ICES REPORT Isogeometric Analysis of Boundary Integral Equations

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1 ICES REPORT 5-2 April 205 Isogeometric Analysis of Boundary Integral Equations by Mattias Taus, Gregory J. Rodin and Tomas J. R. Huges Te Institute for Computational Engineering and Sciences Te University of Texas at Austin Austin, Texas 7872 Reference: Mattias Taus, Gregory J. Rodin and Tomas J. R. Huges, "Isogeometric Analysis of Boundary Integral Equations," ICES REPORT 5-2, Te Institute for Computational Engineering and Sciences, Te University of Texas at Austin, April 205.

2 Report Documentation Page Form Approved OMB No Public reporting burden for te collection of information is estimated to average our per response, including te time for reviewing instructions, searcing existing data sources, gatering and maintaining te data needed, and completing and reviewing te collection of information. Send comments regarding tis burden estimate or any oter aspect of tis collection of information, including suggestions for reducing tis burden, to Wasington Headquarters Services, Directorate for Information Operations and Reports, 25 Jefferson Davis Higway, Suite 204, Arlington VA Respondents sould be aware tat notwitstanding any oter provision of law, no person sall be subject to a penalty for failing to comply wit a collection of information if it does not display a currently valid OMB control number.. REPORT DATE 2 APR REPORT TYPE 3. DATES COVERED to TITLE AND SUBTITLE Isogeometric Analysis of Boundary Integral Equations 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) University of Texas at Austin,Institute for Computational Engineering and Sciences,Austin,TX, PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 0. SPONSOR/MONITOR S ACRONYM(S) 2. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution unlimited 3. SUPPLEMENTARY NOTES. SPONSOR/MONITOR S REPORT NUMBER(S) 4. ABSTRACT Isogeometric analysis is applied to boundary integral equations corresponding to boundary-value problems governed by Laplace s equation. It is sown tat te smootness of geometric parametriza- tions central to computer-aided design can be exploited for regularizing integral operators. As a result one obtains ig-order collocation metods based on superior approximation and numerical integration scemes and well-conditioned systems of linear algebraic equations. It is demonstrated tat te proposed approac allows one to solve boundary-value problems wit an accuracy close to macine precision. 5. SUBJECT TERMS 6. SECURITY CLASSIFICATION OF: 7. LIMITATION OF ABSTRACT a. REPORT unclassified b. ABSTRACT unclassified c. THIS PAGE unclassified Same as Report (SAR) 8. NUMBER OF PAGES 3 9a. NAME OF RESPONSIBLE PERSON Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-8

3 Isogeometric Analysis of Boundary Integral Equations Mattias Taus, Gregory J. Rodin and Tomas J. R. Huges Institute for Computational Engineering and Sciences University of Texas at Austin Austin, TX 7872 USA April 2, 205 Abstract Isogeometric analysis is applied to boundary integral equations corresponding to boundary-value problems governed by Laplace s equation. It is sown tat te smootness of geometric parametrizations central to computer-aided design can be exploited for regularizing integral operators. As a result, one obtains ig-order collocation metods based on superior approximation and numerical integration scemes and well-conditioned systems of linear algebraic equations. It is demonstrated tat te proposed approac allows one to solve boundary-value problems wit an accuracy close to macine precision. Introduction Isogeometric analysis (IgA) [24, 35] is a framework for numerical scemes for solving boundary-value problems (BVPs) in wic te basis functions coincide wit tose used for geometric parametrizations in computer aided design (CAD). Tus, in contrast to conventional finite element metods, IgA relies on Non-Uniform Rational B-splines (NURBS) [43, 46], T-splines [55, 53] or subdivision surfaces [2, 48, 5] rater tan piecewise polynomials as te basis functions. IgA simplifies mes generation, and tus it significantly sortens te design-troug-analysis process for ig-end engineering components. Furtermore, IgA is advantageous because, in contrast to finite element metods, it fully preserves geometry of CAD-generated sapes and involves basis functions wit attractive properties. Tese features ave given rise to accurate and efficient numerical scemes successfully applied to fluid mecanics [, 3, 5, 7, 8, 2, 29], solid mecanics [39], electromagnetism [20], fluid-structure interaction [9, 0,, 60], structural dynamics [25, 26], plates and sells [5, 6, 27, 28, 37, 22, 23], pase-field models [7, 32, 33], and sape optimization [40, 4, 45, 59]. Boundary element metods (BEMs) are numerical scemes for solving boundary integral equations (BIEs). Like finite element metods, BEMs rely on piecewise polynomials for approximating te geometry and field variables. Tus, by replacing piecewise polynomials wit NURBS or T-splines, one can develop isogeometric BEMs. Tis approac as been already undertaken [3, 4, 3, 38, 42, 44, 54, 56]. Te premise of tis paper is tat IgA can radically improve numerical scemes for solving BIEs because of te additional smootness of NURBS and T-splines in comparison to C 0 -continuous piecewise polynomials. We sow tat one can regularize te key singular integral operators, and construct superior approximation and integration scemes and well-conditioned systems of linear algebraic equations. Tese scemes allow one to solve BIEs wit macine precision. We demonstrate tese advantages of IgA by applying it to BIEs corresponding to Laplace s equation. Tis restriction significantly simplifies matematical aspects of our work, but most of te results can be readily extended to equations of classical elasticity and Stokes equations of fluid mecanics.

