Elsevier

Computer-Aided Design

Volume 43, Issue 2, February 2011, Pages 133-144
Computer-Aided Design

Bicubic B-spline blending patches with optimized shape

https://doi.org/10.1016/j.cad.2010.10.003Get rights and content

Abstract

Two constructions of bicubic B-spline patches with fixed boundary conditions are described. Their goal is to minimize functionals taken for measures of patch badness. The first construction is numerically solving the triharmonic equation Δ3p=0. The functional minimized in the second construction is the sum of a term determined by the surface shape (the distribution of mean curvature) and a term introduced to overcome the problem of ambiguity of minimum of the first term. In addition to boundary conditions one can impose constraints, e.g. fix constant parameter curves of the patch.

Research highlights

► Constructions of smooth blending patches by shape optimization are proposed. ► The boundary conditions for the surface make it possible to obtain the curvature continuity. ► It is possible to impose constraints, e.g. interpolation conditions, in order to improve the construction results. ► The hernia effect, related to the optimization criteria, and methods for its compensation, are discussed. ► The numerical optimization procedure has a high convergence rate and it is suitable for a parallel implementation.

Introduction

Undulations on surfaces may be undesirable for three reasons: aesthetics, aero- or hydrodynamic and materials’ strength. Given a set of fixed surfaces it is often necessary to construct a blending surface in order to complete the design. In a typical situation boundary conditions and general topology of the blending surface are given and the problem is to find a surface having a satisfactory shape.

One possible approach is optimization with respect to some quality measure. It may be defined as a functional, whose argument is the surface representation. The parameters of the representation, e.g. coordinates of the control points, are then computed by a numerical optimization procedure. The success depends on the choices of the functional, the surface representation and the optimization algorithm.

Minimization of a functional may often be done by solving its Euler–Lagrange differential equation. The first attempt to construct surfaces by solving boundary problems for PDEs was made by Bloor and Wilson [1]; they solved elliptic equations: Laplace and biharmonic and their generalizations. The functionals minimized by solving these equations depend on the parameterization in the sense that their values for different parameterizations of a given surface may be different. Therefore these functionals may be considered as measures of quality (or, more precisely, badness) of parameterizations. A good shape of a resulting surface is a side effect of minimizing parameterization badness.

Greiner [2] and Sarraga [3] minimized parameterization-independent functionals, taking the same values for any smooth enough parameterization of a given surface. The functionals were defined in terms of the mean and Gaussian curvatures and their derivatives. Currently such functionals are commonly used as the surface optimization criteria. However, their Euler–Lagrange equations are nonlinear. The most popular approach for parameterization-independent functionals uses geometric flow fields: a parabolic equation with the time variable is set up and the initial-boundary value problem is solved. The surface evolves towards a stationary state, which is the goal of the construction (a detailed description may be found e.g. in [4]). A drawback of this approach is the slow (linear) convergence, resulting in a large number of necessary time steps, typically from 50 to 1000.

The requirements of geometric continuity contribute to the difficulty of the construction; in the case of blending surfaces it is necessary to obtain a surface having smooth junctions with the given surfaces. The boundary condition determines the order of the differential equation necessary to form a well defined boundary problem. In the case of tangent plane continuity the boundary of the patch and the tangent plane at each point of the boundary are fixed; to specify the tangent planes one can fix the first order cross derivatives. With such a boundary condition the differential equation has to have order 4. For curvature continuity, the normal curvatures at the surface boundary also have to be given, which may be done by specifying the second order cross derivative. The differential equation must then be of order 6. To solve these differential equations numerically, one of the three approaches may be chosen: searching the solution in a finite-dimensional subspace of the appropriate Sobolev space with the classical Galerkin (finite element) method, taking a wider subspace and using the discontinuous Galerkin method [5], or converting the equation of order 2r to a system of r equations of order 2 (see e.g. [6], [7], [8], [9]). An advantage of the last approach is the simplicity of construction of the surface representation; it may be made of planar triangles forming an arbitrarily irregular mesh. On the other hand, this is also a drawback; the result of the construction is not a smooth surface.

