Constructions of C 1 -Basis Functions for Numerical Solutions of the Fourth-Order Partial Differential Equations

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1 Constructions of C 1 -Basis Functions for Numerical Solutions of the Fourth-Order Partial Differential Equations Hae-Soo Oh 1 and Jae Woo Jeong 2 1 Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC Department of Mathematics, Miami University, Hamilton, OH October 10, 2014 Abstract In [1], Oh et el. introduced the meshfree particles method, a new method to construct global C 1 -basis functions, in which closed form flat-top partition of unity function are multiplied to Lagrange polynomials. Through welding two kinds of smooth functions, patchwise reproducing polynomial particle shape functions of high order were constructed and applied to classical plate problems (Kirchhoff-Love model) to get highly accurate solutions. However, implementing the meshfree particle method may encounter some difficulties for general cases because it requires tracking supports of PU functions that go beyond the boundaries of patches. In this paper, by modifying Bézier polynomials, we construct more directly hierarchical global C 1 -basis basis functions whose implementation is as simple as that of conventional FEM. Hence the proposed method is much easier to use for practical applications than the meshfree particle method. However, the meshfree particle method is applicable to more general plate problems. We show that the Bogner-Fox-Schmit element is a special case of our construction. Keywords: B-spline functions; Bézier polynomials; Biharmonic equation; Collocation Method ; Galerkin Method; Modified B-spline functions. 1 Introduction In this paper, we are concerned with constructions of global C 1 -basis functions by modifying B-spline functions that are almost parallel to that of C 0 -basis functions for conventional p-fem. Thus, the proposed construction of global C 1 -basis functions with compact support is simple, easy to use, and has diverse applicability even though it has some restriction on implementation. Corresponding author. Tel.: ; Fax: ; hso@uncc.edu 1

2 Over the years, several C 1 -finite elements have been suggested. The well known polynomial (or piecewise polynomial) finite elements of class C 1 are the Argyris triangle, the Hsieh-Clough- Tocher triangle, the singular Zienkiewicz triangle, the Bogner-Fox-Schmit rectangle [2] and variants of these types. However, implementing these C 1 -finite elements for the biharmonic equations and the Kirchhoff-Love plate and shell problems is complicated. Thus, the mixed methods or the Reissner-Mindlin plate model have been introduced. The mixed method uses two classes of approximation space such as Raviart-Thomas space and must satisfy the Babuŝka-Brezzi stability condition. The FEM for the Reissner-Mindlin plate model has the locking and the boundary layer problems. Recently, the meshless methods such as RPPM [1], RKPM [3], and PUFEM [4] are used for finite element solutions of the fourth-order partial differential equations by constructing smooth basis functions with aid of partition of unity functions. Most recently, the isogeometric analysis (IGA) introduced by Houghs et al. [5, 6] shed a new light on the methods for numerical solutions of partial differential equations of high order because it is easy to construct highly smooth B-spline basis functions by properly choosing an open knot vector. However, it is not easy to make the patchwise smooth B-spline functions globally smooth functions, when IGA is considered in a patchwise manner for analysis of complicated engineering designs. In this paper, in order to construct C 1 -basis functions to deal with fourth-order partial differential equations in the framework of Galerkin method as well as second order PDEs in the collocation method, we propose to modify the B-spline functions in each patch so that they can be global C 1 -basis functions after they are assembled in the patchwise manner. For this end, we introduce three different medications called, respectively, nodal alteration, side alteration of type I and side alteration of type II. In particular, the modified Bézier polynomials are similar to the the Bogner-Fox-Schmidt element if the polynomial degree is three and similar to C 1 -Q k elements [7] if the polynomial degree is higher than three. This paper is organized as follows: In section two, the construction of B-spline functions and their properties are introduced. We also prove basic theorems that can be used in the modification of B-spline functions, especially Bézier polynomials, so that they can generate global C 1 -functions. In section three, one dimensional C 1 -basis functions are constructed by modifying Bézier polynomials. The construction is generalized for the construction of the hierarchical C 1 - basis functions. These basis functions are tested to various biharmonic equations with clamped boundary conditions. In section four, the constructions of C 1 -basis functions introduced in the previous section is extended to two dimensional cases. We obtain the Bogner-Fox-Schmidt element as a special case of our construction. In section five, we test the proposed methods to two dimensional biharmonic equations with respect to various types of solutions in a rectangular domain as well as an annular domain. Furthermore, the method is also tested to fourth-order equations in polygonal domain. Finally, the concluding remarks are stated in section six and the modification of Bézier polynomials are generalized to piecewise polynomial B-spline functions in appendix. 2

3 1 0.8 N 1,4 N 7,4 N 4,4 0.6 N 2,4 N 3,4 N 5,4 N 6, Figure 1: B-spline functions N i,4 (ξ), i = 1, 2,, 7 of order k = 4 corresponding to the knot vector Ξ = {0, 0, 0, 0, 1/4, 1/2, 3/4, 1, 1, 1, 1}. 2 Construction of Smooth Basis functions A knot vector Ξ = {ξ 1, ξ 2,, ξ m } is a nondecreasing sequence of real numbers in the parameter space [0, 1], and the components ξ i are called knots. An open knot vector of order p + 1 is a knot vector that satisfies ξ 1 = = ξ p+1 < ξ p+2 ξ m p < ξ m p = = ξ m, in which the first and the last p + 1 knots are repeated and the interior knots can be repeated at most p times. The B-spline functions N i,k (ξ) of order k = p + 1 corresponding to the knot vector Ξ = {ξ 1, ξ 2,, ξ m } are piecewise polynomials of degree p which are constructed recursively by the formula (Cox-de Boor): { 1 if ξ i ξ < ξ i+1, N i,1 (ξ) = for 1 i m 1, 0 otherwise, N i,t (ξ) = ξ ξ i N i,t (ξ) + ξ i+t ξ N i+1,t (ξ), for 1 i m t, 2 t k. ξ i+t ξ i ξ i+t ξ i+1 For example, the piecewise cubic B-spline functions N i,4 (ξ), 1 i 7, corresponding to the knot vector Ξ = {0, 0, 0, 0, 0.25, 0.5, 0.75, 1, 1, 1, 1} are depicted in Fig. 1. Properties of B-spline functions (see [6, 8, 9] for details): (1) The B-spline N i,p+1 (ξ) is continuously differentiable up to order p r at an internal knot ξ i, where r is the number of repeats of ξ i in the knot vector. (2) The B-spline functions are useful in design as well as finite element analysis because they have useful properties such as variation diminishing, convex hull, non-negativity, piecewise polynomial, compact support, and partition of unity. For isogeometric analysis (IGA) [5, 6] of an elliptic boundary value problem on non-convex domains [10, 11] and domains with complicated geometry, we would like to apply IGA in a patchwise manner. In such cases, two B-spline functions joined along patch boundaries are not (1) 3

