In the paper [1], the geometrical mapping techniques based on Non-Uniform Rational B-Spline (NURBS) were introduced to solve an elliptic boundary value problem containing a singularity. In the mapping techniques, the inverse function of the NURBS geometrical mapping generates singular functions as well as smooth functions by an unconventional choice of control points. It means that the push-forward of the NURBS geometrical mapping that generates singular functions, becomes a piecewise smooth function. However, the mapping method proposed is not able to catch singularities emerging at multiple locations in a domain. Thus, we design the geometrical mapping that generates singular functions for each singular zone in the physical domain. In the design of the geometrical mapping, we should consider the design of control points on the interface between/among patches so that global basis functions are in <i>C</i> 0 space. Also, we modify the B-spline functions whose supports include the interface between/among them. We put the idea in practice by solving elliptic boundary value problems containing multiple singularities.
It has been introduced to solve multiple crack problems by using various numerical methods. First, converting the multiple crack problems into Fredholm integral equation using two elementary solutions was introduced in [
In this paper, we solve the elliptic boundary value problems with multiple singularities based on the mapping method [
In Section 1 and 2, we briefly review definitions and terminologies that are used throughout this paper. We follow those in the book [
In this section, we introduce B-spline, NURBS, and design geometrical mappings referring to [
A knot vector U = { u 1 , u 2 , ⋯ , u m } is a nondecreasing sequence of real numbers in the parameter space [ a , b ] , and the components u i are called knots. An open knot vector of order p + 1 is a knot vector that satisfies
u 1 = ⋯ = u p + 1 < u p + 2 ≤ ⋯ ≤ u m − p − 1 < u m − p = ⋯ = u m ,
in which the first and the last p + 1 knots are repeated.
The B-spline functions B i , k ( ξ ) of order k = p + 1 corresponding to the knot vector U = { u 1 , u 2 , ⋯ , u m } are piecewise polynomials of degree p which are constructed recursively by the Cox-de Boor recursion formula:
B i ,1 ( ξ ) = ( 1 if u i ≤ ξ < u i + 1 , 0 otherwise , B i , k ( ξ ) = ξ − u i u i + k − 1 − u i B i , k − 1 ( ξ ) + u i + k − ξ u i + k − u i + 1 B i + 1, k − 1 ( ξ ) ,
for 1 ≤ i ≤ ( m − k ) For example, the piecewise quadratic polynomial B-spline functions B i ,5 ( ξ ) corresponding to the open knot vector
U = { 0 , 0 , 0 , 0 , 0 , 0.15 , 0.5 , 0.75 , 0.9 , 1 , 1 , 1 , 1 , 1 }
are depicted in
The B-spline functions are useful in design as well as finite element analysis because they have the following properties: variation diminishing, convex hull, non-negativity, piecewise polynomial, compact support, and partition of unity.
A B-spline curve is defined as follows:
C ( ξ ) = ∑ i = 1 n B i , k ( ξ ) P i ,
where n = m − k and { P i } are control points that make B-spline functions draw a desired curve as shown in
Let U η = { v 1 , ⋯ , v m ′ } be an open knot vector and let p η and k ′ = p η + 1 , respectively, be the polynomial degree and order of B-spline functions B j , k ′ ( η ) . Then a B-spline surface is defined by
S ( ξ , η ) = ∑ i = 1 n ∑ j = 1 n ′ B i , k ( ξ ) B j , k ′ ( η ) P i , j ,
where n ′ = m ′ − k ′ and P i , j are control points that make a bidirectional control net as shown in
In this section, we review the non-uniform rational B-splines (NURBS), NURBS curve and surface, and primary properties of NURBS.
A pth-degree NURBS curve is defined by
C ( ξ ) = ∑ i = 1 n B i , k ( ξ ) w i P i ∑ i = 1 n B i , k ( ξ ) w i , a ≤ ξ ≤ b (1)
where the { P i } are the control points, the { w i } are the weights, k = p + 1 , and the { B i , k ( ξ ) } are the pth-degree B-spline basis functions defined on the nonperiodic (and non-uniform) knot vector
U = { a , ⋯ , a ︸ p + 1 , u p + 2 , ⋯ , u m − k , b , ⋯ , b ︸ p + 1 } .
We assume that a = 0 , b = 1 , and w i > 0 for all i. Setting
R i , k ( ξ ) = B i , k ( ξ ) w i ∑ j = 1 n B j , k ( ξ ) w j (2)
allows us to rewrite Equation (1) in the form
C ( ξ ) = ∑ i = 1 n R i , k ( ξ ) P i (3)
The { R i , k ( ξ ) } are the rational basis functions; they are piecewise rational functions on ξ ∈ [ 0,1 ] .
A NURBS surface of degree p ξ in the ξ direction and degree p η in the η direction is a bivariate vector-valued piecewise rational function of the form
S ( ξ , η ) = ∑ i = 1 n ∑ j = 1 n ′ B i , k ( ξ ) B j , k ′ ( η ) w i , j P i , j ∑ i = 1 n ∑ j = 1 n ′ B i , k ( ξ ) B j , k ′ ( η ) w i , j , 0 ≤ ξ , η ≤ 1 (4)
The { P i , j } form a bidirectional control net, the { w i , j } are the weights, and the { B i , k ( ξ ) } and { B j , k ′ } are the nonrational B-spline basis functions defined on the knot vectors
U = { 0 , ⋯ , 0 ︸ p ξ + 1 , u p ξ + 1 , ⋯ , u m − ( p ξ + 1 ) , 1 , ⋯ , 1 ︸ p ξ + 1 } ,
V = { 0 , ⋯ , 0 ︸ p η + 1 , v p ξ + 1 , ⋯ , v m ′ − ( p η + 1 ) , 1 , ⋯ , 1 ︸ p η + 1 } .
Introducing the piecewise rational basis functions
R i , j ( ξ , η ) = B i , k ( ξ ) B j , k ′ ( η ) w i , j ∑ s = 1 n ∑ t = 1 n ′ B s , k ( ξ ) N t , k ′ ( η ) w s , t
the surface Equation (4) can be written as
S ( ξ , η ) = ∑ i = 1 n ∑ j = 0 n ′ R i , j ( ξ , η ) P i , j .
