Precursors of Isogeometric Analysis

This chapter discusses the meaning of the conventional “integrated CAD/CAE systems,” which is contradicted from the “CAD/CAE integration” (under the umbrella of isogeometric analysis) adopted throughout this book. The history of several important CAD interpolations since 1964 is outlined. Five precursors of the NURBS-based isogeometric analysis are discussed. The general boundary value problem is posed. In order to solve it, three computational methods, i.e., the finite element method, the Boundary Element Method, and the collocation method are presented in brief. The implementation of Coons and Gordon interpolation formulas in mesh generation is discussed. Moreover, the utilization of the closely related transfinite elements in engineering analysis in conjunction with the aforementioned three major computational methods is discussed.

In this chapter, we deal with several important formulas for approximation and interpolation. First we start with the one-dimensional problem and then we extend to the two-dimensional case. In addition to the classical Lagrange and Hermite interpolation, we also focus on some other interpolations which appear in CAGD theory. An easy way to understand the relationship between approximation and CAGD formulas is to consider the graph of the smooth solution \(U(x,y)\) in a boundary value problem (or the graph of the eigenvector in an eigenvalue problem) as a surface patch described by the function \(z = U(x,y)\). Then it is reasonable to approximate the variable U within this patch using any kind of known CAGD surface interpolation formulas. Fifteen exercises clarify the most important issues of the theory.

This chapter explains that Coons interpolation, which chronologically is the first formula for the mathematical representation of surface patches in Computational Geometry, can be used to derive the closed-form analytical expressions of shape functions that appear in classical isoparametric finite elements of the Serendipity family. Moreover, it is shown that not only Lagrange polynomials but also reduced natural cardinal cubic B-splines as well as most of other interpolations discussed in Chap. 2 can be also used as trial functions along the four edges of a Coons patch macroelement (“C-element”). The latter is generally a large isoparametric finite element with nodal points along the boundary of the patch. In the case of using Curry-Schoenberg (de Boor) B-splines or NURBS along each edge in the Coons patch, the nodal points are merely replaced by the control points. The performance of all these elements is thoroughly investigated through ten examples in two-dimensional and axisymmetric potential and elasticity problems.

This chapter discusses transfinite macroelements, which are based on Gordon interpolation formula. The latter extends Coons interpolation formula (see Chap. 3) considering internal nodes as well. It will be shown that the standard tensor-product elements of Lagrange family constitute a subclass of transfinite elements, while one may generally use more or less internal nodes in several configurations. Moreover, true transfinite elements with different pattern in the arrangement of the internal nodes, as well as degenerated triangular macroelements, are discussed. A class of Cij macroelements is introduced, by influencing the trial functions as well as the blending functions. This class is so wide that can include even an assemblage of conventional bilinear elements in a structured \(n_{\xi } \times n_{\eta }\) arrangement. A careful programming of the shape functions and their global partial derivatives resulted in a single subroutine that includes all twelve combinations. The theory is supported by several test cases that refer to potential and elasticity problems in simple domains of primitive shapes where a single macroelement is used. In a couple of cases, somehow more complex domains are successfully treated using two or three Gordon macroelements.

This chapter discusses the derivation of macroelements based on Barnhill’s interpolation formula within triangular CAD patches. Particular attention is paid to the classical triangular elements up to the sixth degree. In both cases, not only boundary nodes but also internal ones will be considered. The performance of a single macroelement is tested in potential boundary value and eigenvalue problems (Laplace equation and acoustics). For the sake of brevity, the discussion restricts to C⁰-continuity only.

Previously, in Chap. 2 the basics of Bézier interpolation have been exposed. This chapter continues with the construction of univariate (1D) and tensor-product (2D) Bézier-based macroelements. The reader is introduced to the fact that, due to a basis change, the nonrational Bézierian elements are equivalent with Lagrangian ones, in all dimensions. The aforementioned higher-order elements are also compared with the well-known p-method. It is shown that, particularly in the 1D problem, all three approaches (Lagrange polynomials, Bernstein–Bézier polynomials, and p-method) are equivalent, in the sense that each of them includes the same monomials. The theory is supported by eleven exercises. Also, the de Casteljau algorithm in curves and the Bernstein–Bézier triangles are explained in detail through three original Appendices.

In this chapter, we present the theoretical background and discuss the numerical performance of CAD-macroelements in which the solution is approximated with B-splines. Univariate as well as two-dimensional tensor-product interpolation will be considered. Not only the usual Curry–Schoenberg B-splines normalized to provide a partition of unity, but also “reduced cardinal B-splines” are studied (to fully explain some older papers of the CAD/CAE group at NTUA since 1989). Numerical experiments of this chapter restrict to the Galerkin–Ritz formulation and refer to domains or structures of simple 2D primitive shapes (rectangles, circles, and ellipses). The numerical analysis is performed using a single macroelement only, without domain decomposition. The results are also compared with the FEM solution of the same mesh density.

This chapter deals with single macroelements in which the approximation of the variable U is based mostly on rational Bézier and less on nonuniform rational B-splines (NURBS). Since univariate rational Bernstein–Bézier polynomials is a special case of univariate NURBS, it becomes obvious that tensor-product rational Bézier is also a specific case of tensor-product NURBS. The major significance of rational elements is that they accurately represent the geometry of conics (circles, ellipses, parabolas, and hyperbolas). In an instructive way, we focus on the analysis of a circular cavity using a single tensor-product macroelement. It is shown that a single quadratic Bézier macroelement, although is capable of accurately representing the entire circle, it leads to a numerical solution of low quality (slightly worse than the classical nine-node finite element of Lagrangian type). In both cases, this is due to its insufficiency to approximate the eigensolutions (e.g., the eigenvectors in dynamics). Nevertheless, after a sufficient degree elevation which maintains the shape of the circle, it is shown that the higher-order Bézier converges to the exact solution. The presentation continues with a very short summary on the NURBS-based dominating IGA, and the reader is advised for further study.