4 smooth enough to be used neither for the collocation method for second order equations nor for the Galerkin method for biharmonic equations. By careful design of a knot vector so that the number of repeats of each interior knot is less than the degree of the B-spline functions, B-spline functions can be constructed to be highly smooth on the interior of the reference domain ˆΩ = [0, 1] (the property (1) above). However, they are only C 0 -functions when B-spline functions are connected along patch boundaries. It is shown in section 3.9 of [8] that two B-spline curves are smoothly connected by selecting control points properly. Instead of modifying the control points, we are going to modify those B-spline functions whose derivatives are not vanishing at end points of ˆΩ. In order to modify B-spline functions to be globally smooth after assembly in the patchwise manner we prove the following useful properties: For 1 < k < n, we inductively construct the following polynomials on [, 1]. h 1 (ξ) = 1 + nξ, h 2 (ξ) = 1 + nξ + n(n + 1) ξ 2, 2 h 3 (ξ) = 1 + nξ +.. h k (ξ) = 1 + k l=1 n(n + 1) ξ n(n + 1)(n + 2) ξ 3, 2 3. n(n + 1)(n + 2)(n + 3) (n + l 1) ξ l l Then we have the following theorem that is used for modifying B-spline functions. Theorem 2.1. For 1 < k < n, let φ R (ξ) = (1 ξ) n h k (ξ) : ξ [0, 1], Then φ L (ξ) = (1 + ξ) n h k ( ξ) : ξ [, 0] φ L () = φ R (1) = 0, φ L (0) = φ R (0) = 1, dφ R (ξ) dx ξ=0 = = dk φ R (ξ) dφ R (ξ) dx k ξ=0 = 0, dx dφ L (ξ) dx ξ=0 = = dk φ L (ξ) dx k ξ=0 = 0, ξ=1 = = dk φ R (ξ) dx k ξ=1 = 0, dφ L (ξ) dx ξ= = = dk φ L (ξ) dx k ξ= = 0. Applying this Theorem, we have the following C k -step functions with k < n: Corollary : ξ (, 0], Φ R (ξ) = φ R (ξ) : ξ [0, 1], 0 : ξ [1, ). 0 : ξ (, ], Φ L (ξ) = φ L (ξ) : ξ [, 0], 1 : ξ [0, ). Moreover, if k = n 1, then they have the partition of unity property (2) (3) (4) (5) Φ R (ξ) + Φ L (ξ 1) = 1 for all ξ (, ). (6) 4

5 Proof. For the proof of the second part, we refer to [4]. Let us consider modifying B-spline functions to be C 1 -basis functions after assembling them in the patchwise manner. For brevity of exposition, we can start with Bézier (Bernstein) polynomials of degree n used for Bézier curves ( in industrial design [8], which are the polynomials n: appearing in the binomial expansion of (1 ξ) + ξ) N 1,n+1 (ξ) = (1 ξ) n N 2,n+1 (ξ) = n C 1 (1 ξ) n ξ N 3,n+1 (ξ) = n C 2 (1 ξ) n 2 ξ 2... N n,n+1 (ξ) = n C n 2 (1 ξ) 2 ξ n 2 N n,n+1 (ξ) = n C n (1 ξ)ξ n N n+1,n+1 (ξ) = ξ n. Except the first two N 1,n+1 (ξ), N 2,n+1 (ξ), and the last two N n,n+1 (ξ), N n+1,n+1 (ξ), the remaining Bézier functions have zero extensions to be global C 1 -basis functions. Bézier functions have many good properties including partition of unity, variation diminishing, convex hull, non-negativity, the symmetry about x = 1/2. Here, the symmetry property means N k,n+1 (ξ) = N n+2 k,n+1 (1 ξ), for all 1 k n + 1. (8) By Theorem 2.1, for 2 k n 1, we modify N 1,n+1 (ξ) and N n+1,n+1 (ξ), respectively to Ñ 1,n+1 (ξ) := (1 ξ) k (1 + kξ) and Ñn+1,n+1(ξ) := ξ k (1 + k kξ) (9) so that they can be C 1 -nodal basis functions when these are assembled in a physical domain in the patchwise manner. That is, if ϕ k : [0, 1] [x k, x k+1 ], is a patch mapping from the reference patch onto a physical patch Ω k = [x k, x k+1 ], then the derivatives of Ñ n+1,n+1 ϕ k and Ñ1,n+1 ϕ k agree at x = x k. In the modifying process, we will keep the following two rules: Symmetry about ξ = 1/2: If the modified shape functions are symmetric about ξ = 1/2, the tensor product of these shape functions can be conveniently assembled to be the global C 1 -basis functions in higher dimensional extensions. Linearly independence of modified shape functions on [0, 1]: For the modified shape functions to have an optimal approximation property, it is desirable that the modified shape functions are linearly independent and span the vector space P n of polynomials of degree n. It is important to note that if a set of modified n + 1 polynomials is weakly linear independent in P n, then those polynomials do not have a good approximation power. In order to modify N k,n+1 (ξ), 1 k n + 1, so that they make the global C 1 -basis functions in the physical domain, it is desirable for the modified polynomial functions Ñk,n+1(ξ) to have the following properties: d dξ Ñk,n+1(0) = d dξ Ñk,n+1(1) = 0, k = 1,, n (7)

6 However, the modified polynomials with these properties can not generate the complete polynomials of degree n. It is easy to check the symmetry about ξ = 1/2, however it is not straightforward to check the linear independence of the modified Bézier functions. For a tool to check linear independence of modified Bézier functions, we consider the inner product on the Hilbert space P n of polynomials of degree n defined by < f(ξ), g(ξ) >= 1 0 f(ξ)g(ξ)dξ, for f(ξ), g(ξ) P n. Let L k, k = 1,, (n + 1) be continuous linear functionals on P n defined by L k (g(ξ)) =< Ñk(ξ), g(ξ) > for g(ξ) P n. (10) It follows from Lemma [12] that L k, 1 k n + 1, are linearly independent in the dual space P n if and only if v(ξ) P n with L k (v) =< Ñk(ξ), v(ξ) >= 0 for k = 1,, n + 1 implies v(ξ) = 0. (11) Suppose a linear combination of Ñk, 1 k n + 1, is zero: n+1 k=1 c kñk = 0. Then ( n+1 n+1 c k L k )(v) =< c k Ñ k (ξ), v(ξ) >= 0, for all v(ξ) P n, k=1 k=1 ( n+1 ) which implies k=1 c kl k = 0 in the dual space Pn. Hence if the linear functionals L k, 1 k n + 1, defined by (10) are linearly independent in the dual space Pn, then c 1 = c 2 =... = c n+1 = 0. Therefore {Ñk 1 k n + 1} are linearly independent in the Hilbert space P n. Similarly, the converse also holds. Since v(ξ) P n is n i=1 C iξ i for some C i R, by (11) it is sufficient to show that L k ( n i=0 C ix i ) = n+1 i=1 C i < Ñi(ξ), ξ (j) >= 0 for k = 1,, n + 1, implies C 1 = C 2 =... = C n+1 = 0. Thus we have the following : Theorem 2.2. The modified Bézier polynomials Ñi,n+1(ξ) are linearly independent if and only if the (n + 1) (n + 1)-matrix with entries < Ñi,n+1(ξ), ξ (j) >, 1 i, j n + 1, is nonsingular. In the following, modifying B-spline functions by using these tools, we construct a family of linearly independent shape functions satisfying the symmetry property and push forwards of the modified polynomials onto physical patches make global C 1 -basis functions. 3 Modifying the Bézier polynomials to construct C 1 -basis functions For brevity, in this section, we present modifications of Bézier polynomials. However, in order to construct C 1 -piecewise polynomial functions, similar modifications can be applied to symmetric 6