4 In te context of BEMs, te proposed approac can be classified as a ig-order collocation metod allowing for weakly-singular, singular, and yper-singular integral operators. Previously tis setting was possible wit Galerkin but not collocation metods. Accordingly, we exploited analytical tools different from tose usually used in analysis of Galerkin metods [49, 58]. In tis regard, we believe tat our approac may be useful for developing matematical foundations for collocation scemes for BIEs. In addition, collocation BEMs dominate engineering applications, tus we believe tat our results are not only of matematical but also practical interest. Te rest of te paper is organized as follows. In Section 2, we define te model BVP, corresponding BIEs, and a proper continuous setting for BVPs defined on domains wit smoot boundaries. In Section 3, we introduce collocation scemes for te integral operators, defined on smoot surfaces. In Section 4, we demonstrate tat all collocated integral operators defined on smoot surfaces can be reduced to weaklysingular integrals. In Section 5, we briefly summarize basic results from IgA and identify extensions needed for solving BIEs. We also describe te extension of our metodology to piecewise smoot surfaces wic allows us to accommodate multiple patces wit C 0 -continuity, degenerate patces resulting in local C 0 -continuity, and extraordinary points or star-points. In Section 6, we consider representative example problems, wic allow us to demonstrate various important matematical and computational aspects. In Section 7, we summarize key results of tis work and briefly discuss directions for future researc. Matematical details are presented in te appendix. 2 Continuous Formulation 2. Model boundary-value problem Consider a bounded domain Ω R 3 wit := Ω. We assume tat is a C 2 -surface ( C 2 ), wic, loosely, means tat can be mapped on R 2, and te inverse of tat map ψ is twice continuously differentiable. For a rigorous definition of C 2 -surfaces we refer to te appendix. Tis restriction C 2 is sufficient for establising matematical foundations for integral equations. A typical CAD parametrization map can be defined as a union of C 2 -surfaces, resulting in a globally Lipscitz ; we denote tis class of surfaces by C 2. Unfortunately, matematical foundations for BIEs defined on C 2 -surfaces are not well developed at tis stage. Tus, in tis and two following sections, we restrict our attention to C 2. On te oter and, te loss of smootness of C 2 -surfaces can be compensated by using appropriate numerical scemes presented in Sections 5 and 6. Te model BVP is formulated for Laplace s equation and mixed boundary conditions, wic include Diriclet and Neumann u = 0 in Ω, () u = g D on D, (2) t := n u = g N on N (3) data. Here n denotes te outward unit normal vector on ; D N = and D N =. At tis stage, we require C 2, but we do not specify te smootness of u, g D, and g N. Tis will be done once we introduce te integral operators. 2

5 2.2 Integral equations Te integral equations equivalent to ()-(3) and teir matematical properties are well known [34, 49, 58]. Tey involve te representation formula tat allows one to determine te solution u in terms of te Caucy boundary data (u, t): u(x) = G(x, y)t(y)ds y n(y) [ y G(x, y)] u(y)ds y x Ω, (4) were G(x, y) = 4π x y is te fundamental solution of Laplace s equation. Te Caucy data can be reconstructed using te singular boundary integral equation (SBIE) ( ) 2 I + K u = Vt on. Here I is te identity, Vt(x) := is te single-layer operator, and Ku(x) := G(x, y)t(y)ds y, x n(y) [ y G(x, y)] u(y)ds y, x is te double-layer operator. Alternatively, te Caucy data can be reconstructed using te yper-singular integral equation (HSBIE) ( ) 2 I K t = Du on, were K t(x) := n(x) [ x G(x, y)] t(y)ds y, x is te adjoint double-layer operator, and Du(x) := n(x) { x n(y) [ y G(x, y)]} u(y)ds y, is te yper-singular operator. x 2.3 Analysis of integral equations Since C 2 one can prove tat te operators K, K : C() C() are compact; see Appendix. Here C() is te space of continuous functions on. Similarly, we adopt te same convention for oter spaces suc as C 2 () wic is te space of twice continuously differentiable functions on. As a consequence of te compactness, Fredolm s alternative implies tat te operators ( ) 2 I + K : C () C (), and ( ) 2 I K : C() C() 3

6 are invertible. Here C () is te space of all functions u C() wit u(y)ds y = 0, and C () is te space of all functions u C() wit u(y)w eq (y)ds y = 0, were w eq = V is te natural weigt, wic as been used in analysis of te yper-singular operator [58]. Furter, compactness of K and K implies tat SBIE and HSBIE sould be solved by inverting 2 I + K and 2 I K, respectively. For te pure Neumann BVP, te SBIE takes te form ( ) 2 I + K u = Vg N on. (5) Since V : C() C() and te boundary data g N must satisfy te solvability condition g N (x)ds x = 0, it follows tat g N C() implies Vg N C (). Terefore, te invertibility of ( 2 I + K) implies tat (5) as a unique solution for u C (). For te pure Diriclet BVP, te HSBIE takes te form ( ) 2 I K t = Dg D on. (6) One can prove tat g D C 2 () implies Dg D C(). Tus te invertibility of ( 2 I K ) implies tat (6) as a unique solution for t C(). Based on analysis of te pure BVPs, it is appropriate to require g D C 2 ( D ) and g N C( N ) for te mixed BVP. Tese restrictions, owever, do not guarantee u C 2 () and t C(). To formulate te BIEs corresponding to te mixed BVP, we define extensions g D C 2 () and g N C(), and express te Caucy data as u = ũ + g D and t = t + g N. (7) By construction, ũ D = 0 and t N = 0. Now we can rewrite te SBIE and HSBIE as te equations for ũ and t: ( ) ( ) V t 2 I + K ũ = 2 I + K g D V g N on, (8) and ( ) ( ) 2 I K t Dũ = D g D 2 I K g N on. (9) Note tat wile g D and g N are not uniquely defined, te structure of (8) and (9) is suc tat u and t are uniquely defined as long as (8) and (9) are uniquely solvable for ũ and t. Since g D C 2 () and g N C(), te rigt-and sides of bot equations are in C(). However, tis is insufficient for establising unique solvability for (8) and (9), under te provisions ũ C 2 () and t C(). 4