An example of the NURBS surface is shown in
Let Ω be a connected open subset of ℝ d . We define the vector space C m (Ω)
to consist of all those functions ϕ which, together with all their partial derivatives ∂ α ϕ ( = ∂ 1 α 1 ⋯ ∂ d α d ϕ ) of orders | α | = α 1 + ⋯ + α d ≤ m , are continuous on Ω . A function ϕ ∈ C m ( Ω ) is said to be a C m -function. If Ψ is a function defined on Ω , we define the support of Ψ as
supp Ψ = { x ∈ Ω | Ψ ( x ) ≠ 0 } ¯ .
For an integer k ≥ 0 , we also use the usual Sobolev space denoted by H k ( Ω ) . For u ∈ H k ( Ω ) , the norm and the semi-norm, respectively, are
‖ u ‖ k , Ω = ( ∑ | α | ≤ k ∫ Ω | ∂ α u | 2 d x ) 1 / 2 , ‖ u ‖ k , ∞ , Ω = max | α | ≤ k { ess . sup | ∂ α u ( x ) | : x ∈ Ω } ; | u | k , Ω = ( ∑ | α | = k ∫ Ω | ∂ α u | 2 d x ) 1 / 2 , | u | k , ∞ , Ω = max | α | = k { ess . sup | ∂ α u ( x ) | : x ∈ Ω } .
Suppose we are concerned with an elliptic boundary value problem on a domain Ω with Dirichlet boundary condition g ( x , y ) along the boundary ∂ Ω . Let
W = { w ∈ H 1 ( Ω ) : w | ∂ Ω = g } and V = { w ∈ H 1 ( Ω ) : w | ∂ Ω = 0 } .
The variational formulation of the Dirichlet boundary value problem can be written as: Find u ∈ W such that
B ( u , v ) = L ( v ) , for all v ∈ V , (5)
where B is a continuous bilinear form that is V -elliptic ( [
‖ u ‖ eng = [ 1 2 B ( u , u ) ] 1 / 2 .
Let W h ⊂ W , V h ⊂ V be finite dimensional subspaces. Since the NURBS basis functions do not satisfy the Kronecker delta property, in this paper we approximate the nonhomogenuous Dirichlet boundary condition by the least squares method as follows: g h ∈ W h such that
∫ ∂ Ω | g − g h | 2 d γ = minimum .
We can write the Galerkin form (a discrete variational equation) of (5) as follows: Given g h , find u h = w h + g h , where w h ∈ V h , such that
B ( u h , v h ) = L ( v h ) , for all v h ∈ V h ,
which can be rewritten as: Find the trial function w h ∈ V h such that
B ( w h , v h ) = L ( v h ) − B ( g h , v h ) , for all test functions v h ∈ V h . (6)
In elasticity, the displacement field is denoted by { u } = { u x ( x , y ) , u y ( x , y ) } T , and the stress field is denoted by { σ } = { σ x , σ y , τ x y } T . Let { ε } = { ε x , ε y , γ x y } T be the strain field. Then the strain-displacement and the stress-strain relations are given by
{ ε } = [ D ] { u } , { σ } = [ E ] { ε } , (7)
respectively, where [ D ] is the differential operator matrix,
[ D ] = [ ∂ ∂ x 0 0 ∂ ∂ y ∂ ∂ y ∂ ∂ x ]
and [ E ] is the 3 × 3 symmetric positive definite matrix of material constants. Material constants are classified by the property of the material. For an isotropic elastic body,
[ E ] = E 1 − ν 2 [ 1 ν 0 ν 1 0 0 0 1 − ν 2 ] for plane stress,
[ E ] = [ ζ + 2 μ ζ 0 ζ ζ + 2 μ 0 0 0 μ ] for plane strain .
Here,
μ = E 2 ( 1 + ν ) , ζ = ν E ( 1 + ν ) ( 1 − 2 ν ) ,
where E is the Young’s modulus of elasticity and ν ( 0 ≤ ν ≤ 1 / 2 ) is Poisson’s ratio.
The equilibrium equations of elasticity are
[ D ] T { σ } ( x , y ) + { f } ( x , y ) = 0, ( x , y ) ∈ Ω , (8)
where { f } = { f x ( x , y ) , f y ( x , y ) } T is the vector of internal sources representing the body force per unit volume.
The equilibrium Equation (8) can be expressed in terms of the displacement field { u } through the relations (7). Then we consider the following system of elliptic differential equations in terms of the displacement field,
[ D ] T [ E ] [ D ] { u } ( x , y ) + { f } ( x , y ) = 0 , ( x , y ) ∈ Ω , (9)
subject to the boundary conditions,
[ N ] { σ } ( s ) = { T ˜ } ( s ) = { T ¯ } ( s ) = { T ¯ x ( s ) , T ¯ y ( s ) } T , s ∈ Γ N , { u } ( s ) = { u ¯ } ( s ) = { u ¯ x ( s ) , u ¯ y ( s ) } T , s ∈ Γ D , (10)
where Γ N ∪ Γ D = ∂ Ω ,
[ N ] = [ n x 0 n y 0 n y n x ] ,
{ n x , n y } T is a unit vector normal to the boundary ∂ Ω of the domain Ω .
For the Galerkin approximation to the equilibrium equations in terms of the displacement field (9), the variational form of (9) through (10) is:
find the vector { u } such that u x , u y ∈ H 1 ( Ω ) , { u } = { u ¯ } on Γ D , and
B ( { u } , { v } ) = F ( { v } ) , for all { v } ∈ H 0 1 ( Ω ) , (11)
where
B ( { u } , { v } ) = ∫ Ω ( [ D ] { v } ) T [ E ] ( [ D ] { u } ) d x d y ,
F ( { v } ) = ∫ Ω { v } T { f } d x d y + ∮ Γ N { v } T { T ¯ } d s (12)
The finite element approximation of the solution of (11) is to construct approximations of each component of the vector { u } .