7 B-spline functions corresponding to a symmetric open knot vector whose interior knots are symmetric about ξ = 1/2 as shown in appendix A. From now on, we use Greek letters ξ, η for variables in the reference domain ˆΩ = [0, 1] and Roman letters x, y for the variables in the physical domain Ω = [a, b]. Let us note that the first two of Bézier polynomials shown in (7) have non zero derivatives at ξ = 0 and the last two have non zero derivatives at ξ = 1. In order to construct C 1 -basis functions, we modify the first two and the last two of the Bézier polynomials of degree n (order n + 1): N 1,n+1 (ξ) = (1 ξ) n Ñ1,n+1(ξ) Nodal Alteration, N 2,n+1 (ξ) = n C 1 (1 ξ) n ξ Ñ2,n+1(ξ) Side Alteration I or II, N 3,n+1 (ξ) = n C 2 (1 ξ) n 2 ξ 2 N 3,n+1 (ξ) No modification,..... (12) N n,n+1 (ξ) = n C n 2 (1 ξ) 2 ξ n 2 N n,n+1 (ξ) No modification, N n,n+1 (ξ) = n C n (1 ξ)ξ n Ñn,n+1(ξ) Side Alteration I or II, N n+1,n+1 (ξ) = ξ n Ñn+1,n+1(ξ) Nodal Alteration, where Ñ1,n+1(Nodal Alteration) is constructed in (9) and Ñ2,n+1(Side Alteration) is constructed below: We apply Theorem 2.1 for modification of the first and the last Bézier functions as shown in Fig. 2 (a), which shows assembling of two nodal shape functions before and after the modification. Side Alteration I: Noticing that the derivative of N 2,n+1 (ξ) at ξ = 0 is n and the derivative of N n,n+1 (ξ) at ξ = 1 is n, we can join push forward of N n,n+1 (ξ) and pushforward of () N 2,n+1 (ξ) on the adjacent patch to make one global C 1 -function as shown in Fig. 2 (b). However, if they are push forwarded to patches of different patch sizes, their derivatives do not agree. Suppose the linear patch mapping ϕ k from the reference patch ˆΩ = [0, 1] to a physical patch Ω k = [x k, x k+1 ] is defined by ϕ k (ξ) = (x k+1 x k )ξ + x k. For example, if Ω k = [0, 1] and Ω k+1 = [1, 1.25], the derivatives of N 4,5 ϕ k (x) and N 2,5 ϕ k+1 (x) at x k+1 = 1, respectively, are 4 and 6 (lowerly starched red-green ) curve over [1, 1.25] in Fig. 2 (b)). However, derivative of ( J(ϕ k+1 ) N 2,5 ϕ (x) (upper k dotted blue curve over [1, 1.25] in Fig. 2 (b)) at x k+1 beomes 4 since J(ϕ k+1 ) = 1/4. Thus two elemental shape functions have the same derivatives, 4, at the common vertex x k+1 = 1. Here J(ϕ k ) (= x k+1 x k ) is the determinant of the Jacobian of the patch mapping ϕ k. In general, since we have the following relation: J(ϕ k ) d dx N n,n+1(ϕ k (x)) x=x k+1 = J(ϕ k+1 ) d dx N 2,n+1(ϕ k+1 (x)) x=x k+1 (13) 7

8 the scaling factors J(ϕ k ), J(ϕ k+1 ) make the assembled global basis function, defined by J(ϕ k ) N n,n+1 (ϕ k (x)) ( J(ϕ k+1 ) )N 2,n+1 (ϕ k+1 (x)) J(ϕ k ) N n,n+1 (ϕ k (x)) if x ϕ k([0, 1]), = ( J(ϕ k+1 ) )N 2,n+1 (ϕ k+1 (x)) if x ϕ k+1([0, 1]), (14) to be C 1 even for non uniform patches. Side Alteration II: The second method is modifying N 2,n+1 (ξ) and N n,n+1 (ξ) so that their derivatives vanish at both ends of the reference patch [0, 1]. It goes as follows: With a small positive number δ 1, we have the following left and right C 1 -step functions Ψ L (ξ) = Ψ R (ξ) = ( ξ δ ) 2 (3 2( ξ δ ) ) : ξ [0, δ] 1 : ξ [δ, ) 0 : ξ (, 0], ( 1 ξ ) 2 ( 3 2( 1 ξ ) ) : ξ [1 δ, 1] δ δ 1 : ξ (, 1 δ] 0 : ξ [1, ]. Ψ L (ξ) N 2,n+1 (ξ) (shown in Fig. 3) and Ψ R (ξ) N n,n+1 (ξ) become global C 1 -basis functions in the reference patch. Since they are not polynomials, overall accuracy will be significantly decreased if these modifications were applied to all of the physical patches. Thus, we will use these modifications minimally. That is, we use the Side Alteration II only when a patch has non-void intersection with boundaries of a physical domain. Since Ñn+1,n+1(ξ) = Ñ1,n+1(1 ξ) and Ñn,n+1(ξ) = Ñ2,n+1(1 ξ), there are essentially only two alterations in the modification of Bézier functions stated above. For brevity, in this section, we use the same notations for the modified Bézier functions as the original Bézier functions unless there is notational confusion. 3.1 Modified Bézier polynomials [A: Shape functions constructed by modifying Bézer polynomials of Degree 6] The following are shape functions defined on the reference patch ˆΩ = [0, 1] which are linearly independent and their first derivatives are zero at both ends of their supports except N 2,7 (ξ) (15) (16) 8

9 Ñ (1,5) ( x) Ñ (1,5) (x) N (4,5) on [0,1] N (1,5) ( x) J N (2,5) on [1, 1.25] N (2,5) on [1, 1.25] 0 N (1,5) (x) (a) N 5,5ϕ k N1,5ϕ k+1 : (b) N 4,5ϕ k N2,5ϕ k+1 : Figure 2: (a) (Nodal Alteration) Joins of N 1,5 ϕ k and N 1,5 ϕ k+1 before (inner red curve) and after (outer dotted blue curve) modification, where ϕ k ([0, 1]) = [, 0], ϕ k+1 ([0, 1]) = [0, 1]. (b) (Side Alteration I) Join of N 2,5 ϕ k+1 (lower red curve) and N 4,5 ϕ k ; Join of J(ϕ k+1) N 2,5 ϕ k+1 (upper dotted blue curve) and N 4,5 ϕ k on two non uniform patches [0, 1] = [x k, x k+1 ] and [1, 1.25] = [x k+1, x k+2 ]. Here J(ϕ k+1 ) = 1/ N (2,5) (x) ΨL(x)N(2,5)(x) (a) N 2,5(ξ) step function: Figure 3: (Side Alteration II) N 2,5 (ξ) before modifying and N 2,5 (ξ) Left step function (lower red curve) after modifying by multiplying the left step function Ψ L (ξ) with δ = 0.25 defined by (15). 9