7 Remark 2.. It is straigtforward to extend our analysis to te pure Robin BVP in wic te boundary data are prescribed as t + κu = g R on, were κ L () and g R is a prescribed function in C(). Tis problem is similar to te pure Neumann problem. Remark 2.2. Te regularity requirements on, g D and g N can be sligtly relaxed. Te results of tis section can be extended to continuously differentiable, g D wit Lipscitz continuous derivatives, and g N L. Remark 2.3. Wile it appears tat te condition g D C 2 () is too restrictive, in practice g D is often a constant. For example, in te context of eat conduction, Diriclet boundary conditions represent a situation in wic Ω is placed in a constant-temperature environment wose temperature is not affected by Ω. If it is te case, a constant g D is simply extended to te entire boundary, so tat g D is constant. 3 Collocation discretization Let us consider te mixed BVP. Approximations for ũ and t are constructed as ũ (x) = n D A= ũ[a]n D A (x), and t (x) = n N A= t[a]n N A (x), respectively, were NA D(x) and N A N(x) are te basis functions and ũ and t are column-vectors. Since ũ(x) = 0 for x D and t(x) = 0 for x N, te basis functions are suc tat NA D(x) = 0 for x D and NA N (x) = 0 for x N. Accordingly, we define collocation points x D A on N and x N A on D. Tis assignment of te superscripts may be somewat confusing, but it simply reflects te fact tat te basis functions NA D (x) and NA N(x) are supported on N and D, respectively. Upon collocating (8) at x D A and (9) at xn A, one generates te system of linear algebraic equations for ũ and t: ( ) V t 2 I D + K ũ = f S (0) and ( ) 2 I N K t Dũ = f H. () Here te components of te system matrices are defined as V [A, B] := VNB N (x D A) = G(x D A, y)nb N (y)ds y, D (2) I D [A, B] := NB D (x D A), (3) K[A, B] := KNB D (x D A) = n(y) [ y G(x D A, y)]nb D (y)ds y, N (4) I N [A, B] := NB N (x N A ), (5) K [A, B] := K NB N (x N A ) = n ( x N ) A [ x G(x N A, y)]nb N (y)ds y, (6) D D[A, B] := DNB D (x N A ) = n ( x N ) { [ A x n(y) y G(x N A, y) ]} NB D (y)ds y. (7) N 5

8 Te rigt-and-side vectors are defined as f S [A] := 2 g D(x D A) + K g D (x D A) V g N (x D A), f H [A] := D g D (x N A ) 2 g N(x N A ) + K g N (x N A ). For te pure Neumann BVP, all collocation points are x D A, and terefore it is sufficient to solve (0) by setting t = 0, g D = 0, and g N = g N : ( ) 2 I D + K ũ = f S. (8) Te compactness of K can be used to sow tat (8) is uniquely solvable as long as ũ (x)ds x = 0. (9) Furtermore, under condition (9), ũ converges to te exact solution u of (5) at optimal rates. For te pure Diriclet BVP, all collocation points are x N A, and terefore it is sufficient to solve () by setting ũ = 0, g N = 0, and g D = g D : ( ) 2 I N K t = f H. (20) Te compactness of K can be used to sow tat (20) is uniquely solvable and t converges to te exact solution t of (6) at optimal rates. Remark 3.. It is straigtforward to extend te collocation sceme to te pure Robin BVP following te prescription for te Neumann BVP. Also, for te pure Robin BVP, unique solvability and optimal convergence rates can be establised in a way similar to te pure Neumann BVP. Remark 3.2. In all cases, te system of governing algebraic equations is constructed so tat one as to invert matrices associated wit te operators 2 I+K and 2 I K. Tis construction results in well-conditioned linear algebraic systems [2], and it is superior to alternative formulations; see Section 6 for numerical examples. 4 Regularization of operators In general, te SBIE and HSBIE involve integrals containing weakly-singular, singular, and yper-singular kernels. In tis section, we establis tat for C 2 and sufficiently smoot functions, as establised in Section 2, all integrals can be evaluated as weakly singular ones. Tis property is essential as it allows one to develop numerical integration scemes wit spectral accuracy [52]. Te single-layer operator V is naturally weakly singular for te continuous data; te same is true for K and K (see Appendix). To regularize te yper-singular operator D we begin wit approximating u C 2 () in te vicinity of x: u(y) = u(x) + ( T u)(x) (y x) + O( x y 2 ), (2) 6

9 were ( T u)(x) is te tangential gradient of u on. Wit tis approximation, te yper-singular operator can be expressed as Du(x) = n(x) x {n(y) [ y G(x, y)]} u(y)ds y = n(x) x {n(y) [ y G(x, y)]} [u(y) u(x) ( T u)(x) (y x)]ds y u(x) n(x) x {n(y) [ y G(x, y)]} ds y n(x) x {n(y) [ y G(x, y)]} ( T u)(x) (y x)ds y. Since constant and linear functions are armonic, te HSBIE implies tat n(x) x {n(y) [ y G(x, y)]} ds y = 0, n(x) { x n(y) [ y G(x, y)]} ( T u)(x) (y x)ds y = n(x) x [G(x, y)] n(y) ( T u)(x)ds y. Now te last line of (22) can be rewritten as Du(x) = n(x) x {n(y) [ y G(x, y)]} [u(y) u(x) ( T u)(x) (y x)]ds y n(x) x [G(x, y)] n(y) ( T u)(x)ds y. In tis equation, te first term on te rigt-and side is weakly singular because of (2) and te second term is weakly singular because it is equal to K [n(y) ( T u)(x)]. Our development closely follows tat in [36]. An important aspect of te regularization sceme is computing te tangential gradient. Tis issue will be addressed in Section 5 once IgA parametrizations ave been introduced. 5 Isogeometric Analysis 5. CAD geometry In IgA it is presumed tat te surface is described using a CAD tool. Invariably tose descriptions rely on B-splines. A one-dimensional B-spline of degree p is a piecewise polynomial function of degree p. Te smootness between polynomials can be controlled locally and can be up to C p. Tus, if desired, one can construct a B-spline of degree p, wic is C p globally. Multi-dimensional B-splines are constructed as tensor products of one-dimensional B-splines. T-splines are constructed as linear combinations of B-splines witout te need for te tensor-product structure. Tis feature is very attractive, as it allows local refinement wit anging nodes, and terefore we use T-splines. For furter details on T-splines we refer to [53, 54]. (22) First, let us assume tat can be mapped on a rectangular parametric domain ˆ R 2. In CAD te map ϕ : ˆ is constructed in terms of a set of control points P A R 3, weigts w A > 0, and T-splines ˆN A T defined on ˆ: A x = ϕ(ξ, ξ 2 ) := P T Aw A ˆN A (ξ, ξ 2 ) B w ˆN B B T (ξ (ξ, ξ 2 ) ˆ. (23), ξ 2 ) Tis map as numerous advantages over te simpler map A P ˆN A A T (ξ, ξ 2 ). In particular, it can represent quadric surfaces exactly. Furter, since T-splines ˆN A T are a superset of B-splines, te map (23) can be restricted to NURBS, wic are currently an industrial standard. Note tat in principle te map smootness can be controlled by coosing appropriately smoot T-splines. However, in practice it is standard to set 7