We introduce a geometrical mapping from the parameter space Ω ^ = [ 0,1 ] × [ 0,1 ] to ℝ 2 that generates singular basis functions [
In particular, we first show how a B-spline curve F ( η ) : [ 0,1 ] × [ 0,1 ] → ℝ handles effectively one-dimensional singularities. Let U η = { 0, ⋯ ,0,1, ⋯ ,1 } be an open knot vector of order k ′ = p η + 1 . Then the B-spline functions B j , k ′ ( η ) corresponding to U η are
B j , k ′ ( η ) = ( p η j − 1 ) η j − 1 ( 1 − η ) p η − j + 1 for j = 1, ⋯ , k ′ . (13)
Here, B j , k ′ , j = 1 , ⋯ , k ′ , are also called the Bernstein polynomials of degree p η . Let
P j = ( 0,0 ) , for j = 1, ⋯ , k ′ − 1, and P k ′ = (0,γ)
be control points, for a constant γ . Then the B-spline geometrical mapping
F ( η ) = ∑ j = 1 k ′ B j , k ′ ( η ) B j (14)
= B 1, k ′ ( η ) ( 0,0 ) + ⋯ + B p η , k ′ ( η ) ( 0,0 ) + B k ′ , k ′ ( η ) ( 0, γ ) (15)
= ( 0, γ η p η ) (16)
maps the parameter space [ 0,1 ] onto the physical space { 0 } × [ 0, γ ] ⊂ ℝ 2 and its inverse is
η = F − 1 ( 0, y ) = ( 1 / γ ) 1 / p η y 1 / p η .
Thus, the approximation space V h = span { B i , k ″ ∘ F − 1 | i = 1, ⋯ , k ″ } , where k ″ is an integer greater than or equal to k ′ and N i , k ″ are the Bernstein polynomials (B-spline functions) of degree k ″ − 1 and contain the following singular as well as smooth functions:
y 1 / p η , l = 0 , 1 , ⋯ , k ″ − 1.
In other words, the geometrical mapping F is able to generate the singularity of type r λ , where 0 < λ = 1 / p η < 1 .
For example, if p η = 2 , then the Bernstein polynomials of degree 2 are
B 1 , 3 = ( 1 − η ) 2 , B 2 , 3 = 2 η ( 1 − η ) , B 3 , 3 = η 2 .
and
P 1 = ( 0,0 ) , P 2 = ( 0,0 ) , P 3 = (0,γ)
are control points. Then the geometrical mapping obtained by these control points and its inverse, respectively, are
F ( η ) = ( 0, γ η 2 ) and F − 1 ( 0, y ) = 1 / γ y .
Suppose S η h = span { B j ,5 | j = 1, ⋯ ,5 } where B j ,5 are the Bernstein polynomials corresponding the the open knot vector U = { 0 , 0 , 0 , 0 , 0 , 1 , 1 , 1 , 1 , 1 } of order 5, then S η h contains 1, η , ⋯ , η 4 . Hence the approximation space V y h = span { B j ,5 ∘ F − 1 : j = 1, ⋯ ,5 } for isogeometric analysis contains
1, y , y , y 3 / 2 , y 2 .
The argument which is the construction of geometrical mapping that generates singular basis functions, can be extended to NURBS surface from the parameter space Ω ^ = [ 0,1 ] × [ 0,1 ] to Ω ⊂ ℝ 2 . To end this, we construct a NURBS surface to deal with monotone singularity of type r q ψ ( θ ) , where q is a rational number with 0 < q < 1 , ψ ( θ ) is a piecewise smooth function, ( r , θ ) is the polar coordinates. the construction of the NURBS surface from Ω ^ to the unit disk in [
We now consider a NURBS surface from the parameter space Ω ^ to the physical domain Ω . Consider the knot vectors:
U ξ = { 0 , ⋯ , 0 ︸ p ξ + 1 , ζ 1 , ⋯ , ζ l , 1 , ⋯ , 1 ︸ p ξ + 1 } , U η = { 0 , 0 , ⋯ , 0 ︸ p η + 1 , 1 , 1 , ⋯ , 1 ︸ p η + 1 } . (17)
where ζ i = { ξ i , 1 , ξ i , 2 , ⋯ , ξ i , p ξ } , ξ i , j = ξ i , j + 1 , i = 1 , ⋯ , l , j = 1 , ⋯ , p ξ − 1 , and ζ 1 ≠ ζ 2 ≠ ⋯ ≠ ζ l .
Let m and m ′ be the number of knots in the knot vectors U ξ and U η , respectively. Also, let k and k ′ be p ξ + 1 and p η + 1 , respectively. Here, if the function to be approximated has a singularity of type O ( r q ) with 0 < q = n q / m q < 1 , where n q , m q ∈ ℤ , then the polynomial degree of B-spline functions corresponding to U η is p η = m q .
Let B i , k ( ξ ) , i = 1 , ⋯ , m − k be the B-splines corresponding to the knot vector U ξ and let B j , k ′ ( η ) , j = 1 , ⋯ , p η + 1 be the B-splines corresponding to the knot vector U η . Here, the B-spline functions B j , k ′ , j = 1 , ⋯ , p η + 1 , corresponding to the open knot vector U η are the Bernstein polynomials of degree p η .
Consider the control points P i , j and the weights w i , j for 1 ≤ i ≤ n = m − k , 1 ≤ j ≤ p η + 1 , that are listed in
F ( ξ , η ) = ∑ i = 1 n ∑ j = 1 k ′ R i , j ( ξ , η ) P i , j . (18)
Here R i , j ( ξ , η ) , 1 ≤ i ≤ n , 1 ≤ j ≤ p η + 1 , are NURBS basis functions defined by
R i , j ( ξ , η ) = B i , k ( ξ ) B j , k ′ ( η ) w i , j W ( ξ , η ) , (19)
where
W ( ξ , η ) = ∑ s = 1 n ∑ t = 1 k ′ B s , k ( ξ ) B t , k ′ ( η ) w s , t .
