prohibitive. The approach adopted in this paper enables the stochastic characterization of a fine mesh problem based on the high dimensional polynomial chaos solution of a coarse mesh analysis. After relatively reducing the dimensionality of the problem through a Karhunen–Loeve representation of the stochastic variables, the SSFEM solution consisting of a high dimensional polynomial in Gaussian independent variables is obtained for the coarse mesh problem. Then the attained solution is used to define a new basis for solving the fine mesh problem. The paper presents some new algorithms for the estimation of chaos coefficients in the presence of complex non-Gaussian dependencies. A numerical convergence study is presented together with a discussion of the results.

Computational fluid dynamics simulation of pedestrian wind in urban area with the effects of tree

The purpose of this paper is to find a more accurate method to evaluate pedestrian wind by computational fluid dynamics approach. Previous computational fluid dynamics studies of wind environmental problems were mostly performed by simplified models, which only use simple geometric shapes, such as cubes and cylinders, to represent buildings and structures. However, to have more accurate and complete evaluation results, various shapes of blocking objects, such as trees, should also be taken into consideration. The aerodynamic effects of these various shapes of objects can decrease wind velocity and increase turbulence intensity. Previous studies simply omitted the errors generated from these various shapes of blocking objects. Adding real geometrical trees to the numerical models makes the calculating domain of CFD very complicated due to geometry generation and grid meshing problems. In this case the function of Porous Media Condition can solve the problem by adding trees into numerical models without increasing the mesh grids. The comparison results between numerical and wind tunnel model are close if the parameters of porous media condition are well adjusted.

Field modeling with sampled distances

Traditional mesh-based approaches to the modeling and analysis of physical fields within geometric models require some form of topological reconstruction and conversion in the mesh generation process. Such manipulations tend to be tedious and error-prone manual processes that are not easily automated. We show that most field problems may be solved directly by using approximate distance fields computed from designed or sampled geometric data, thus avoiding many of the difficult reconstruction and meshing problems. With distances we can model fields that satisfy boundary conditions while approximating the governing differential equations to arbitrary precision. Because the method is based on sampling, it provides natural control for multi-resolution both in geometric detail of the domain and in accuracy of the computed physical field. We demonstrate the field modeling capability with several heat transfer applications, including a typical transient problem and a ‘scan and solve’ approach to the simulation of a physical field in a real-world artifact.

Construction of the discrete geometric conservation law for high-order time-accurate simulations on dynamic meshes

The formulation and implementation of higher-order accurate temporal schemes for dynamic unstructured mesh problems which satisfy the discrete conservation law are presented. The general approach consists of writing the spatially-discretized equations for an arbitrary-Lagrange–Eulerian system (ALE) as a non-homogeneous coupled set of ODE’s where the dependent variables consist of the product of the flow variables with the control volume. Standard application of backwards difference (BDF) and implicit Runge–Kutta (IRK) schemes to these ODE’s, when grid coordinates and velocities are known smooth functions of time, results in the design temporal accuracy of these schemes. However, in general, these schemes do not satisfy the GCL and are therefore not conservative. Using a suitable approximation of the grid velocities evaluated at the locations in time prescribed by the specific ODE time integrator, a GCL compliant scheme can be constructed which retains the design temporal accuracy of the underlying ODE time integrator. This constitutes a practical approach, since the grid velocities are seldom known as continuous functions in time. Numerical examples demonstrating design accuracy and conservation are given for one, two, and three-dimensional inviscid flow problems.

Comparison of the AC losses of BSCCO and YBCO conductors by means of numericalanalysis

This paper presents a quantitative comparison of the AC loss performance of two BSCCOand two YBCO conductors, characterized by the same self-field critical current of 150 A. Wecompare a 37-filamentary BSCCO tape, a 16-filamentary BSCCO square wire, a standardYBCO tape, and a stack of four narrower YBCO tapes. The comparison is made using anumerical technique, based on the finite-element method, which employs a non-linearE–J relation and a dependence of the local critical current densityJc on the local magnetic field. For the simulation of YBCO conductors, a new ‘shell-region’model is utilized. This overcomes the geometry and mesh problems typical ofsuperconductors with very high geometric aspect ratio. Different AC working conditions aresimulated: self-field, applied external field, and combined transport current and externalfield of varying orientation. We outline the advantages of using BSCCO or YBCOconductors for the different applications. Various magnetic field and current density profilesare investigated in order to illustrate the reasons for the loss difference in the fourconductors. Particular attention is drawn to the YBCO tape and the YBCO stack, whoseAC loss characteristics are less well known than those of BSCCO conductors.

Generation of finite element mesh with variable size over an unbounded 2D domain

With the advance of the finite element, general fluid dynamic and traffic flow problems with arbitrary boundary definition over an unbounded domain are tackled. This paper describes an algorithm for the generation of finite element mesh of variable element size over an unbounded 2D domain by using the advancing front circle packing technique. Unlike the conventional frontal method, the procedure does not start from the object boundary but starts from a convenient point within an open domain. The sequence of construction of the packing circles is determined by the shortest distance from the fictitious centre in such a way that the generation front is more or less a circular loop with occasional minor concave parts due to element size variation. As soon as a circle is added to the generation front, finite elements are directly generated by properly connecting frontal segments with the centre of the new circle. In contrast to other mesh generation schemes, the domain boundary is not considered in the process of circle packing, this reduces a lot of geometrical checks for intersection with frontal segments, and a linear time complexity for mesh generation can be achieved. In case the boundary of the domain is needed, simply generate an unbounded mesh to cover the entire object. As the element adjacency relationship of the mesh has already been established in the circle packing process, insertion of boundary segments by neighbour tracing is fast and robust. Details of such a boundary recovery procedure are described, and practical meshing problems are given to demonstrate how physical objects are meshed by the unbounded meshing scheme followed by the insertion of domain boundaries.