10 p = 3 and use C 2 T-splines, so tat ϕ C 2 (ˆ). By adopting a global definition for te map ψ := ϕ, we conclude tat defined by ϕ is a C 2 -surface. Te map defined in (23) requires a rectangular ˆ and terefore it is rater limited. For example, it cannot be used for constructing a cube. In te context of NURBS, tis issue is usually addressed by allowing te parametric domain to consist of multiple rectangular patces. Tis creates a new ost of problems associated wit imposing continuity conditions across patces. Tis issue is naturally resolved wit T-splines, as tey allow for anging nodes and smoot basis functions across patces. Neverteless, even wit T-splines, one as to address extraordinary points. By definition, tose points are intersections of tree or more tan four patces. At extraordinary points, te parametrization map ϕ is only C 0, and as a result C 2. For details we refer to [54]. Anoter way of generalizing (23) is by allowing rectangles to be mapped on triangles by collapsing edges. Tis approac involves two ingredients: (i) in te parametric domain, all T-splines, supported on te edge to be collapsed, are constructed as C 0 functions across te edge, and (ii) control points corresponding to T-splines supported on te edge to be collapsed are assigned to te same position. Like te treatment of extraordinary points, tis construction yields locally C 0 -parametrizations. Tus generalized maps, involving multiple rectangular patces, give rise to C 2. parametric space pysical space ˆ ϕ ξ 2 x 3 x 2 ξ x Figure : Parametric and pysical spaces for a torus. 5.2 Basis functions Let us suppose tat control points P A, weigts w A, and T-splines ˆN A T, prescribing via (23) are given. Ten in te parametric domain te basis functions are constructed using te partition of unity ˆN A (ξ, ξ 2 ) := w A ˆN T A (ξ, ξ 2 ) n B= w B ˆN T B (ξ, ξ 2 ). (24) Te basis functions N A in te pysical domain are constructed via te standard map N A := ˆN A ϕ. (25) Tis construction includes extraordinary points but not collapsed edges. For te latter cases, te basis functions ˆN A supported on a collapsed edge are replaced by a single basis function ˆN = ˆNA. A 8

11 After tat, te corresponding basis function N is constructed via (25). Note tat in our work, bot extraordinary points and collapsed edges give rise to basis functions wic are locally C 2 but globally continuous. For extraordinary points, one could use a constrained optimization framework tat gives rise to C -basis functions; for details see [54]. For solving BIEs, one also needs discontinuous basis functions for approximating t, wic can be discontinuous and even singular even if C 2 and prescribed Caucy data are smoot. It is straigtforward to T,disc define discontinuous T-splines ˆN A. However, CAD parametrizations involve te weigts for continuous T-splines only. Wile, in principle, one can compute te weigts for discontinuous T-splines, and ten use (24), we adopt a simpler construction involving unweigted scaled basis functions: ˆN A (ξ, ξ 2 ) := ˆN T,disc A (ξ, ξ 2 ) n B= w B ˆN T B (ξ, ξ 2 ). (26) [ n Te scaling by B= w ˆN B B T (ξ, ξ 2 )] is motivated by numerical examples rater tan teory. Te construction in (26) is not a partition of unity, but tis property is not required for analysis of BIEs. 5.3 Collocation points Te Greville abscissa of a B-spline is a point in te parametric domain ˆ wose coordinates are defined as te average of te coordinates of te knots. Tese points often correspond to te maximum value of te B-spline. It as been sown tat tey are ideally suited for interpolation and tat tey can be naturally extended to T-splines. Furter, Greville abscissae ave been widely used as collocation points for te basis functions (25) in various numerical metods [6, 4, 50], including tose for BIEs [38, 54, 56, 57]. However, it as been recognized [54] tat for discontinuous T-splines Greville s abscissae may coincide, and terefore one needs to modify te construction. Tis issue as been addressed by introducing 2-ring collocation points [54] for discontinuous cubic T-splines. In tis work, we generalize te definition of te 2-ring collocation points to T-splines of degree p. To tis end, let us consider a one-dimensional B-spline B(ξ) of degree p wit a ξ b. If B(ξ) C ([a, b]), ten te 2-ring collocation point is simply Greville abscissa. If B(ξ) is discontinuous at ξ = a ten te 2-ring collocation point is ξ 2 ring := a + b a p + 2 ; if te discontinuity is at ξ = b, ten ξ 2 ring := b + a b p + 2. To construct te collocation point for a two-dimensional discontinuous T-spline, we can exploit tat locally (rater tan globally) T-splines are tensor products of one-dimensional B-splines. Terefore once a twodimensional discontinuous T-spline is represented by T ij (ξ, ξ 2 ) = B i (ξ )B j (ξ 2 ), one can find te coordinates of te 2-ring collocation point by treating B i (ξ ) and B j (ξ 2 ) separately. 5.4 Numerical Integration It as been establised in Section 4 tat, upon collocation, all operators can be evaluated as weakly-singular integrals on C 2 -surfaces. In tis section, we focus on numerical integration scemes applicable to te operators on C 2 -surfaces. First, let us establis tat all operators can be evaluated as weakly-singular integrals on C 2 -surfaces, as long as approximations are allowed to include discontinuous basis functions. Tis provision is essential as it allows one to move te collocation points away from surface irregularities, associated wit eiter CAD parametrizations of smoot surfaces (extraordinary points and collapsed edges) or non-smoot surfaces. 9