10 and N 6,7 (ξ) : N 1,7 (ξ) = (1 ξ) 2 (1 + 2ξ) { () 6(1 ξ) N 2,7 (ξ) = ξ J(ϕ k+1 ) when ϕ k+1 (ˆΩ) does not intersect the boundary, () 6(1 ξ) 5 ξ Ψ L (ξ) when ϕ k+1 (ˆΩ) intersects the boundary N 3,7 (ξ) = 15(1 ξ) 4 ξ 2 N 4,7 (ξ) = 20(1 ξ) 3 ξ 3 N 5,7 (ξ) = 15ξ 4 (1 ξ) { 2 6ξ N 6,7 (ξ) = (1 ξ) J(ϕ k ) when ϕ k (ˆΩ) does not intersect the boundary, 6ξ 5 (1 ξ) Ψ R (ξ) when ϕ k (ˆΩ) intersects the boundary N 7,7 (ξ) = ξ 2 (3 2ξ) (17) 1. Let us note that in order to impose constraint boundary conditions: u(a) = u (a) = u(b) = u (b) = 0, instead of multiplying step functions, one may constrain N 2,7 (ξ) and N 6,7 (ξ) to be zero at the left and right end points of the domain, respectively. 2. Theorem 2.1 implies that the possible choice in P 6 for modification of N 1,7 (ξ) are C 1 : (1 ξ) k (1 + kξ), ( ) for k = 2,, 5 C 2 : (1 ξ) k (1 + kξ + (k + 1)k/2 ξ 2 ), for k = 3, 4 Since we are concerned with C 1 modification, we choose a modification from the first group. [B: Shape functions constructed by modifying Bézier polynomials of Degree 5] The following are modified Bézier functions defined on the master patch [0, 1] which are linearly independent and their first derivatives are zero at both ends except N 2,6 (ξ) and N 5,6 (ξ) : N 1,6 (ξ) = (1 ξ) 2 (1 + 2ξ) { () 5(1 ξ) N 2,6 (ξ) = 4 ξ J(ϕ k+1 ) when ϕ k+1 (ˆΩ) does not intersect the boundary, () 5(1 ξ) 4 ξ Ψ L (ξ) when ϕ k+1 (ˆΩ) intersects the boundary N 3,6 (ξ) = 5(1 ξ) 3 ξ 2 N 4,6 (ξ) = 10(1 ξ) 2 ξ { 3 5(1 ξ)ξ N 5,6 (ξ) = 4 J(ϕ k ) when ϕ k (ˆΩ) does not intersect the boundary, 5(1 ξ)ξ 4 Ψ R (ξ) when ϕ k (ˆΩ) intersects the boundary N 6,6 (ξ) = ξ 2 (3 2ξ) (18) Note that if the first and the last were modified to N 1,6 (ξ) = (1 ξ) 2 (1 + 2ξ) and N 6,6 (ξ) = ξ 2 (3 2ξ), then the set of six polynomials are linearly independent and hence span P 5. [C: Shape functions constructed by modifying Bézier polynomials of Degree 4] The following are shape functions defined on the reference element [0, 1] which are linearly independent and their first derivatives become zero at both end of the reference patch [0, 1] 10

11 except N 2,5 (ξ) and N 4,5 (ξ) : N 1,5 (ξ) = (1 ξ) 2 (1 + 2ξ) { () 4(1 ξ) N 2,5 (ξ) = 3 ξ J(ϕ k+1 ) when ϕ k+1 (ˆΩ) does not intersect the boundary, () 4(1 ξ) 3 ξ Ψ L (ξ) when ϕ k+1 (ˆΩ) intersects the boundary N 3,5 (ξ) = 6(1 ξ) 2 ξ { 2 4(1 ξ)ξ N 4,5 (ξ) = 3 J(ϕ k ) when ϕ k (ˆΩ) does not intersect the boundary, 4(1 ξ)ξ 3 Ψ R (ξ) when ϕ k (ˆΩ) intersects the boundary N 5,5 (ξ) = ξ 2 (3 2ξ) (19) By Theorem 2.1, the eligible C 1 modification for N 1,5 (ξ) within P 4 are : (1 ξ) 3 (1 + 3ξ), (1 ξ) 2 (1 + 2ξ). One can see that either choice yields an optimal solution. In each case, N 5,5 (ξ) should also be modified to N 1,5 (1 ξ). [D: Shape functions constructed by modifying Bézier polynomials of Degree 3] The set of modified Bézier polynomials of the lowest degree are N 1,4 (ξ) = (1 ξ) 2 (1 + 2ξ) { () 3(1 ξ) N 2,4 (ξ) = 2 ξ J(ϕ k+1 ) when ϕ k+1 (ˆΩ) does not intersect the boundary, () 3(1 ξ) 2 ξ Ψ L (ξ) when ϕ k+1 (ˆΩ) intersects the boundary { 3(1 ξ)ξ N 3,4 (ξ) = 2 J(ϕ k ) when ϕ k (ˆΩ) does not intersect the boundary, 3(1 ξ)ξ 2 Ψ R (ξ) when ϕ k (ˆΩ) intersects the boundary N 4,4 (ξ) = ξ 2 (3 2ξ) (20) which are linearly independent, symmetric, and span P 3. In this case, there is only one eligible choice for C 1 modification for N 1,4 (ξ) = (1 ξ) Construction of Hierarchical Shape functions that can be assembled to be C 1 -basis functions in physical space. A: Non symmetric hierarchical construction: The shape functions defined by (20) are hierarchically extended to linearly independent shape functions of degree n 4 as follows: N 1,4 (ξ) = (1 ξ) 2 (1 + 2ξ) { 3(1 ξ) N 2,4 (ξ) = 2 ξ J(ϕ k+1 ) when ϕ k+1 (ˆΩ) does not intersect the boundary, 3(1 ξ) 2 ξ Ψ L (ξ) when ϕ k+1 (ˆΩ) intersects the boundary { 3(1 ξ)ξ N 3,4 (ξ) = 2 J(ϕ k ) when ϕ k (ˆΩ) does not intersect the boundary, 3(1 ξ)ξ 2 Ψ R (ξ) when ϕ k (ˆΩ) intersects the boundary N 4,4 (ξ) = ξ 2 (3 2ξ) ) Nj+1 (ξ) = 6(1 ξ)2 ξ 2 (ξ j 4, j = 4, 5,, n (21) 11

12 However, this hierarchical extension does not satisfy the symmetry condition. One can inductively prove that these (n + 1) modified shape functions are linearly independent in the vector space P n. For example, if n = 6, the modified hierarchic shape functions of (21) can be written as follows: f 1 (ξ) = (1 ξ) 2 (1 ( + 2ξ) ) f 2 (ξ) = 3(1 ξ) 2 ξ () J(ϕ k+1 ) ( ) f 3 (ξ) = 3(1 ξ)ξ 2 J(ϕ k ) f 4 (ξ) = ξ 2 (3 2ξ) f 5 (ξ) = 6(1 ξ) 2 ξ 2 f 6 (ξ) = 6(1 ξ) 2 ξ 3 f 7 (ξ) = 6(1 ξ) 2 ξ 4. (22) One can use Theorem 2.2 to show these monomials are linearly independent. However, in this simple case, it is easier to prove directly. Suppose a linear combination of these functions of (22) is zero: Then for all ξ [0, 1], 7 A k f k (ξ) = 0, for all ξ [0, 1]. k=1 A 1 + 3A 2 ξ + ( 3A 1 6A 2 + 3A 3 + 3A 4 + 6A 5 )ξ 2 +(2A 1 + 3A 2 + 3A 3 2A 4 12A 5 + 6A 6 )ξ 3 +(6A 5 12A 6 + 6A 7 )ξ 4 + (6A 6 12A 7 )ξ 5 + 6A 7 ξ 6 = 0 if and only if all of the coefficients are identically zero, which implies A k k = 1,, 7. Hence {f k : 1 k 7} is a base of P 6. = 0, for all Moreover, the last two columns of Table 2 showing test results implies that the approximation power of the basis functions assembled by the hierarchic shape functions of (22) is the same as that of the modified Bézier functions N k,7 (ξ), 1 k 7, of (17). B: Symmetric modified Shape functions that are partially hierarchical: The shape functions defined by (20) can be extended in a partial hierarchical manner to linearly independent shape functions of degree n 4 as follows: 1. Hierarchic extension to five shape function of degree 4: ( N 1,4, N 2,4, N 3,5 = 6(1 ξ) 2 ξ 2), N 3,4, N 4,4. (23) 12