This chapter deals with plate bending analysis applying several CAD-based interpolations. First the performance of boundary-only Coons interpolation is studied; it will be shown that its simplest form coincides with the well-known BFS element of mid-1960s. Then Gordon interpolation is used (i.e., internal nodes are inserted) in order to improve the accuracy of the numerical solution; it will be shown that the Hermite tensor-product element is a special case. The applicability of Bernstein–Bézier interpolation, as a substitute of Lagrange and Hermite polynomials, is discussed in detail. Also, the use of B-splines is examined and it is clearly shown that the barriers are broken when a control points-based tensor product is applied to curvilinear domains. Numerical examples include rectangular and circular thin plates which are solved using a single macroelement.

This chapter deals with three-dimensional macroelements (large solid bricks) based on several CAD-based interpolation formulas. First, three alternative expressions are derived for the boundary-only Coons interpolation; the first of them is complete, whereas the next two cover particular cases. It will be shown that the classical eight-node trilinear and the twenty-node triquadratic solid elements are the simplest ones of the Coons family. Second, Gordon interpolation in conjunction with internal nodes is fully explained, and it is shown that the classical 27-node tensor-product Lagrangian element is the simplest element of this class. Based on the two aforementioned CAD interpolations (i) simplified edge-only Coons macroelements will be developed, which are hierarchically improved (ii) through enhanced boundary-only formulation (based on the entire faces) as well as (iii) full tensor-product Coons–Gordon macroelements. Nonrational Bézier elements as well as elements based on B-splines will be presented. For the sake of brevity, numerical results reduce to the acoustic analysis of rectangular and spherical cavities, where all formulations are compared to the closed-form exact solution and are thoroughly criticized. The reader is also referred to previously published numerical examples that include potential (Laplace and Poisson’s equations) and elasticity problems in bodies of cuboidal (parallelepiped with six rectangular faces), cylindrical and spherical shape, using a single macroelement.

The CAD-based global approximation of the approximate solution within a patch (or a volume block) leads to large matrices, and therefore, high computer effort is required. The involved matrices may be either fully populated (as happens when using Lagrange and Bernstein–Bézier polynomials) or partially populated (thanks to the compact support of B-splines and NURBS). This fact is the motivation for preserving the global basis functions but replacing the Galerkin–Ritz with a collocation method which is here called the “global collocation method.” In the latter method, each element of the large matrices can be calculated without performing domain integration, since only a substitution of the basis functions into the partial differential operator is needed. Nevertheless, the numerical solution is highly influenced by the location of the so-called collocation points, and this is an open topic for research. Through a number of test problems, it will be shown that the global collocation method performs equally well in 1D, 2D, and 3D, static and dynamic, problems. Numerical results are presented for a broad spectrum of test problems, such as eigenvalue analysis of rods and beams, transient thermal analysis, plane elasticity, rectangular and circular acoustic cavities and plates.

This chapter deals with the numerical solution of three-dimensional boundary value problems using the Boundary Element Method (BEM) in conjunction with CAD-based macroelements. In more details, the boundary is discretized into a certain number of CAD-based patches (Coons, Gordon, Bernstein–Bézier, B-splines, NURBS, Barnhill, etc.), where both the geometry \({\mathbf{x}}(\xi ,\eta )\) and the variable \(U(x,y,z)\) are interpolated through the same CAGD formula. Each of the aforementioned patches is a single isoparametric (or isogeometric) macroelement to which a global approximation of the variable U is applied. The theory is accompanied with numerical results in elasticity problems and acoustics.

While previous chapters focused on the performance of single macroelements, here we study the case of assembling adjacent CAD-based macroelements in which the computational domain has been decomposed. The discussion starts with Coons–Gordon interpolation using Lagrange polynomials, in which no difficulty appears provided the same nodes are used along an interface being at the same time an entire side in both adjacent patches. Obviously, the aforementioned easiness appears to all tensor-product CAD-based macroelements (Bézier, B-splines). If, however, the interface between two adjacent patches is not an entire edge, then Gordon interpolation has to be extended in a proper way using artificial external nodes. The case of two macroelements that share the same edge but do not have the same number of nodes along it is also studied. A short discussion is devoted to issues such as closed surface patches and local control. Numerical examples refer to potential problems (heat flow and acoustics) and elasticity problems (tension and bending).

In this chapter, we summarize the most important issues which we expect that the reader should have learned from the previous chapters. We repeat the six most important CAGD interpolations, and then we sketch a draft picture on the evolution of the computational methods which are based on the first five of them (older than isoGeometric Analysis). As the author happened to have been involved in these older CAD-based methodologies since 1982 and has somehow contributed in a large part of the whole CAE spectrum (FEM, BEM, and global collocation), the short history of this research is given from his point of view; a full list of the sixty papers, properly classified (in FEM, BEM, and collocation categories), is given in Appendix. In some places, the text is inspired by self-criticism. We believe that the compact information provided in this chapter, as well as the details in the whole book, will strengthen and broaden the horizon of researchers and postgraduate students in the field of Computational Mechanics. In more detail, we anticipate that the gap between CAD and CAE approaches and communities will be further reduced, bridging the older with the contemporary ideas.