12 (a) 0,0, ,4,4 0,0, ,4,4 (b) 0,0, ,4,4 Figure 2: Greville abscissae (large grey circles) and 2-ring Greville abscissae (small black circles) for a single patc in (a) -D and (b) 2-D. Note tat te abscissae coincide except for te patc boundaries. For discontinuous basis functions te collocation points are restricted to te smoot part of. Ten among te operators defined in (2), K and D can be evaluated using weakly-singular integrals because tey operate on functions tat are smoot in neigboroods of te collocation points. Ten local Taylor expansions can be exploited for regularization, similar to te way it is done in Section 4. Furter, te operator V is naturally weakly singular on C 2 -surfaces. To establis weak singularity of (σi + K) u, let us substitute u(x) = in te SBIE to obtain and te regularization (σi + K) u(x) = (σi + K) = 0 n(y) y G(x, y)u(y) [u(y) u(x)] ds y. Note tat we replaced /2 wit σ to reflect te fact tat is not smoot near certain x. Tis regularization is not sufficient for making te integral weakly singular for u C(). On te oter and if u is Lipscitz continuous, so tat u(y) u(x) < C y x, ten te integral becomes weakly singular. Tis additional restriction on u(x) does not pose problems witin te context of IgA. As usual, in IgA all integrals are evaluated in te parametric domain ˆ. Tus numerical integration scemes ave to be developed, so tat tey properly take into account te geometric parametrization. In particular, te weak singularity, establised in te pysical domain, sould be preserved for integrals defined over te parametric domain ˆ. Furter, since ˆ is partitioned into Bézier elements, defined suc tat witin eac element te supported T-splines are C, locally, te map ϕ and te basis functions are also C. Tis is sufficient for developing numerical integration scemes wit spectral accuracy [52]. If te integrations were carried out in te pysical domain, ten it would be natural to use curvilinear coordinates attaced to te surface. Accordingly, if one uses parametric coordinates, one needs to rely on differential geometry. Alternatively, one can avoid explicit use of curvilinear coordinates by constructing a transformation based on te singular-value decomposition teorem [9]. Tis transformation is constructed as follows:. At a given collocation point, compute te vectors τ and τ 2 as τ = ϕ(η, η 2 ) η and τ 2 = ϕ(η, η 2 ) η 2, were (η, η 2 ) are te coordinates of te collocation point in ˆ. 0

13 2. Form a Jacobian matrix τ τ 2 J = τ2 τ2 2, τ3 τ3 2 were te subscripts refer to te Cartesian components in a global coordinate system, in wic te control points are prescribed. 3. Compute te reduced singular-value decomposition J = UΣV, so tat U is a 3 2 matrix and Σ is a 2 2 diagonal matrix wit no zeros on te diagonal. 4. Define te reparametrization ϕ( ξ, ξ 2 ) of ϕ(ξ, ξ 2 ) via were ξ = ( ξ, ξ 2 ) T. 5. Compute te ortonormal vectors τ and τ 2 as ϕ(ξ, ξ 2 ) = ϕ( ξ, ξ 2 ) = ϕ(v Σ ξ), τ = ϕ( ξ, ξ 2 ) ξ and τ 2 = ϕ( ξ, ξ 2 ) ξ 2, were ( ξ, ξ 2 ) are te transformed parametric coordinates of te collocation point. Tis reparametrization sceme, leading to ortonormal basis vectors, gives rise to optimal numerical integration scemes for weakly-singular integrals [52], [9]. Furter, based on numerical results presented in Section 6, it appears tat te reparametrization sceme is critical for numerical integration for CAD parametrizations involving collapsed edges. Numerical integration in te parametric domain is carried out Bézier-element-wise. If te collocation point is inside te element, ten we apply te reparametrization sceme (ξ, ξ 2 ) ( ξ, ξ 2 ), wic transforms a rectangular element into a parallelogram, on wic one can implement a standard singular integration sceme using a polar coordinate transformation [52]. If te collocation point is outside te element, ten te integrand is regular, and one can use Gaussian quadratures, provided tat te collocation point is not too close to te element. Typically, Gaussian quadratures work well for points located at a distance d > 3, were is an element size, and bot dimensions are defined in te pysical space. If d < 3 one can carry out recursive element subdivision, wic reduces until d > 3. Tus, in effect, Gaussian quadratures are applied wenever te point is outside of te element, but for cases wen d < 3 one needs to combine Gaussian quadratures wit recursive element subdivision. 6 Numerical Examples 6. Overview In tis section, we present numerical examples empasizing various important matematical and computational aspects of IgA of BIEs. All examples involve tree sapes: a torus, a spere, and a cube. Te torus is a C -surface wic allows a C 2 -parametrization; actually one can sow tat te torus parametrization is C but tis additional smootness is not exploited. Te spere is a C -surface wit a C 2 -parametrization, because it involves collapsed edges at te poles. Te cube is a C 2 -surface wit a C 2 -parametrization. All sapes were constructed using standard parametrizations based on 6 (torus), 8 (spere), and 6 (cube) NURBS patces. For eac sape, te patces were used to generate five meses via uniform refinement in te parametric domain, so tat at eac level of refinement, eac element was divided into four. Figure