13 2. Hierarchic extension to six shape functions of degree 5: ) N 1,4, N 2,4, (N 3,6, N 4,6, N 3,4, N 4,4. (24) 3. Hierarchic extension to seven shape functions of degree 6: ) N 1,4, N 2,4, (N 3,6, N 4,7, N 4,6, N 3,4, N 4,4. (25) Note that extensions are not hierarchical whenever the degrees are odd numbers. 3.3 Tests of modified Bézier functions to one-dimensional biharmonic equations Consider the following 1D biharmonic equation with clamped BC: The variational form is d 4 u dx 4 = f in Ω = (0, 1), (26) u = du = 0 at x = 0, 1. (27) dx 1 B(u, v) = F(v) for all test function, v H 2 (Ω) H 1 0 (Ω), 1 d 2 u d 2 v where B(u, v) = 0 dx 2 dx, F(v) = f(x)v(x)dx, and u H 2 (Ω) H dx2 0 1(Ω). 0 In this paper, the relative error in the energy norm is defined as follows: u u h 2 eng,rel = u eng u h eng u eng, (28) where u 2 eng = 1 2 B(u, u) is the strain energy of u. u h and u are an approximate solution and the true solution respectively. The relative error in the maximum norm is defined by u u h,rel = Max( u u h / u ). Example 3.1. We test the modified Bézier functions of degrees 6, 5, 4, and 3, listed in (17), (18), (19), and (20), respectively, to the biharmonic equation (26)-(27) for the following two cases: (1) The exact solution is u(x) = (x x 2 ) 3 and f(x) = x 360x 2. (2) The exact solution is u(x) = e x (x x 2 ) 6 and f(x) = x 2 e x (x) 2 (x 8 +44x x x x x x x+360). 13

14 Testing Galerkin method using modified Bézier polynomials to two biharmonic equations, we obtained the relative errors in the maximum norm and the energy norm in Tables 1 and 2, respectively. From the results in the two tables, we observe the following: 1. Comparing the columns Deg 3 and Deg 5 in Tables 1 and 2, one can see that the higher degree yields better results than the low degree at the same degrees of freedom. Since the proposed method is a high order method, the modified shape functions of higher degree have a stronger approximation power. 2. Oh et al. [4] introduced Reproducing Polynomial Particle Method (RPPM) with use of a flat-top partition of unity and tested to these two biharmonic equations. Their accuracy (Table 4 of [4]) is virtually the same as those in Tables 1 and 2. RPPM is as simple as the proposed method in one dimensional case. However, in the higher dimension, implementing RPPM is more complicated than that of the modified Bézier polynomials. 3. In the last two columns of Table 2, we compared non-hierachical modified Bézier polynomials with hierarchical modified Bézier polynomials to see that they have virtually the same accuracy. Table 1: Relative errors in the maximum norm and in the energy norm ( E,rel = eng,rel ) of biharmonic equation whose true solution is (x x 2 ) 3. u(x) = (x x 2 ) 3 Relative Error in Maximum Norm Deg 3 Deg 4 Deg 5 Deg 6 h-size dof err,rel dof err,rel dof err,rel dof err,rel 1/ E E E E-16 1/ E E E / E E E / E E E Relative Error in Energy Norm h-size dof err E,rel dof err E,rel dof err E,rel dof err E,rel 1/ E E E / E E E / E E E / E E E Two Dimensional Extension For brevity of exposition, we consider the two dimensional extension of modified Beziér blending polynomials of degree 4 listed in (19). A simple extension to the two dimension case is as follows: 14

15 Table 2: Relative errors in the maximum norm and in the energy norm ( E,rel = eng,rel ). The column (Hierach) are relative errors of the solutions obtained by using hierarchical modified Bézier polynomials. u(x) = e x (x x 2 ) 6 Relative Error in Maximum Norm (Hierach) Deg 3 Deg 4 Deg 5 Deg 6 Deg 6 h-size dof err,rel dof err,rel dof err,rel dof err,rel err,rel 1/ E E E E E-05 1/ E E E E E-07 1/ E E E E E-09 1/ E E E E E-11 Relative Error in Energy Norm h-size dof err E,rel dof err E,rel dof err E,rel dof err E,rel err E,rel 1/ E E E E E-03 1/ E E E E E-05 1/ E E E E E-06 1/ E E E E E-06 (1) Consider the following (non-uniform) partitions of (a, b) and (c, d): a = x 1 < x 2 < < x n < x n+1 = b, and c = y 1 < y 2 < < y m+1 = d, such that mesh sizes are δ x i = x i+1 x i, for i = 1, 2,, m; δ y j = y j+1 y j, for j = 1, 2,, n (2) Now by the product of two partitions, Ω = (a, b) (c, d) is partitioned into m n non uniform rectangles. The patchwise and coordinate-wise linear mapping is given by ϕ ij (ξ, η) : ˆΩ = [0, 1] [0, 1] Ω ij = [x i, x i+1 ] [y j, y j+1 ] (x, y) = ϕ ij (ξ, η) = (x i + δi x ξ, y j + δ y j η). (29) (3) For the reference shape functions on the reference patch ˆΩ = [0, 1] [0, 1], we consider the tensor product of the following sets of modified Bézier polynomials: Ñ 1,5 (ξ), N 2,5 (ξ) δi x, N 3,5(ξ),, N 4,5 (ξ) δi x, Ñ 5,5 (ξ) Ñ 1,5 (η), N 2,5 (η) δ y j, N 3,5(η),, N 4,5 (η) δ y j, Ñ 5,5 (η). (30) 15

16 Suppressing the scale factors and denoting N k,5 by N k, the tensor product of two sets of modified Bézier polynomials yields the following 25 reference shape functions: 4 nodal shape functions: Ñ 1 (ξ) Ñ5(η), Ñ 1 (ξ) Ñ1(η), Ñ 5 (ξ) Ñ5(η), Ñ 5 (ξ) Ñ1(η). (31) 12 side shape functions: Ñ 1 (ξ) N 2 (η), Ñ 1 (ξ) N 3 (η), Ñ 1 (ξ) N 4 (η), N 2 (ξ) N 5 (η), N 3 (ξ) Ñ5(η) N 4 (ξ) N 5 (η), Ñ 5 (ξ) N 2 (η), Ñ 5 (ξ) N 3 (η), Ñ 5 (ξ) N 4 (η), N 2 (ξ) Ñ1(η), N 3 (ξ) Ñ1(η), N 4 (ξ) Ñ1(η). (32) 9 inside shape functions: N 2 (ξ) N 4 (η), N 3 (ξ) N 4 (η), N 4 (ξ) N 4 (η), N 2 (ξ) N 3 (η), N 3 (ξ) N 3 (η), N 4 (ξ) N 3 (η), N 2 (ξ) N 2 (η), N 3 (ξ) N 2 (η), N 4 (ξ) N 2 (η). (33) Some of these reference shape functions are depicted in Figs (i) For an internal node, push-forwards of four nodal shape functions are assembled together in the physical domain for one global C 1 -basis function. For each node on Ω, two elemental shape functions make one global C 1 -basis function. These nodal basis functions are those k, k = 1,, 16 in Diagram (48). (ii) Among the inside shape functions of (33), (a) N 3 (ξ) N 3 (η) is the only global C 1 -function. (b) push-forward of each of N 2 (ξ) N 4 (η), N 2 (ξ) N 2 (η), N 4 (ξ) N 2 (η), N 4 (ξ) N 4 (η) makes a global basis function together with three adjacent push-forward shape functions which are the global basis functions numbered 37, 50, 76, 84 in Diagram (48). (c) push-forward of each of N 3 (ξ) N 2 (η), N 4 (ξ) N 3 (η), N 2 (ξ) N 3 (η), N 3 (ξ) N 4 (η) makes a global basis function together with two adjacent push-forward shape functions which are the global basis functions numbered 36, 34, 49, 48, 62, 75, 73, 83, 82, 91, 102, 111 in Diagram (48). (iii) Push-forward of two side shape functions makes one global basis function as shown in Diagram (48). (4) If the domain were partitioned into a rectangular structured grid as shown in Fig. 7 and local stiffness matrices are assembled as shown in Diagram (48), we do not consider an orientation in assembling side elemental shape functions. That is, we do not consider general cases where we have to consider the orientation reversing for the patchwise (elemental) shape functions on the opposite sides like those arguments shown in Chapter 6 in [13] and the Stress Check code (p-version FEM). (5) Imposing essential boundary conditions: u = 0 and u n = For u = 0 along Ω, nodal and side basis functions can be constrained in the standard manner. 16