Since B j , k ′ ( η ) is the Bernstein polynomial and P i , j = ( 0 , 0 ) unless j = k ′ , substituting Equations (13) into (19) the NURBS surface mapping (18) becomes
F ( ξ , η ) = p η η p η [ ∑ i = 1 n B i , k ( ξ ) w i , k ′ P i , k ′ ] / W ( ξ , η ) .
| 1 ≤ j ≤ p η | j = p η + 1 | |||
|---|---|---|---|---|
| i | P i , j | w i , j | P i , j | w i , j |
| 1 | ( 0,0 ) | β 1 | ( x 1 , y 1 ) | β 1 |
| 2 | ( 0,0 ) | β 2 | ( x 2 , y 2 ) | β 2 |
| ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |
| m − k | ( 0,0 ) | β m − k | ( x m − k , y m − k ) | β m − k |
Now, we derive the derivative of the mapping F ( ξ , η ) by using formulas in Chapter 4.5 in [
Let
F ( ξ , η ) = W ( ξ , η ) F ( ξ , η ) W ( ξ , η ) = A ( ξ , η ) W ( ξ , η ) ,
where A ( ξ , η ) is the numerator of F ( ξ , η ) .
Denoting ϕ ( α 1 , α 2 ) ( ξ , η ) = ∂ α 1 + α 2 ∂ ξ α 1 ∂ η α 2 ϕ ( ξ , η ) , the derivative of F ( ξ , η ) can be expressed as
F ( α 1 , α 2 ) = [ A ( ξ , η ) W ( ξ , η ) ] ( α 1 , α 2 ) = [ W ( ξ , η ) F ( ξ , η ) W ( ξ , η ) ] ( α 1 , α 2 )
Then
A ( ξ , η ) ( α 1 , α 2 ) = [ W ( ξ , η ) F ( ξ , η ) ] ( α 1 , α 2 ) = ∂ α 2 ∂ η α 2 [ ∑ i = 0 α 1 ( α 1 i ) W ( i ,0 ) F ( α 1 − i ,0 ) ] = ∑ i = 0 α 1 ( α 1 i ) ∑ j = 0 α 2 ( α 2 j ) W ( i , j ) F ( α 1 − i , α 2 − j ) = W ( 0,0 ) F ( α 1 , α 2 ) + ∑ i = 1 α 1 ( α 1 i ) W ( i ,0 ) F ( α 1 − i , α 2 ) + ∑ j = 1 α 2 ( α 2 j ) W ( 0, j ) F ( α 1 , α 2 − j ) + ∑ i = 1 α 1 ( α 1 i ) ∑ j = 1 α 2 ( α 2 j ) W ( i , j ) F ( α 1 − i , α 2 − j ) (20)
Solving the Equation (20) for F ( ξ , η ) , we obtain
F ( ξ , η ) ( α 1 , α 2 ) = 1 W ( ξ , η ) [ A ( α 1 , α 2 ) − ∑ i = 1 α 1 ( α 1 i ) W ( i ,0 ) F ( α 1 − i , α 2 ) − ∑ j = 1 α 2 ( α 2 j ) W ( 0, j ) F ( α 1 , α 2 − j ) − ∑ i = 1 α 1 ( α 1 i ) ∑ j = 1 α 2 ( α 2 j ) W ( i , j ) F ( α 1 − i , α 2 − j ) ] (21)
We employ the lemma below from Chapter 3 in [
Lemma 1 Let P i ( 0 ) = P i , and C ( ξ ) = C ( 0 ) ( ξ ) = ∑ i = 1 n B i , k ( ξ ) P i ( 0 ) . Then
C ( α 1 ) ( ξ ) = ∑ i = 1 n − α 1 B i , k − α 1 ( ξ ) P i (α1)
with
P i ( α 1 ) = ( P i , α 1 = 0 k − α 1 u i + k − u i + α 1 ( P i + 1 ( α 1 − 1 ) − P i ( α 1 − 1 ) ) , α 1 > 0
and the knot vector corresponding to C ( 0 ) ( ξ ) is
U ( α 1 ) = { 0, ⋯ ,0 ︸ k − α 1 , u k + 1 , ⋯ , u m − k , 1, ⋯ ,1 ︸ k − α 1 } .
Applying the lemma 1 into (21), we have
A ( α 1 , α 2 ) = p η ! ( p η − α 2 ) ! η p η − α 2 [ ∑ i = 1 n − α 1 B i , k − α 1 ( ξ ) P i , k ′ ( α 1 ) ] ,
where
P i , k ′ ( α 1 ) = ( P i , k ′ , α 1 = 0 k − α 1 u i + k − u i + α 1 ( P i + 1, k ′ ( α 1 − 1 ) − P i , k ′ ( α 1 − 1 ) ) , α 1 > 0.
The derivative of the total weight function W ( ξ , η ) , also, can be described in detail by substituting the Bernstein polynomial into B j , k ′ ( η ) .
W ( ξ , η ) ( α 1 , α 2 ) = ∑ i = 1 n B i , k ( ξ ) ( α 1 ) [ ∑ j = 1 k ′ ( p η j − 1 ) { η j − 1 ( 1 − η ) k ′ − j } ( α 2 ) w i , j ] = ∑ i = 1 n B i , k ( ξ ) ( α 1 ) [ p η ! ( p η − α 2 ) ! ( 1 − η ) p η − α 2 w i , 1 + ∑ j = 2 p η ( p η j − 1 ) { η j − 1 ( 1 − η ) k ′ − j } ( α 2 ) w i , j + p η ! ( p η − α 2 ) ! η p η − α 2 w i , k ′ ] .
