Abstract
Radial Basis Functions (RBF) mesh morphing, its theoretical basis, its numerical implementation, and its use for the solution of industrial problems, mainly in Computer Aided Engineering (CAE), are introduced. RBF theory is presented showing the mathematical framework for a basic RBF fit, its MathCAD implementation, and its usage. The equations required for a 2D case comparing RBF smoothing and pseudosolid smoothing based on Finite Elements Method (FEM) structural solution are given; RBF exhibits excellent performance and produces high quality meshes even for very large deformations. The industrial application of RBF morphing to Computational Fluid Dynamics (CFD) is covered presenting the RBF Morph software, its implementation, and a description of its working principles and performance. Practical examples include: physical problems that use CFD, shape parameterisation strategy, and modelling guidelines for setting-up a well posed RBF problem. Future directions explored are: transient shape evolution, fluid structure interaction modelling, and shape parameterization in multi-physics, multi-objective design optimization.
TopIntroduction
Morphing is the ability to change one thing into another smoothly. In modern computer graphics 3D character animation uses morphing for several reasons. The movement of characters is usually gained through motion capture techniques while the backbone of the character is considered to be constrained to rigid motions. However the characters are not rigid, with the obvious exclusion of characters representing mannequins or humanoid robots. Morphing transforms the surfaces of the original model into a new position or shape (see Figure 1 where the motorbike driver position is adjusted in two steps).
Figure 1. An example of a mesh morphing application. A motorbike driver’s position is adjusted in two steps prescribing a rigid motion to the helmet and to the bike while leaving the driver’s body free to deform. In the first step a rotation around the ankle is imposed to change the hunching angle and in the second step a rotation around the neck is added to correct the positions of the head so it maintains a look-ahead view.
In computer graphics however the accuracy of the movement is not important because it just needs to look good. Morphing the mesh required for a numerical simulation is a more complicated and quite a delicate task, especially for a 3D CFD mesh. In this case morphing, also termed smoothing, is not limited to the surface but has to be extended to the entire volume of the mesh and the solver suffers dramatically. Although the concept is basically the same as morphing in computer graphics, i.e. morphing defines the motion of a set of points and moves them accordingly to the action of a motion field. The way the task is accomplished, depends on which smoothing algorithm is selected and on the definition of the control points criteria which can substantially change the result. A good morpher is one that preserves the exact shape that the user wants (i.e. it undergoes a rigid motion where there is a steady object and a null rigid motion prescribed) and gently deforms the surface and volume elements that are within the deformation field and minimises the distortion of each element (i.e. distributing the motion across the entire domain).
In the literature plenty of smoothing algorithms are available. There are so many that they can be difficult to classify. However a key feature that can be used to subdivide them is how they are related to the original mesh. Some methods are defined on the internal discretised domain adopting some physical rule to propagate the movement of the boundary, as for example in the pseudosolid where an elastic FEM solution is used to propagate the deformation inside. Exotic variations of this concept exist: a solid mesh can be replaced with a network of lumped node to node springs, or an equation different to the elasticity one can be solved. The main advantage of mesh related methods is the ability to exactly prescribe surface movements when the field is known; very good quality can be achieved, especially using the pseudosolid method. However a general implementation of such a method can be very complicated because of all the CFD solver mesh features (i.e. the element type or presence of special elements as connectors or interfaces) have to be managed; furthermore it is very difficult to manage free-surface deformation without the help of a geometric modeller.
Other methods are meshless, i.e. define the internal field as an analytic function adopting a point valued vector function. A common and well established technique, the Free Form Deformation (FFD, Sederberg 1986) method, deforms volumes and controls their shape using a trivariate Bernstein polynomial; in this case a network of deforming volumes (typically hexahedron) is built around the model; the deforming action is defined by prescribing the movement of the points that brings the volume and its mesh into a new configuration; the overall deformation field is composed of the field defined within each volume interpolated by means of Bernstein polynomials.
Key Terms in this Chapter
Mesh Morphing: a method for changing the shape of a meshed surface while preserving the topology. Only node positions are updated.
Fluent User Defined Functions (UDF): a set of functions (usually written in C) that allows the user to customize the Fluent CFD solver.
Finite Element Method (FEM): a numerical method used to solve continuum problems replacing the original domain with a set of simple sub-domains (finite elements) connected at nodal points. It is known for its success in solving structural problems.
Computational Fluid Dynamics (CFD): numerical methods for the solution of fluid dynamic/flow problems.
Mesh Smoothing: a method for changing the shape of a volume mesh while preserving the original topology and node distribution. The new shape is obtained by updating nodal positions.
Radial Basis functions (RBF): an interpolation technique for scattered data, it recovers the exact values of a function at given source points; the interpolation is available everywhere as a point function and its behaviour between points depends on the kind of radial function used.
Moving Encap: according to the RBF Morph software notation this is a special volume used to control the RBF problem. It allows the user to place RBF points onto the surfaces of the problem domain with a prescribed resolution and with a field congruent with the one defined (in a simple closed form) inside the volume. Gometric information is also used to specify to the smoother how to move a node, if the node falls inside the Moving Encap it is updated with a straightforward formula. Moving Encaps with null movement (i.e. the default) can be used to protect parts of the volume mesh that fall inside the deforming volume, minimizing the number of RBF points required in the problem.
Pseudosolid: a smoothing technique for a volume mesh that usees a fictitious FEM solution to obtain the desired smoothing field.
Domain Encap: according to the RBF Morph software notation a special volume is used to control the RBF problem. It allows the user to place RBF points onto surfaces in the problem domain with a prescribed resolution and with a null field so that the smoothing field vanishes at the Encap Boundary. The geometric information is also used to specify to the smoother whether a node has to be moved or not. The use of Domain Encaps can dramatically improve the smoother performance especially if only a small detail of a very large mesh has to be updated.
Fluent Text User Interface (TUI): a command line interface based on the Scheme language that allows the user to customize the Fluent CFD solver.