14 4 sows te two coarsest meses for eac sape. Continuous basis functions of degree p were constructed so tat, upon refinement, tey remained C p locally and continuous over patc boundaries. In contrast to continuous basis functions, teir discontinuous counterparts were C p locally and discontinuous over patc boundaries. Unless oterwise noted, all regular approximations involved p = 2, and degree elevated approximations involved p = 3. Te number of integration points in numerical integration scemes involving te polar coordinate transformation was dependent on te refinement level. At te coarse level, we used 5 points in te radial direction and 0 points in te angular direction. Wit eac refinement, te number of points in te radial direction stayed te same, wile te number of points in te angular direction was increased by tree. For regular integrals, we used te 5 5 Gaussian quadrature rule on eac subelement. Tese rules were establised empirically and no attempts were made to optimize tem. Te majority of numerical examples involved manufactured exact solutions in te form u(x) = x x 0. (27) For eac sape te source point x 0 was cosen far away from te sape center (Fig. 3), so tat u(x) was an analytic function wic did not involve near-singular beavior. Tese manufactured solutions were cosen in order to demonstrate te necessity of numerical scemes even for problems wit smoot solutions. Te function u(x) was used to construct te boundary data g D and/or g N for various BVPs. After tat, appropriate BIEs were solved numerically to reconstruct te full Caucy data. Te quality of numerical solutions was measured using te L 2 ()-error for te Caucy data. To evaluate te order of convergence, we defined te mes size as te square root of te area of te largest Bézier element in te pysical space. Te estimated order of convergence (eoc) for eac refinement was computed as ( ef log eoc = ( ), f log c were te subscripts f and c refer to te fine and coarse meses, respectively. For p = 2 te optimal order of convergence for te L 2 ()-error for te Caucy data is equal to p + = 3. e c ) Unless stated oterwise, all arising algebraic problems were solved using a preconditioned GMRES metod [47] wit a tolerance of 0 2. Eac preconditioner was constructed as te inverse of te interpolation matrix corresponding to te basis functions. As a result, we were able to reveal te spectral properties of te collocated operators and significantly reduce te iteration count. In te remainder of tis section, we present six case studies. Eac study demonstrates te importance of a particular aspect. Tose studies focus on (i) te recursive subdivision sceme for near-singular integration (Section 6.2), (ii) local surface reparametrization (Section 6.3), (iii) exponential convergence of te adopted integration sceme (Section 6.4), (iv) spectral properties of te collocated operators (Section 6.5), (v) discontinuous basis functions (Section 6.6), and (vi) approximations for mixed BVPs (Section 6.7). 6.2 Recursive subdivision for near-singular integration Te objective of tis section is to demonstrate tat te recursive subdivision sceme (Section 5.4) for evaluating near-singular integrals is essential. To tis end, we considered te manufactured pure Neumann BVP on te torus and establised tat te optimal order of convergence and accurate results can be attained only if te subdivision sceme was employed. Te approximate solutions to tis problem were obtained by solving (8) and (9), using continuous basis functions. 2

15 (a) (b) x 0 0; 0; 60 z y x 0 0; 0; 20 z y x x (c) x 0 0.5; 0.5; 0 z y x Figure 3: Tree representative sapes: (a) torus (inner radius r = and outer radius R = 3), (b) spere (radius r = ), and (c) cube (edge lengt a = ). Te source points for te manufactured solutions: x 0 = (0, 0, 60) for te torus, x 0 = (0, 0, 20) for te spere, and x 0 = (/2, /2, 0) for te cube. 3

16 (a) (b) (c) Figure 4: Meses for te first two refinement levels for te (a) torus, (b) spere, and (c) cube. 4

17 Figure 5 presents te L 2 ()-error of ũ for two numerical integration scemes, wit and witout recursive subdivision. It is clear tat recursive subdivision is necessary for attaining te optimal order of convergence. Furtermore, recursive subdivision significantly reduced te magnitude of te errors. Te numerical example is representative of te oter sapes and boundary conditions. 0. L 2 -error e-05 e-06 Witout subdivision Wit subdivision 3.08 Wit subdivision Witout subdivision L 2 -error eoc L 2 -error eoc 2.87E-0 2.4E E-03.5E-0 2.2E E E E E E E E E E E e Figure 5: L 2 ()-errors for two near-singular integration scemes: wit and witout subdivision. 6.3 Surface reparametrization sceme Te objective of tis section is to demonstrate te importance of te surface reparametrization sceme (Section 4) for evaluating singular integrals, particularly wen collapsed edges are involved. To tis end, we considered te manufactured pure Diriclet BVP on te spere. Te approximate solutions to tis problem were first obtained by using te SBIE and discontinuous basis functions. Te corresponding linear algebraic problem is V t = f S. (28) Note tat te SBIE requires one to invert te matrix corresponding to te single-layer operator, wic is not optimal, as far as te spectral properties are concerned. Neverteless, it allows us to demonstrate tat te surface reparametrization sceme is necessary even for a naturally weakly-singular operator. Figure 6 presents te L 2 ()-error for te solutions of (28), using two numerical integration scemes, wit and witout te reparametrization. It is clear tat te reparametrization is necessary for attaining te optimal order of convergence and small errors. Tis example is representative of oter CAD parametrizations involving collapsed edges. Alternatively, one can solve te manufactured problem using te HSBIE (20) and te discontinuous basis functions. In tis case, similar to equation (28), te reparametrization sceme is necessary for accurate integration at collocation points near collapsed edges. Furter te reparametrization sceme is natural for computing te tangent gradient required for regularizing te yper-singular operator. Figure 7 presents te L 2 ()-error of t on te spere using SBIE and HSBIE; results for te SBIE are identical to tose presented in Figure 6. It is clear tat te two approaces yield similar results, and terefore bot are acceptable. Tus we can conclude tat te surface reparametrization sceme is necessary, and it is capable of delivering optimal and accurate numerical solutions even if te yper-singular operator is involved. 5