The mapping method proposed was implemented in the paper [
Throughout this paper, we measure the error ( u − u h ) of the computed solutions obtained by isogeomtric analysis using the proposed mapping method in the following norms:
· The relative error in the maximum norm in %:
‖ u − u h ‖ ∞ , rel ( % ) = ‖ u − u h ‖ ∞ ‖ u ‖ ∞ × 100
· The relative error in L 2 norm in %:
‖ u − u h ‖ L 2 , rel ( % ) = ‖ u − u h ‖ L 2 ‖ u ‖ L 2 × 100
· The relative error in energy norm in %:
‖ u − u h ‖ eng,rel ( % ) = [ | ‖ u ‖ eng 2 − ‖ u ‖ eng 2 | ‖ u ‖ eng 2 ] 1 2 × 100
Assuming that the Young’s modulus E = 1000 and the Poisson’s ratio ν = 0.3 in a sector of the unit circle whose the central angle is 270˚, plane strain isotropic elastic body, we consider that the following analytic stress field,
σ x = λ r λ − 1 [ ( 2 − q ( λ + 1 ) ) cos ( ( λ − 1 ) θ ) − ( λ − 1 ) cos ( ( λ − 3 ) θ ) ] , σ y = λ r λ − 1 [ ( 2 − q ( λ + 1 ) ) cos ( ( λ − 1 ) θ ) − ( λ − 1 ) cos ( ( λ − 3 ) θ ) ] , τ x y = λ r λ − 1 [ ( λ − 1 ) sin ( ( λ − 3 ) θ ) + q ( λ + 1 ) sin ( ( λ − 1 ) θ ) ] , (22)
where λ = 2 / 3 , and q = cos [ ( λ − 1 ) 0.75 π ] cos [ ( λ + 1 ) 0.75 π ] . Then, the stress field (22) satisfies the equilibrium equations of elasticity on the sector shaped domain Ω L . And the displacement field has the singularity of the form r 2 / 3 ϕ ( θ ) where ϕ ( θ ) is a smooth function.
For the design of the physical domain Ω L , we set p ξ = 2 , p η = 3 and ζ 1 = { 1 / 3 , 1 / 3 } , ζ 2 = { 2 / 3 , 2 / 3 } in the knot vector (17) so that the open knot vector corresponding to ξ -direction is as follows:
U ξ = { 0 , 0 , 0 , 1 / 3 , 1 / 3 , 2 / 3 , 2 / 3 , 1 , 1 , 1 } (23)
We construct the open knot vector corresponding to η -direction using the form of the knot vector U η in (17):
U η = { 0 , 0 , 0 , 0 , 1 , 1 , 1 , 1 } , (24)
which make Bernstein polynomials in [ 0,1 ] on the parameter space. We choose ( 0,0 ) for control points P i , j , i = 1, ⋯ , k ( = 7 ) , j = 1, ⋯ , p η ( = 3 ) , and set the other control points as depicted in
F L ( ξ , η ) : Ω ^ L ↦ Ω L , Ω ^ L = [ 0,1 ] × [ 0,1 ] ,
and the inverse of the design mapping generates the singularity of the form r 1 / 3 ϕ ( θ ) along the radial direction on the Ω L in
In order to enrich the NURBS or B-spline basis functions without failing the structure of the mapping technique, we employ refinements [
In the case of that a physical domain contains multiple cracks, we re-design the
| p ξ = p η | DOF | ‖ u − u h ‖ ∞ , rel | ‖ v − v h ‖ ∞ , rel | ‖ u − u h ‖ L 2 , rel | ‖ v − v h ‖ L 2 , rel |
|---|---|---|---|---|---|
| 2 | 60 | 3.816E−00 | 2.333E−00 | 3.142E−00 | 2.103E−00 |
| 3 | 96 | 1.435E−00 | 1.945E−01 | 6.447E−01 | 1.812E−01 |
| 4 | 176 | 2.284E−01 | 1.017E−01 | 1.659E−01 | 8.127E−02 |
| 5 | 280 | 7.493E−02 | 1.142E−02 | 2.754E−02 | 1.069E−01 |
| 6 | 408 | 1.822E−02 | 6.621E−03 | 7.580E−03 | 3.255E−03 |
| 7 | 560 | 5.590E−03 | 6.702E−04 | 1.180E−03 | 5.394E−04 |
| 8 | 736 | 1.264E−03 | 4.110E−04 | 3.269E−04 | 1.306E−04 |
| 9 | 936 | 3.618E−04 | 5.643E−05 | 4.977E−05 | 2.551E−05 |
| 10 | 1160 | 8.230E−05 | 2.636E−05 | 1.364E−05 | 5.271E−06 |
| 11 | 1408 | 2.154E−05 | 3.159E−06 | 2.054E−06 | 1.164E−06 |
| 12 | 1680 | 4.766E−06 | 1.446E−06 | 5.569E−07 | 2.153E−07 |
| 13 | 1976 | 1.230E−06 | 3.281E−07 | 8.242E−08 | 5.206E−08 |
| 14 | 2296 | 1.688E−07 | 1.783E−07 | 2.231E−08 | 8.811E−09 |
| p ξ = p η | DOF | ‖ σ x − σ x h ‖ ∞ , rel | ‖ σ y − σ y h ‖ ∞ , rel | ‖ τ x y − τ x y h ‖ ∞ , rel | Strain Energy |
|---|---|---|---|---|---|
| 2 | 60 | 6.334E+03 | 9.922E+03 | 1.483E+04 | 88.98117013046564 |
| 3 | 96 | 3.670E+02 | 1.818E+02 | 2.030E+02 | 89.14930415746228 |
| 4 | 176 | 2.556E+02 | 3.853E+02 | 9.833E+02 | 89.15648698854705 |
| 5 | 280 | 6.185E+01 | 5.250E+01 | 1.091E+02 | 89.15771576771605 |
| 6 | 408 | 2.072E+01 | 3.242E+01 | 8.419E+01 | 89.15782485559064 |
| 7 | 560 | 5.732E−00 | 3.710E−00 | 6.687E−00 | 89.15782012258868 |
| 8 | 736 | 1.457E−00 | 2.070E−00 | 5.457E−00 | 89.15781818638276 |
| 9 | 936 | 5.276E−01 | 3.246E−01 | 5.544E−01 | 89.15781843245076 |
| 10 | 1160 | 1.049E−01 | 1.372E−01 | 3.657E−01 | 89.15781850351074 |
| 11 | 1408 | 3.942E−02 | 2.152E−02 | 3.261E−02 | 89.15781849595437 |
| 12 | 1680 | 6.713E−03 | 7.708E−03 | 2.116E−02 | 89.15781849355913 |
| 13 | 1976 | 2.915E−03 | 1.782E−03 | 2.515E−03 | 89.15781849380155 |
| 14 | 2296 | 6.877E−04 | 9.680E−04 | 2.107E−03 | 89.15781849389655 |
| ∞ | 89.15781849384732 |
geometrical mapping by using both standard NURBS mappings and the proposed mappings. Then it simplifies things to describe these sub-domains by different patches. We describe how to construct the set of global basis functions crossing interfaces between patches. Throughout the following examples, we show that the patchwise mapping method is effective in dealing with a problem containing multiple singularities.
