Isogeometric boundary element analysis using unstructured T-splines

TLDR: This work extends the definition of analysis-suitable T-splines to encompass unstructured control grids and develops basis functions which are smooth (rational) polynomials defined in terms of the Bezier extraction framework and which pass standard patch tests.

About: This article is published in Computer Methods in Applied Mechanics and Engineering.The article was published on 2013-02-01 and is currently open access. It has received 366 citations till now. The article focuses on the topics: Isogeometric analysis & Boundary knot method.

Figures

Figure 14: (a) Radial displacement contours and (b) the errors in radial displacement, |u u h| ||u||1 .

Figure 25: The original Bézier mesh for the propeller model and an exaggerated displacement profile with displacement magnitude superimposed.

Figure 20: The values of spherical stress components along the line 0 ⇡, ✓ = 0 and R = 0.5. It should be noted that all other stress components are zero along this line.

Figure 8: Representation of local tangent and normal vectors at a point s on an element, along with a local orthonormal coordinate system.

Figure C.30: A two-dimensional uniaxial tension problem

Figure 5: The T-spline basis function corresponding to the valence five extraordinary point in Figure 1. (a) The C0 bicubic patches whose control points are shown in Figure 4. (b) The G1 biquartic patches

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Isogeometric boundary element analysis using unstructured t-splines" ?

The authors extend the definition of analysis-suitable T-splines to encompass unstructured control grids ( unstructured meshes ) and develop basis functions which are smooth ( rational ) polynomials defined in terms of the Bézier extraction framework and which pass standard patch tests. The authors believe this work clearly illustrates the power of combining new analysis-suitable computer aided design technologies with established analysis methodologies, in this case, the boundary element method. 

Q2. What have the authors stated for future works in "Isogeometric boundary element analysis using unstructured t-splines" ?

In future work, the authors plan on developing adaptive quadrature schemes which properly account for nearly singular and singular integrals and which take advantage of the smoothness of the underlying T-spline basis [ 110 ]. The authors also plan on extending the range of applicability of the method to encompass other areas of application. 

Q3. What are the basic properties of analysis-suitable T-splines?

Analysis-suitable T-splines are a canonical class of T-splines which possess the basic mathematical properties of NURBS (linear independence, partition of unity, etc.) while maintaining the local refinement property and design flexibility of general T-spline descriptions. 

Q4. What is the importance of smoothness in the analysis?

When used in analysis, all inter-patch connectivity and smoothness must be enforced explicitly in the analysis to ensure consistent (at least continuous) deformations of the geometry. 

Q5. How many NURBS patches are required to model a complicated engineering design?

In traditional NURBS-based design, modeling a complicated engineering design often requires hundreds, if not thousands, of NURBS patches which are usually discontinuous across patch boundaries. 

Q6. What is the common practice for setting the remaining knot intervals to zero?

If a T-mesh boundary is crossed or an extraordinary point is encountered before n knot intervals are constructed, it is common practice to set the remaining knot intervals to zero. 

Q7. What are the properties of analysis-suitable T-splines?

Analysis-suitable T-splines preserve the important mathematical properties of NURBS while providing an efficient and highly localized refinement capability [43, 24]. 

Q8. What are the other tests that are not satisfied by their method?

Other standard patch tests such as shear and rotation are also satisfied by their method but are not shown for the sake of brevity. 

Q9. What is the simplest way to achieve G1 continuity?

Since C0 is not acceptable for most geometric modeling applications, some adjustments to the Bezier control points of the one-neighborhood elements must be made in order to achieve G1 continuity. 

Q10. What was the surprising development in the initial NURBS-based isogeometric?

A surprising development in the initial NURBS-based isogeometric investigations was the tremendous computational advantages that smoothness offers over standard finite elements [3, 4]. 

Q11. How many polynomials are used in the T-spline basis?

The authors focus their developments in this paper on defining the T-spline basis over irregular Bézier elements using one polynomial patch per element. 

Q12. What is the meaning of greville abscissae?

In CAGD, Greville abscissae commonly refer to particular control point positions in physical space which induce a linear geometric map. 

Q13. What are the properties of boundary element methods?

These geometric properties are especially critical in the context of boundary element methods where the behavior and accuracy of the method are strongly influenced by the watertightness, smoothness, and the ability to refine the surface mesh while maintaining exact geometry. 

Q14. What is the way to represent a NURBS model?

Any trimmed NURBS model can be represented by a watertight trimless T-spline [40] and multiple NURBS patches can be merged into a single watertight T-spline [39, 41]. 

Q15. What is the knot interval for spoke edges of an extraordinary point?

In this paper, the authors require that the knot intervals for spoke edges of an individual extraordinary point either be all non-zero or all zero. 

Q16. What is the simplest way to evaluate a weakly singular integral?

It relies on the following expression:C(s) = Z T(s,x) d (x), (44)which is derived in Appendix D. By substituting this expression into (1), the following BIE is obtained:ZT(s,x)(u(x) u(s)) d (x) = Z U(s,x)t(x) d (x). (45)The integral on the left of (45) is now weakly singular and the technique described in Section 8.2 can be used. 

Q17. What is the key feature of the boundary element method?

These early formulations outlined the key feature of the boundary element method (BEM): Discretization is restricted to the boundary of the physical domain.