A trouble with the functionals, whose values are determined by the surface shape, is that the solutions of their minimization problems are not unique in the appropriate Sobolev spaces, which may lead to the numerical instability of constructions. There are various ways to deal with this problem. For example, the procedure of constructing a surface made of triangles may include a step introducing corrections for the triangles close to degeneration (i.e. having one or two angles too small). Another possibility is to restrict the form of the solution; e.g. in [10], [5] a smooth surface filling a polygonal hole is assumed to be a graph of a scalar function of two variables.

In this paper two constructions of smooth bicubic B-spline blending patches with equidistant knots are described. The first construction is numerically solving the triharmonic equation, which is the Euler–Lagrange equation of a functional T, being a quadratic form, described in Section 3. The second construction finds a minimum of a functional S determined by the shape of the surface. This functional is the integral of the square of length of the mean curvature gradient with respect to the surface measure (the same functional was minimized in [8] by solving a geometric flow equation). Both functionals are chosen so as to penalize undulations; the value of the functional T increases with the growth of undulations of the constant parameter curves of the surface, and the value of S is greater for greater undulations of the surface.

The problem of ambiguity of solution of the minimization problem is overcome by adding a stabilization term, measuring the badness of the parameterization. The order of convergence of the numerical optimization method used in this construction is quadratic. It turns out that shapes of the minimal surfaces of the shape-only-dependent functional may be not satisfactory, and the presence of the stabilization term may improve it. There are also possibilities of introducing corrections, by using affine pre-transformations and by imposing constraints. Examples are discussed in Section 6.

Section snippets

Preliminaries

We are going to construct surface patches described by the formula p(u,v)=i=0N4j=0M4dijNi3(u)Nj3(v),(u,v)Ω=(3,N3)×(3,M3), where dijR3 and the functions Ni3 and Nj3 are cubic B-spline functions defined with the equidistant knot sequences 0,1,,N and 0,1,,M. For any choice of the control points dij the parameterization p has continuous derivatives up to the order 2. The linear space spanned by the functions Ni3Nj3 is a subspace of the Sobolev space H3(Ω).

A more concise form of Formula 

Construction of triharmonic surfaces

The first construction is done by numerically solving the triharmonic equation Δ3p=0in Ω with boundary condition (3). Using Green’s theorem, it is easy to prove that Eq. (4) is the Euler–Lagrange equation of the functional T(p)=defΩΔpF2dΩ. The integrand above is the square of Frobenius norm (sum of squares of all coefficients) of the 3×2 matrix, whose rows are gradients of Laplacians of three scalar functions, x(u,v), y(u,v), z(u,v), which describe the coordinates of the parameterization p

The shape quality measure

A good shape of triharmonic blending surfaces, if achieved, is only a side effect of optimization of the parameterization. To explicitly measure the shape quality, the following functional may be used: S(p)=defMMH22dM. Here M is the surface in R3, whose parameterization is p, H denotes the mean curvature of M and MH is the gradient of the mean curvature on the surface M. Note that the actual parameterization is irrelevant, as the value of this functional is determined by the shape of the

Imposing constraints

Sometimes minimization of a functional in the set of all parameterizations satisfying a given boundary condition does not produce a satisfactory result. One remedy is imposing constraints, i.e. searching for a minimum of the functional in a restricted set of parameterizations. Typically it may be desirable to formulate some interpolation conditions in addition to the boundary condition, for example demanding that the surface pass through a fixed curve. Another possibility is modifying the

Results

Results of both constructions depend on the choice of resolution, determined by the numbers N and M; given an initial patch q it is easy to increase the resolution using the Lane–Riesenfeld algorithm, as described in Section 4.3. Results of the construction based on minimization of the functional F are also influenced by the choice of the constant C (see Section 4.3).

The first example illustrates these influences. Nine patches satisfying a fixed boundary condition are shown in Fig. 3 using

Conclusion

A surface shape optimization procedure may be a powerful design tool. On the other hand, no fixed optimization criterion may replace the designer’s taste and skill, which is also fortunate. The functionals taken for the optimization criterion were used before, but the novelties of the constructions described in this paper include constructing smooth B-spline surfaces instead of sets of triangles, using the parameterization quality term to overcome the ambiguity of minimization of a functional

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