First, We apply the mapping method for the elliptic boundary value problems with multiple singularities of type
r i λ ψ i ( θ ) , where Ω 1, i , and ψ i ’s are smooth functions.
Example 1. Let
u 1 = r 1 1 / 2 cos ( θ 1 / 2 ) + r 2 1 / 2 sin ( θ 2 / 2 ) , f = − Δ u 1 , and g = u 1 | Γ
where
Ω 1 = [ − 1 , 1 ] × [ 0 , 1 + 2 ] r 1 = x 2 + y 2 , r 2 = ( x − 1 ) 2 + ( y − 1 − 2 ) 2 θ 1 = cos − 1 ( x r 1 ) , θ 2 = − cos − 1 ( x − 1 r 2 ) (25)
Then u 1 is the analytic solution of the Poisson equation:
− Δ u = f in Ω 1 and u = g on Γ = ∂ Ω 1 (26)
and has two singularities at ( 0,0 ) and ( 1,1 + 2 ) .
In Example 1, we divide the physical domain into three patches:
Ω 1 , 1 = [ − 1 , 1 ] × [ 0 , 1 ] Ω 1 , 2 = [ 0 , 1 ] × [ 1 , 1 + 2 ] Ω 1 , 3 = [ − 1 , 0 ] × [ 1 , 1 + 2 ]
Let Ω 1, i ’s are physical patches. We construct NURBS geometrical mappings F 1,1 and F 1,2 that generate singularities of the type r 1 1 / 2 ψ ( θ 1 ) and r 2 1 / 2 ϕ ( θ 2 ) , respectively. They are also the design maps from the parameter space Ω ^ 1, i to the physical patch Ω 1, i , for each i = 1 , 2 . To build up F 1 , i , i = 1 , 2 we use the following knot vectors: For F 1,1 ,
U ξ = { 0 , 0 , 0 , ζ 1 , ζ 2 , ζ 3 , 1 , 1 , 1 } U η = { 0 , 0 , 0 , 0.5 , 0.5 , 1 , 1 , 1 } , (27)
where ζ 1 = { 0.25 , 0.25 } , ζ 2 = { 0.5 , 0.5 } , and ζ 3 = { 0.75 , 0.75 } . For F 1,2 ,
U ξ = { 0 , 0 , 0 , ζ 1 , 1 , 1 , 1 } U η = { 0 , 0 , 0 , 0.5 , 0.5 , 1 , 1 , 1 } ,
where ζ 1 = { 0.5 , 0.5 } .
In the design mappings, we observe the following:
1) We employ control points and weights from Example 5.3 in [
2) We design the NURBS geometrical mapping F ^ 1,2 ( ξ , η ) that generates a singularity ( x 2 + y 2 ) 1 / 4 ψ ( cos − 1 ( x ( x 2 + y 2 ) 1 / 2 ) ) using the control points as depicted in
3) Using the affine transformation we define
F 1 , 2 ( ξ , η ) ≡ F ^ 1 , 2 ( ξ + 1 , η + ( 1 + 2 ) ) .
In
4) Since a singularity does not appear in the patch Ω 1,3 , we employ the standard NURBS design technique to build the mapping F 1,3 ( ξ , η ) from the parameter space Ω ^ 1,3 to Ω 1,3 .
Now, we construct an approximation space by using B-spline functions which were used in the design mapping F 1, i . First, we consider connectivity among B-spline functions defined on different patches and are nonzero along the interface as depicted in
an interface between two different patches, we merge two B-spline local basis functions defined on different patches that have the same nonzero value on the interface between the two patches. In
F 1, j − 1 ( F 1, i ( I i , j ) ) = I j , i .
To construct global basis functions which are nonzero functions on I i , j ∪ I j , i , i ≠ j , we merge the nonzero basis function in I i , j and the nonzero basis function in I j , i , where i ≠ j such that they are reflection about the interface F 1 , i ( I i , j ) = F 1 , j ( I j , i ) in the physical domain. In
B t , k [ 1 ] | ξ ∈ I 1,3 ≠ 0 when t = p ξ + 1, ⋯ ,2 ( p ξ + 1 ) − 1
B t , k [ 3 ] | ξ ∈ I 3,1 ≠ 0 when t = 1, ⋯ , p ξ + 1.
Because
· B 4 , k ′ [ 1 ] ( η ) | I 1 , 3 = B 1 , k ′ [ 3 ] ( η ) | I 3 , 1 = 1 and
· ( B t 1 , k [ 1 ] ⋅ B 4 , k ′ [ 1 ] ) ∘ F 1 , 1 − 1 ( x , y ) | F 1 , 1 ( I 1 , 3 ) = ( B t 2 , k [ 3 ] ⋅ B 1 , k ′ [ 3 ] ) ∘ F 1 , 1 − 1 ( x , y ) | F 1 , 3 ( I 3 , 1 ) ,
· ( t 1 , t 2 ) = ( p ξ + 1 , 1 ) , ( p ξ + 2 , 2 ) , ⋯ , ( 2 ( p ξ + 1 ) − 1 , p ξ + 1 ) ,
· ( B t 1 , k [ 1 ] ⋅ B 4 , k ′ [ 1 ] ) = ( B t 2 , k [ 3 ] ⋅ B 1 , k ′ [ 3 ] ) on the inter face F 1 , 1 ( I 1 , 3 ) ( = F 1 , 3 ( I 3 , 1 ) ) .