17 2. For basis functions to have nu = 0 along Ω, using the smooth step functions (Side Alteration method II), we should modify internal basis functions, for example, those corresponding to basis functions in ND 1 and ND 2 which are push-forwards of N 2,5 (ξ) N j,5 (η), 1 j 5, N 4,5 (ξ) N j,5 (η), 1 j 5, and those functions corresponding to ND 3 ND 4 which are push-forwards of N i,5 (ξ) N 2,5 (η), 1 i 5, N i,5 (ξ) N 4,5 (η), 1 i 5, where the following numbers are the basis functions shown in Diagram (48): ND 1 = {29, 30, 31, 20, 45, 46, 40, 46, 58, 59}, ND 2 = {29, 32, 35, 25, 71, 74, 68, 100, 103}, ND 3 = {59, 61, 63, 56, 90, 92, 87, 119, 121}, ND 4 = {103, 104, 105, 94, 112, 113, 107, 120, 121}. (34) Remark 4.1. The Bogner-Fox-Schmit rectangle (Theorem of [2]) which has the the following degrees of freedom Σ K on a square K with verices (0, 0), (1, 0), (1, 1), (0, 1): φ(0, 0), φ ξ (0, 0), φ η (0, 0), φ ξη (0, 0), φ(1, 0), φ ξ (1, 0), φ η (1, 0), φ ξη (1, 0), φ(1, 1), φ ξ (1, 1), φ η (1, 1), φ ξη (1, 1), φ(0, 1), φ ξ (0, 1), φ η (0, 1), φ ξη (0, 1), is a special case of our construction. Suppose Bézier functions of degree 3 are modified as follows: N 1 (ξ) = (1 ξ) 2 (1 + 2ξ), N 2 (ξ) = (1 ξ) 2 ξ, N 3 (ξ) = N 2 (1 ξ) = (1 ξ)ξ 2, N 4 (ξ) = N 1 (1 ξ) = ξ 2 (3 2ξ). Then the set {N i (ξ) N j (η) : 1 i, j 4} is the Bogner-Fox-Schmit elements with respect to the degrees of freedom listed above. Similar constructions for rectangular C 1 -finite elements can also be found in [7, 14]. 4.1 Pullback of the bilinear form for 4th order problems onto the reference patch Let Φ : ˆΩ Ω be a mapping from the parameter space to the physical space. Then we have û = u Φ, where u is a differentiable function defined on Ω. (35) ( x u) Φ = J(Φ) ξ û; or (36) u x Φ = J11 + J12 u y Φ = J21 + J22 where Φ(ξ, η) = (x(ξ, η), y(ξ, η)), J 11 = x dξ ; J 12 = y dξ, J 21 = x dη ; J 12 = y dη. 17

18 (a) N 4,5(ξ) N 1,5(η): (b) N 2,5(ξ) N 1,5(η): (c) N 1,5(ξ) N 1,5(η): (d) N 5,5(ξ) N 1,5(η): Figure 4: Shape functions (a) side shape function : N 4,5 (ξ) N 1,5 (η) (b) side shape function : N 2,5 (ξ) N 1,5 (η) (c) nodal shape function: N 1,5 (ξ) N 1,5 (η) (d) nodal shape function : N 5,5 (ξ) N 1,5 (η) 18

19 ( ) ( (a) N 4,5(ξ) N 1,5(η) ( N 2,5(ξ)) ) N 1,5(η) : ( ) ( (b) N 5,5(ξ) N 1,5(η) N 1,5(ξ) ) N 1,5(η) : Figure [ 5: (a) Join of two shape ] functions shown in (a) and (b) of Fig. 4: [ ] push-forward of (N 4,5 (ξ) N 1,5 (η)) push-forward of ( N 2,5 (ξ) N 1,5 (η)) ; (b) Join of [ two shape functions shown in (c) and (d) of Fig. 4: push-forward of (N 5,5 (ξ) ] [ ] N 1,5 (η)) push-forward of (N 1,5 (ξ) N 1,5 (η)) (a) N 3,5(ξ) N 1,5(η): (b) N 3,5(ξ) N 3,5(η): Figure 6: (a) Side shape function: N 3,5 (ξ) N 1,5 (η) (b) Inside shape function: N 3,5 (ξ) N 3,5 (η). 19

20 Using (36), we have ( x u x ) Φ = J(Φ) ξ (u x Φ) = J(Φ) ξ (J 11 ûξ + J 12 ûη) (37) [ uxx Φ u xy Φ ] [ = J(Φ) (J 11 ûξ + J12 ûη) ξ (J11 ûξ + J12 ûη) η ] (38) Similarly we have ( x u y ) Φ = J(Φ) ξ (u y Φ) = J(Φ) ξ (J 21 ûξ + J 22 ûη) (39) [ uyx Φ u yy Φ ] [ = J(Φ) (J 21 ûξ + J22 ûη) ξ (J21 ûξ + J22 ûη) η ] (40) Suppose ϕ(x, y) = ˆϕ Φ (x, y). Then ( xx ϕ) Φ = J 11 ( yy ϕ) Φ = J 21 ( xy ϕ) Φ = J 21 ( yx ϕ) Φ = J 11 ξ ξ ξ ξ (J 11 (J 21 (J 11 (J 21 ξ ξ ξ ξ ˆϕ + J 12 ˆϕ + J 22 ˆϕ + J 12 ˆϕ + J 22 η η η η ˆϕ) + J 12 ˆϕ) + J 22 ˆϕ) + J 22 ˆϕ) + J 12 η η η η (J 11 (J 21 (J 11 (J 21 ξ ξ ξ ξ ˆϕ + J 12 ˆϕ + J 22 ˆϕ + J 12 ˆϕ + J 22 η ˆϕ), η ˆϕ). η ˆϕ). η ˆϕ). (41) Suppose u(x, y) = û ϕ i (x, y) and v(x, y) = ˆv ϕ i (x, y), and ϕ i : ˆΩ Ω i is a coordinatewise linear elemental mapping (29). Then by these formula, the bilinear form defined on physical domain can be computed by the reference shape functions as follows: ( ûξξ u vdxdy = Ω i ˆΩ (δ y ) 2 + ûηη )( ˆvξξ (δ x ) 2 (δ y ) 2 + ˆv ) ηη (δ x ) 2 δ x δ y dξdη. (42) 5 Tests of two dimensional biharmonic equations In the framework of Galerkin method, C 1 -basis functions constructed by modified Bézier polynomials are tested to the biharmonic equations of various types. 5.1 Biharmonic equations on rectangular domains In Example 5.1, the square domain [0, 1] [0, 1] is uniformly partitioned for finite element solutions. 20