18 L 2 -error e-05 e-06 Witout reparametrization Wit reparametrization 3.0 Wit reparametrization Witout reparametrization L 2 -error eoc L 2 -error eoc 3.54E E E E-0.5E E E E E E E E E E E Figure 6: Te manufactured pure Diriclet BVP for te spere: L 2 ()-errors for two singular integration scemes, wit and witout reparametrization. 0. L 2 -error e SBIE HSBIE L 2 -error eoc L 2 -error eoc 3.54E E-03.08E E-0.5E E E E E E E E E E E e Figure 7: Te manufactured pure Diriclet BVP for te spere: L 2 ()-errors for two approaces, one based on te SBIE (dased line) and te oter based on te HSBIE (solid line). 6

19 6.4 Exponential convergence of te integration sceme Te objective of tis section is to demonstrate tat te adopted numerical integration sceme is exponentially convergent wit respect to te number of integration points. Te demonstration involves all tree sapes. Tus te integration sceme was tested on problems involving non-smoot surfaces and parametrizations. For eac sape, te test problem was a pure Neumann BVP wit te exact solution u(x) = x + x 2 + x 3, x Ω. For tis coice, one can prove tat te approximation error is exactly equal to zero for every sape, and terefore te cosen test problems are ideal for assessing numerical integration errors. Figure 8 presents te L 2 ()-error of te solution u. Te results were obtained by using te coarsest meses, wile te number of integration points in eac direction was increased. Te results confirm an exponential convergence for every sape. Note tat we were able to reac te macine precision for te spere and te cube, wile for te torus te error stagnated near 0 0 due to round-off errors in te adopted numerical integration sceme. If desired, tis issue can be resolved by using a more sopisticated singular integration sceme proposed in [9]. 6.5 Spectral properties of integral operators It is well-known tat, upon discretization, te operators 2 I + K and 2 I K give rise to well-conditioned matrices. Te objective of tis section is to demonstrate tat one sould coose te governing BIEs so tat one takes advantage of tis property. Accordingly, for pure Neumann BVPs, te SBIE is a natural coice. In contrast, for pure Diriclet BVPs, one sould coose te HSBIE. Tis coice would be problematic for conventional collocation BEMs but it is legitimate for IgA. Tis point will be supported by numerical examples presented in Section Furter, we sow tat for mixed boundary-value problems one sould coose te SBIE on N and te HSBIE on D. Numerical examples presented in Section sow tat, as far as spectral properties and iteration counts are concerned, tis coice is superior tan uniform use of SBIE. Numerical examples involved te manufactured solutions for all tree sapes and results are presented for bot iteration counts and condition numbers Diriclet problems In principle, a pure Diriclet BVP can be solved using eiter te SBIE or HSBIE. Te corresponding linear algebraic problems are and V t = f S ( ) 2 I N K t = f H. Tus te SBIE-based approac requires one to invert V, wereas te HSBIE-based approac requires one to invert 2 I N K. Figure 9 presents te iteration counts and spectral condition numbers κ as functions of for te tree sapes. It is clear tat te HSBIE is a better coice tan te SBIE bot in terms of te iteration counts and spectral properties. For te torus and spere, te iteration counts and spectral properties for te HSBIE are independent of te mes size. Tis is in agreement wit teoretical results based on compactness of K. In tis regard, let us observe tat upon discretization te torus remains a C 2 -surface, wereas te spere becomes a C 2 -surface. Tus te numerical results for te torus are fully expected, wile te results for te spere need additional teoretical considerations. For te cube, bot te iteration count and te spectral condition number sow a logaritmic dependence on. For te cube, te matematical foundations are not well-establised because K is not compact. For te SBIE, te iteration count sould grow as / wereas κ sould grow as /. Surprisingly, tese scalings old only for te cube. It is unclear to us wy te results for te torus are better tan expected and for te spere worse tan expected. 7

20 (a) (b) (c) L 2 -error L 2 -error L 2 -error e-06 e-08 e-0 e-2 e-4 e Integration Points e-06 e-08 e-0 e-2 e-4 e Integration Points e-06 e-08 e-0 e-2 e-4 e Integration Points Int. Pts. L 2 -error.58e E E E E E E E E E E-0 Int. Pts. L 2 -error 2.0E E E E E E E E E E E-5 Int. Pts. L 2 -error.94e E E E E E E E E E E-4 Figure 8: Numerical integration L 2 ()-errors for te (a) torus (b) spere, and (c) cube. 8

21 (a) Number of iterations 00 0 SBIE HSBIE SBIE Number of iterations κ HSBIE Number of iterations 2.87E E E E E κ (b) Number of iterations 00 0 SBIE HSBIE SBIE Number of iterations κ HSBIE Number of iterations 3.54E E E E E κ (c) Number of iterations 00 0 SBIE HSBIE SBIE Number of iterations HSBIE Number of iterations 4.08E E E E E κ κ Figure 9: Iteration counts of te preconditioned GMRES metod and condition numbers κ for te (a) torus (b) spere, and (c) cube. 9

22 6.5.2 Mixed boundary-value problem In tis section, we present numerical examples suggesting tat one sould coose te SBIE on N and te HSBIE on D, as opposed to using te SBIE on te entire. For our purposes, we coose boundary conditions as sown in Figure 0. For te torus and spere, te Diriclet (Neumann) boundary conditions are prescribed on te upper (lower) alves. For te cube, te Neumann boundary conditions are prescribed on te top and bottom faces, and te Diriclet boundary conditions on te oter faces. Similar to pure Figure 0: Mixed boundary conditions for te torus, spere, and cube. Diriclet BVPs, mixed BVPs can be solved wit our witout te HSBIE. We refer to te former metod as SBIE/HSBIE and to te latter one SBIE/SBIE. In te SBIE/HSBIE, te natural domain for te HSBIE is D, as in Diriclet BVPs, wile te SBIE is natural for N. In te SBIE/SBIE, te SBIE must be applied in its unnatural domain D. In terms of linear algebra, SBIE/HSBIE allows one to avoid inverting matrices wit unfavorable spectral properties. Figure presents te iteration counts for te mixed BVPs for te tree sapes. It is clear tat (i) te SBIE/HSBIE is superior to SBIE/SBIE, and (ii) te iteration counts grow wit mes-refinement for all tree problems; for te SBIE/HSBIE sceme, te iteration counts grow logaritmically wit. We do not present comparisons for te spectral properties because suc comparisons strongly depend on te definition of spectral properties. Tat is, te spectral properties for te entire matrices are different from tose obtained using Scur complements; in contrast, te use of Scur complements as minimal effects on iteration counts. 6.6 Discontinuous basis functions Te objective of tis section is to demonstrate tat discontinuous basis functions are critical for approximating t wen it is discontinuous. Tose problems include not only non-smoot domains like a cube, but also mixed BVPs defined on smoot domains; in te latter case, t may be discontinuous at te interface between D and N. For demonstration purposes, we solved te manufactured pure Diriclet BVP on te cube. In tis problem, t is discontinuous at te edges and vertices because of te discontinuous normal. Furter, te normal discontinuity does not allow us to collocate te operators 2 I K and D at te edges and vertices. Tis creates two options: (i) one can use te SBIE wit eiter continuous or discontinuous basis functions, or (ii) one can use eiter te SBIE or HSBIE wit discontinuous basis functions because tey require collocation points off te edges and vertices. We pursue te first option as it allows us to compare continuous versus discontinuous basis functions. 20