We merge two B-spline basis functions ( B t 1 , k [ 1 ] ⋅ B 4, k ′ [ 1 ] ) and ( B t 2 , k [ 3 ] ⋅ B 1, k ′ [ 3 ] ) so that we count the new function as one global basis function. The new function has a nonzero value on two distinct patches. Here, we should carefully set ( t 1 , t 2 ) , and we apply the same degree of p-refinement into each parameter space.
For a space with the non-homogeneous boundary condition in Example 1,
W 1 = { w ( x , y ) ∈ H 1 ( Ω 1 ) : w | ∂ Ω 1 = g , Ω 1 ⊂ ℝ 2 } (28)
We decompose the space (28) into
W 1 , 1 = { w ( x , y ) ∈ H 1 ( Ω 1 ) : w | ∂ Ω 1 = 0 , Ω 1 ⊂ ℝ 2 }
and
W 1,2 = { w ( x , y ) ∈ H 1 ( Ω 1 ) : w | ∂ Ω 1 = g , Ω 1 ⊂ ℝ 2 } .
The finite dimensional subspace i.e. approximation space of the Poisson equation (26) is
W 1 h = W 1,1 h ⊕ W 1,2 h = { w 1 + w 2 : w 1 ∈ W 1,1 h , w 2 ∈ W 1,2 h } ,
W 1,1 h ⊂ W 1,1 , W 1,2 h ⊂ W 1,2 ,
W 1, i h = span [ S i ,1 ∪ S i ,1 n e w ∪ S i ,2 ∪ S i ,2 n e w ∪ S i ,3 ] , i = 1,2
S 1,1 = { ( B i , k [ 1 ] ⋅ B j , k ′ [ 1 ] ) ∘ F 1,1 − 1 : i = 2, ⋯ , n 1 − 1, j = 2, ⋯ , n ′ 1 − 1 } ,
S 1,2 = { ( B i , k [ 2 ] ⋅ B j , k ′ [ 2 ] ) ∘ F 1,2 − 1 : i = 2, ⋯ , n 2 − 1, j = 2, ⋯ , n ′ 2 − 1 } ,
S 1,3 = { ( B i , k [ 3 ] ⋅ B j , k ′ [ 3 ] ) ∘ F 1,3 − 1 : i = 2, ⋯ , n 3 − 1, j = 2, ⋯ , n ′ 3 − 1 } ,
S 2,1 = { ( B i , k [ 1 ] ⋅ B j , k ′ [ 1 ] ) ∘ F 1,1 − 1 : j = 1, ⋯ , n ′ 1 when i = 1, n 1 and i = 1, ⋯ , p ξ , n 1 − ( p ξ − 1 ) , ⋯ , n 1 when j = n ′ 1 } , (29)
S 2,2 = { ( B i , k [ 2 ] ⋅ B j , k ′ [ 2 ] ) ∘ F 1,2 − 1 : j = 1, ⋯ , n ′ 2 − 1 when i = 1, n 2 } ,
S 2,3 = { ( B i , k [ 3 ] ⋅ B j , k ′ [ 3 ] ) ∘ F 1,3 − 1 : j = 2, ⋯ , n ′ 3 when i = 1 and i = 2, ⋯ , n 3 − 1 when j = n ′ 3 } ,
S 1,1 n e w = { ( B i , k [ 1 ] , n e w ⋅ B j , k ′ [ 1 ] , n e w ) ∘ F 1,1 − 1 : i = p ξ + 2, ⋯ , n 1 − ( p ξ + 1 ) , j = n ′ 1 } ,
S 1,2 n e w = { ( B i , k [ 2 ] , n e w ⋅ B j , k ′ [ 2 ] , n e w ) ∘ F 1,2 − 1 : i = p ξ + 2, ⋯ , n 2 − 1, i ≠ p ξ + 1, j = n ′ 2 } ,
S 2,1 n e w = { ( B i , k [ 1 ] , n e w ⋅ B j , k ′ [ 1 ] , n e w ) ∘ F 1,1 − 1 : i = p ξ + 1, n 1 − p ξ , j = n ′ 1 } ,
S 2,2 n e w = { ( B n 2 , k [ 2 ] , n e w ⋅ B n ′ 2 , k ′ [ 2 ] , n e w ) ∘ F 1,2 − 1 } ,
where
1) n i and n ′ i are the number of B-spline functions in ξ- and η-direction of the patch Ω 1, i , respectively.
2) B s , k [ i ] , n e w , B t , k ′ [ j ] , n e w are new global basis functions by merging two B-spline functions in Ω 1, i and Ω 1, j , respectively of ξ and η , respectively.
3) S 1, i and S 1, i n e w are the set of B-spline basis functions composition with the inverse of NURBS surface mapping F 1, i on the physical domain Ω 1 satisfying homogeneous boundary condition.
4) S 2, i and S 2, i n e w are the set of B-spline basis functions composition with the inverse of NURBS surface mapping F 1, i on the physical domain Ω 1 satisfying non-homogeneous boundary condition.