21 Example 5.1. (Biharmonic Equation on Square with Uniform Mesh) u = f in Ω = (0, 1) (0, 1) u = u = 0 along Ω n where u(x, y) = sin 6 (πx) sin 6 (πy) is the exact solution (test problem of [7]). We test modified Bézier polynomials in the frameworks of h-fem as well as p-fem to get results in Table 3 (h-version) and Table 4 (p-version), respectively. Comparing results in two tables, we conclude that the p-version of modified Bézier polynomials is superior to the h-version. Table 3: (h-version test) Relative errors in the maximum norm and in the energy norm, and computed energy norm. In the column dog = 81(121), 121 is the total number of the global C 1 -basis functions used, and 81 is the actual number of unknowns. The numbers in the column Deg represent the polynomial degrees, and represents the true solution. Num. of Patches Deg dof err,rel err eng,rel Computd Energy (64) 5.09E E E (196) 1.03E E E (676) 4.23E E E (2500) 1.48E E E E+02 Table 4: (p-version test) Relative errors in the maximum norm and in the energy norm, and computed energy norm. Num. of Patches Deg dof err,rel err eng,rel Computed Energy (64) 5.09E E E (100) 2.90E E E (144) 2.79E E E (196) 1.22E E E (256) 1.40E E E (324) 4.98E E E (400) 5.63E E E E+02 21

22 (, 1) (1, 1) (, ) (1, ) Figure 7: Domain with non-uniform mesh Remark 5.1. In order to impose the clamped boundary conditions, Galerkin method using Bogner-Fox-Schmit elements or C 1 -Q k elements removes those basis functions whose normal derivatives are not zero along boundary, whereas those basis functions are modified to have zero normal derivatives on boundaries in modified Bézier polynomials (Side Alteration II). However, numerical experiments show that retaining those basis functions with non-vanishing normal derivatives (after modified by Side Aleteration II) does not allow much gains in accuracy for this example. In next example, we test the modified Bézier polynomials to a biharmonic equation on the rectangular domain with non-uniform mesh. Example 5.2. (Biharmonic Equation on Square with Non-Uniform Mesh) Consider a biharmonic equation with clamped boundary condition u = f in Ω = (, 1) (, 1), (43) u = u = 0 along Ω. n The non-uniform mesh on Ω used for numerical solutions in Table 5 is shown in Fig. 7. To get relative errors in the maximum norm and the energy norm shown in Table 5, we test Galerkin method with use of modified Bézier polynomials to three biharmonic equations with the following three exact solutions, respectively: Test 1: u(x, y) = (1 x 2 ) 2 (1 y 2 ) 2, Test 2: u(x, y) = (1 x 2 ) 3 (1 y 2 ) 3, Test 3: u(x, y) = e x+y (1 x 2 ) 3 (1 y 2 ) 3. Results in Table 5 show that two dimensional extensions of modified Bézier polynomials yield as accurate results as those in one dimensional cases. In next example, we consider biharmonic equations on annular domains. 22

23 Table 5: Relative errors in the maximum norm and in the energy norm. In the column dof = 81(121), 121 is the total number of global C 1 -basis functions used, and 81 is the actual degrees of freedom used for numerical solutions after imposing boundary conditions. Relative error in the maximum norm Test 1 Test 2 Test 3 Deg dof err,rel err,rel err,rel 4 81(121) 6.44E E E (196) E E (289) E E (400) E (1681) E-13 Relative error in the energy norm Test 1 Test 2 Test 3 Deg dof err eng,rel err eng,rel err eng,rel 4 81(121) 1.17E E E (196) E E (289) E E (400) E (1681) E-07 Example 5.3. (Biharmonic Equation on Annulus) Consider the biharmonic equation (43) when Ω = {(x, y) : 1 x 2 + y 2 4, x 0, y 0} is one quarter of an annulus (Fig. 8), and the true solution is ( 2. u(x, y) = xy(x 2 + y 2 1)(x 2 + y 2 4)) Suppose a patch mapping ϕ I : [0, 1] 2 Ω I is defined by ϕ I (ξ, η) = (x(ξ, η), y(ξ, η)), where ( x(ξ, η) = (0.7η + 1) cos π 6 ξ + π ) 2 ( and y(ξ, η) = (0.7η + 1) sin π 6 ξ + π ). 2 Then J(ϕ I ), the Jacobian of ϕ I, is [ x ξ = ( ) ( π 6 (0.7η + 1) sin π 6 ξ + π ) 2, y ξ = ( π ) ( 6 (0.7η + 1) cos π 6 ξ + π ) 2, x η = (0.7) cos ( π 6 ξ + π ) 2, y η = (0.7) sin ( π 6 ξ + π ) 2 ]. 23

24 y Ω II Ω I 1 Ω III Ω IV O π x Figure 8: One quarter annulus Ω and annular partition. and the inverse J (ϕ I ) is [ 6 (0.7) sin ( π 6 ξ + π ) ( π ) ( 2 6 (0.7η + 1) cos π 6 ξ + π ) 2 0.7π(0.7η + 1) ( 0.7) cos ( π 6 ξ + π ) ( π ) ( 2 6 (0.7η + 1) sin π 6 ξ + π ) 2 ]. Thus, for a reference shape function ˆR defined on [0, 1] 2, we have ( ) xy ˆR ϕ I ϕ I (ξ, η) = J (ϕ I ) ξη ˆR(ξ, η). For example, if ˆR k4 = N k,5 (ξ)n 4,5 (η) = N k,5 (ξ) ( 4(1 η)η 3) is a product of modified Bézier polynomials, then, along η = 1 (the edge [0, 1] {1} of the reference domain), we have ( ) [ ( xy R ˆ k4 ϕ 6 (1.7) π ) ( I ϕ I (ξ, 1) = 6 cos π 6 ξ + π ) 2 Nk,5 (ξ)n 4,5 (1) ] 0.7π(1.7) (1.7) ( ) ( π 6 sin π 6 ξ + π ) 2 Nk,5 (ξ)n 4,5 (1) = 1 [ ( cos π 6 ξ + π ) 2 Nk,5 (ξ)n 4,5 (1) ] 0.7 sin ( π 6 ξ + π ) 2 Nk,5 (ξ)n 4,5 (1). Now, let us consider the counterparts of the patch mapping ϕ I and ˆR k4, which are ϕ II : [0, 1] 2 Ω II and ˆR k,2 = N k,5 (ξ)n 2,5 (η) = N k,5 (ξ) ( 4(1 η) 3 η ), respectively. The patch mapping ϕ II is defined by ϕ II (ξ, η) = (x(ξ, η), y(ξ, η)), where ( x(ξ, η) = (0.3η + 1.7) cos π 6 ξ + π ) 2 Thus, along η = 0, we have ( ) xy R ˆ k2 ϕ II ϕ II (ξ, 1) = 1 [ ( cos π 0.3 ( and y(ξ, η) = (0.3η + 1.7) sin π 6 ξ + π ) ξ + π ) 2 Nk,5 (ξ)n 2,5 (0) ] sin ( π 6 ξ + π ) 2 Nk,5 (ξ)n 2,5 (0).