23 (a) Number of iterations 00 0 SBIE HSBIE SBIE/SBIE SBIE/HSBIE 2.87E-0 9.5E E E E (b) Number of iterations 00 0 SBIE SBIE/SBIE SBIE/HSBIE 3.54E E E E E HSBIE (c) Number of iterations 00 0 SBIE HSBIE SBIE/SBIE SBIE/HSBIE 4.08E E E E E Figure : Iteration counts of te preconditioned GMRES metod for te (a) torus, (b) spere, and (c) cube. 2

24 Figure 2 presents te L 2 ()-error of t for two approximations, one involves only continuous basis functions and te oter only discontinuous ones. It is clear tat for te continuous basis functions t converges very slowly. In contrast, for te discontinuous basis functions, t converges at te optimal rate and delivers very accurate solutions. Continuous L 2 -error e-06 e-08 Discontinuous 3.00 Continuous Discontinuous L 2 -error eoc L 2 -error eoc 4.08E E E E E E E E E E-02.94E E E-02.38E E e Figure 2: L 2 ()-errors for two approximations, wit and witout te discontinuous basis functions. 6.7 Approximations for mixed boundary-value problems In approximation teory for BIEs, it is well establised tat for mixed boundary-value problems one sould use different approximations for te Caucy data u and t. In particular, to attain te optimal convergence rate for te Caucy data, approximations for u sould be one degree iger tan tose for t [58]. In tis section, we present numerical results supporting tis statement. Furter, we present results suggesting tat approximations of te same degree are capable of delivering te optimal convergence rate for te torus and spere. For demonstration purposes, we considered te same mixed BVPs as in Section 6.5. Tese problems were solved using te SBIE/HSBIE sceme, discontinuous basis functions for approximating t, and continuous regular (p = 2) and degree elevated (p = 3) basis functions for approximating u. Figure 3 presents te L 2 ()-errors obtained using regular basis functions for u. It is clear tat te L 2 ()-error for u converges optimally for all tree cases. In contrast, te rate of convergence for t is optimal for te torus and spere, but not for te cube. Peraps te results for te torus and spere are more surprising tan tose for te cube, as we expected suboptimal convergence rates for t for all tree cases. Figure 4 presents te L 2 ()-errors obtained using degree elevated basis functions for u. It is clear tat in all tree cases te L 2 ()-errors for bot u and t exibit optimal convergence rates. Note tat for te cube te errors corresponding to te degree elevated basis functions are significantly smaller tan tose corresponding to te regular basis functions. For te torus and spere, te errors in u corresponding to te degree elevated basis functions are also significantly smaller tan tose corresponding to te regular basis functions, wic is not surprising simply because we used iger-order approximations. In contrast, for te torus and spere, te degree elevated approximation for u ad a minor impact on te errors for t. 7 Summary In tis paper, we adopted IgA as te foundation for solving BIEs corresponding to Laplace s equation. Accordingly, we focused on problems defined on C 2 -surfaces, wic are common in IgA. Our teoretical 22

25 (a) L 2 -error e-06 e-08 Flux t Potential u Flux Potential L 2 -error eoc L 2 -error eoc 2.87E E E-04.5E E E E E E E E E E E E e (b) L 2 -error e-06 e-08 Flux t Potential u Flux Potential L 2 -error eoc L 2 -error eoc 3.54E-0.04E-02.20E E-0.E E E-0.94E E E-02.99E E E E E e (c) L 2 -error e-06 e-08 Flux t Potential u Flux Potential L 2 -error eoc L 2 -error eoc 4.08E E E E E E E-0.46E E E E E E E E e Figure 3: L 2 ()-errors for te flux t and potential u on te (a) torus (b) spere, and (c) cube. Te approximations involved continuous basis functions degree p = 2 for u and discontinuous basis functions of degree p = 2 for t. 23

26 (a) L 2 -error e-06 e-08 Flux t Potential u Flux Potential L 2 -error eoc L 2 -error eoc 2.87E-0.87E-03.58E-05.5E E E E E E E E E E E E e (b) L 2 -error e-06 e-08 Flux t Potential u Flux Potential L 2 -error eoc L 2 -error eoc 3.54E E E E E E E-0.69E E E-02.86E E E E E e (c) L 2 -error e-06 e-08 e-0 Flux t Potential u Flux Potential L 2 -error eoc L 2 -error eoc 4.08E-0.64E E E E E E-0.90E E E-02.47E E E-02.44E E Figure 4: L 2 ()-errors for te flux t and potential u on te (a) torus (b) spere, and (c) cube. Te approximations involved continuous basis functions degree p = 3 for u and discontinuous basis functions of degree p = 2 for t (solid lines). Te dased lines correspond to te results presented in Figure 3. 24