25 Hence, we can modify the reference shape functions as follows: ˆR k,4 = N k,5 (ξ)n 4,5 (η) N k,5 (ξ)(0.7n 4,5 (η)) ˆR k,2 = N k,5 (ξ)n 2,5 (η) N k,5 (ξ)( 0.3N 2,5 (η)). Then the global shape function defined by joining two shape functions ˆR k,4 ϕ I and ˆR k,2 ϕ II becomes a C 1 -function. Similarly, between the subdomains Ω I and Ω III, we modify the reference shape functions (that is, Side Alteration I) ( N 4,5 (ξ)n k,5 (η) 6 ) π N 4,5(ξ) N k,5 (η) = modified ˆR 4,k ( ) 3 N 2,5 (ξ)n k,5 (η) π N 2,5(ξ) N k,5 (η) = modified ˆR 2,k. Then the function defined by joining two push-forward of modified shape functions, ˆR 4,k ϕ I and ˆR 2,k ϕ III, also becomes a C 1 -function on Ω I Ω III. Since the true solution is a polynomial of degree 10 in each variable, one can not expect a highly accurate solution when basis functions of lower degree are employed as shown in Table 6. Table 6: Relative errors in the maximum norm and in the energy norm. In the column dof, the number in parentheses is the total number of global C 1 -basis functions before imposing boundary conditions. Deg dof err,rel err eng,rel Computed energy 6 100(144) 1.24E E E (256) 1.32E E E (400) 3.47E E E (576) 6.81E E E (784) 5.32E E E E+04 Remark 5.2. This problem can be solved in the framework of Isogeometric Analysis (IGA). Consider a design mapping ϕ : ˆΩ = [0, 1] 2 Ω(one quarter of an annulus) defined by ϕ(ξ, η) = (1 η)(cos ξπ/2, sin ξπ/2) + η(2 cos ξπ/2, 2 sin ξπ/2). Then ϕ is a bijective C -mapping. Now we can use tensor product of modified B-spline functions constructed in section A of appendix for basis functions on the reference domain. Even though this setting needs more work for implementing, we expect to have similar results to those in Table 6. 25

26 5.2 Biharmonic equations on polygonal domains In the previous subsection, we tested biharmonic equations on rectangles and annuli. In this subsection, we introduce two approaches to deal with biharmonic equations on polygonal domains Cutting rectangular patches by slant smooth step functions We can relax the domain restriction by considering the polygonal boundaries as cut off rectangles by using smooth step functions as shown in Fig. 9. Example 5.4. (Polygonal Domain) Consider the biharmonic equation u = f in Ω, u = u = 0 on Ω, n where Ω = {(x, y) : 0 x, y 2, x + y 1} is a polygon shown in Fig. 9. The true solution is ( 2. u(x, y) = x(x 2)y(y 2)(x + y 1)) y 1 Ω I Ω III Ω II O x Figure 9: The domain Ω and partition into eight patches. The function values of the step function is between 0 and 1 on the shaded region. The width of the strip is δ. For numerical integrations, the domain has been divided into 12 sub-domains. Let ϕ I : [0, 1] 2 Ω I be the coordinate-wise linear patch mapping, where Ω I is the rectangle containing Ω I. Then the pull-back of Ω I to the reference patch is as shown in Fig. 10. By the translation (ξ, η) (ξ, η η ) and the rotation by θ 1, we define an affine mapping A as follows: A(ξ, η) = (ξ cos θ 1 (η η ) sin θ 1, ξ sin θ 1 + (η η ) cos θ 1 ), (44) 26

27 η 1 (0, η + δ 1 + (η ) 2 ) η (1, δ 1 + (η ) 2 ) 0 θ 1 1 ξ Figure 10: Pullback of the patch Ω I into the reference domain by the coordinate-wise linear patch mapping. where sin θ 1 = η 1 + (η ) 2, cos θ 1 = (η ) 2, tan θ 1 = η (45) Define ˆΨ L (ξ, η) = Ψ L (η), where Ψ L (η) is the left step function defined by (15). Next we define an oblique step function by Ψ obq = ˆΨ L A. Then we have Ψ obq (ξ, η) = ( ˆη δ ) 2 (3 2( ˆη δ ) ) : ˆη [0, δ] 1 : ˆη [δ, ) 0 : ˆη (, 0], (46) where ˆη = ξ sin θ 1 + (η η ) cos θ 1 = (ξη + (η η ))/ 1 + (η ) 2. Then the oblique step function Ψ obq has we have the following properties. 1. If (ξ, η) lies on the line η = η ξ + (η + δ 1 + (η ) 2 ), then ˆη = δ. 2. If (ξ, η) lies on the line η = η ξ + η, then ˆη = Outward normal vector along the line η = η ξ + η is n = (sin θ 1, cos θ 1 ). 4. The gradient of Ψ obq is as follows: ( Ψobq ξ = 6ˆη sin θ 1 (δ ˆη)/δ 3, Ψ obq η = 6ˆη cos θ 1 (δ ˆη)/δ 3). 27

28 Table 7: Relative errors in the maximum norm and in the energy norm, and computed energy. In the column dof, the number in parentheses is the total number of global C 1 -basis functions. The numbers in the column δ represent the δ-values in the construction of the step function. Deg dof err,rel err eng,rel Computed Energy δ 6 209(264) 4.42E E E E (264) 3.52E E E E (264) 3.41E E E E (264) 3.40E E E E (480) 3.78E E E E (480) 3.51E E E E (480) 3.39E E E E (480) 3.42E E E E (760) 3.56E E E E (760) 3.51E E E E (760) 3.39E E E E (760) 4.67E E E E E Thus, the normal derivative of Ψ obq is ( Ψ obq ) n = 6ˆη(δ ˆη)/δ By the property (2) above, the normal derivative of Ψ obq along the line η = η ξ + η is vanishing. Now the modified reference shape functions ˆN(ξ) ˆN(η) used for biharmonic equation rectangular domains are modified further by Ψ obq (ξ, η) ˆN(ξ) ˆN(η). Similarly, the patchwise shape functions corresponding to Ω II and Ω III can be moved back to the reference domain for the calculation of the bilinear form. The numerical solutions obtained by these modifications are listed in Table Modifying shape functions by step functions so that their derivatives are vanishing at both ends Consider the following modifications of Bézier function of degree 4. N 1,5 (ξ) = (1 ξ) 3 (1 + 3ξ) N 2,5 (ξ) = 4(1 ξ) 3 ξ Ψ L (ξ) N 3,5 (ξ) = 6(1 ξ) 2 ξ 2 N 4,5 (ξ) = 4(1 ξ)ξ 3 Ψ R (ξ) N 5,5 (ξ) = ξ 3 (4 3ξ). (47) 28

29 Ω I Ω IV Ω II Ω III Figure 11: Quadrangular patch and rectangular patches Here Ψ L (ξ) and Ψ R (ξ) are the left and the right step functions, respectively, defined by (15). The tensor products N i (ξ)n j (η), 2 i, j 4 and their normal derivatives of these shape functions are vanishing along ˆΩ. Hence, if Ω k is a quadrangular patch and ϕ k : ˆΩ Ω k is a bilinear patch mapping, the push-forwards of N i (ξ)n j (η), 2 i, j 4, have vanishing normal derivatives along the boundary of Ω k. Thus construction of C 1 -basis functions on a quadrangular patch becomes easier, however, approximation power of the resulting new basis functions becomes much weaker because non polynomial step functions are involved in N 2,5 and N 4,5. Moreover, for an accurate gaussian quadrature, the patch is divided into nine subregions where shape functions become polynomials. Because of these difficulties, we limit the use of (47) to quadrangular boundary patches. Suppose a quadrangular patch Ω I is surrounded by rectangular patches Ω II, Ω III, Ω IV as shown in Fig. 11. Then we use the following mixed sets of reference shape functions: 1. Since the normal derivatives along the intersection between quadrangular (Ω I ) and rectangular (Ω II, Ω IV ) patches are not constants, Side Alteration I (Fig. 2) is not applicable. Thus we apply Side Alteration II (Fig. 3) along both sides of the intersecting sides ( in Fig. 11). 2. Alteration I is allowed along the sides between Ω II and Ω III, Ω III and Ω IV ( in Fig. 11). 6 Concluding Remarks The C 1 -basis functions are constructed hierarchically for numerical solutions of the fourth-order partial differential equations. All arguments and results presented in sections 4 and 5 can be extended to deal with the Kirchhoff-Love plate model. We expect that by using modified Bernstein polynomials on triangles in a similar manner, we can ease the difficulties arising in dealing with biharmonic equations on curved domains or polygonal domains. Additionally, it is expected that we can reduce the number of internal shape functions like serendipity basis functions in the conventional finite element analysis [15]. 29