Example 2. Let Ω 2 = ∪ i = 1 3 Ω 2 , i be the unit disk, where Ω 2, i ’s are minor sectors whose central angles are 120˚ for each i = 1 , 2 , 3 , and
u 2 ( x , y ) = ∑ i = 1 3 r i 1 / 2 cos ( θ i / 2 ) , f = − Δ u 2 and g = u 2 | Γ
| p ξ = p η | DOF | ‖ u − u h ‖ ∞ , rel | ‖ u − u h ‖ L 2 , rel | ‖ u − u h ‖ eng,rel | Strain Energy |
|---|---|---|---|---|---|
| 2 | 71 | 5.376E−01 | 4.968E−01 | 1.219E−00 | 1.9224816355887329 |
| 3 | 145 | 1.358E−01 | 4.062E−02 | 1.014E−00 | 1.9223933783586342 |
| 4 | 245 | 2.655E−02 | 1.083E−02 | 3.504E−01 | 1.9221720544391365 |
| 5 | 371 | 6.469E−03 | 3.201E−03 | 1.482E−01 | 1.9221998800087665 |
| 6 | 523 | 1.613E−03 | 4.345E−04 | 6.126E−02 | 1.9221949350016059 |
| 7 | 701 | 4.077E−04 | 6.766E−05 | 2.950E−02 | 1.9221958237730838 |
| 8 | 905 | 1.031E−04 | 2.624E−05 | 1.424E−02 | 1.9221956174946071 |
| 9 | 1135 | 2.433E−05 | 6.642E−06 | 6.643E−03 | 1.9221956649732888 |
| 10 | 1391 | 5.681E−06 | 1.028E−06 | 3.238E−03 | 1.9221956544748704 |
| 11 | 1673 | 1.302E−06 | 2.510E−07 | 1.508E−03 | 1.9221956569276313 |
| 12 | 1981 | 2.926E−07 | 9.886E−08 | 8.398E−04 | 1.9221956563547675 |
| 13 | 2315 | 6.736E−08 | 2.099E−08 | 1.512E−04 | 1.9221956564947542 |
| 14 | 2675 | 2.232E−08 | 4.300E−09 | 4.330E−04 | 1.9221956564543110 |
| ∞ | 1.9221956564903575 |
where
r i = t i x 2 + t i y 2 , { t i x t i y } = T α i ( { x + x i y + y i } ) ,
T α i ( { x y } ) = [ cos ( α i ) − sin ( α i ) sin ( α i ) cos ( α i ) ] { x y } , i = 1 , 2 , 3 (30)
x 1 = − 0.68 cos 12 ∘ , y 1 = − 0.68 sin 12 ∘ , α 1 = 12 ∘
x 2 = − 0.7 cos 150 ∘ , y 2 = − 0.7 sin 150 ∘ , α 2 = 150 ∘
x 3 = 0 , y 3 = 0.5 sin 90 ∘ , α 3 = 90 ∘
θ i = ( cos − 1 ( x + x i r i ) if y > 0 , − cos − 1 ( x + x i r i ) if y ≤ 0
Then u 2 solves the following elliptic boundary value problem:
− Δ u = f in Ω 2 and u = g on Γ = ∂ Ω 2 , (31)
and the solution u 2 has three crack singularities at ( x i , y i ) , i = 1 , 2 , 3 .
In Example 2, we divide the unit disk into three sectors Ω 2 , i , i = 1 , 2 , 3 including each crack as depicted in
( Ω 2 , 1 = { ( r 1 , θ 1 ) | 0 ≤ r 1 ≤ 1 , 150 ∘ ≤ θ 1 ≤ 270 ∘ } Ω 2 , 2 = { ( r 1 , θ 1 ) | 0 ≤ r 1 ≤ 1 , 270 ∘ ≤ θ 1 ≤ 390 ∘ } Ω 2 , 3 = { ( r 1 , θ 1 ) | 0 ≤ r 1 ≤ 1 , 30 ∘ ≤ θ 1 ≤ 150 ∘ }
Then, we build a design mapping F ^ 2, i ( ξ , η ) from the parameter space Ω ^ 2, i to a quasi-physical sector Q i using the proposed mapping method, for i = 1 , 2 , 3 . Here, we define three physical patches Ω 2, i by using quasi-physical sectors Q i as follows:
Ω 2 , i = { ( x + x i , y + y i ) | ( x , y ) ∈ Q i } ,
which means that Q i ’s are sectors having the same radii and the central angles as these of Ω 2, i ’s through the transformation (30) but the position of the crack tip in Q i is the origin other than ( x i , y i ) in Ω 2, i for i = 1 , 2 , 3 . A structural drawing detailed for Q 1 and Q 2 is shown in
F 2, i ( ξ , η ) ≡ F ^ 2, i ( ξ , η ) + ( x i , y i ) .
Similar to that of Example 1, considering the continuity of the basis functions and the construction of the basis functions on interfaces, we merge two basis
functions defined on different patches that are nonzero along the interface as depicted in
W 2, i h = span [ S i ,1 ∪ S i ,1 n e w ∪ S i ,2 ∪ S i ,2 n e w ∪ S i ,3 ] , i = 1,2
where
1) S 1, i and S 1, i n e w are the set of B-spline basis functions composited with the inverse of the NURBS surface mapping F 2, i on the physical domain Ω 2 satisfying the homogeneous boundary condition.
2) S 2, i and S 2, i n e w are the set of B-spline basis functions composited with the inverse of the NURBS surface mapping F 2, i on the physical domain Ω 2 satisfying the non-homogeneous boundary condition.
In this paper, the physical domains of the elliptic boundary value problems containing multiple singularities, were re-designed by the patchwise mapping methods. In the patchwise mapping method, the construction of the approximation space is different from that in the conventional mapping method [
One of the advantages of the patchwise NURBS mapping method including the NURBS mapping technique is to not only generate singular functions but also preserve the properties of IGA. The properties are the followings [
| p ξ = p η | DOF | ‖ u − u h ‖ ∞ , rel | ‖ u − u h ‖ L 2 , rel | ‖ u − u h ‖ eng,rel | Strain Energy |
|---|---|---|---|---|---|
| 2 | 127 | 4.672E−01 | 1.908E−01 | 9.413E−00 | 1.0530977213691153 |
| 3 | 262 | 1.616E−01 | 5.245E−02 | 3.107E−00 | 1.0448557851141744 |
| 4 | 445 | 6.882E−02 | 1.994E−02 | 9.807E−01 | 1.0439482790063588 |
| 5 | 676 | 9.847E−03 | 2.542E−03 | 1.508E−01 | 1.0438454972029929 |
| 6 | 955 | 6.620E−03 | 1.562E−03 | 8.457E−02 | 1.0438471270008702 |
| 7 | 1282 | 2.285E−03 | 5.035E−04 | 1.872E−02 | 1.0438478370927871 |
| 8 | 1657 | 4.608E−04 | 9.087E−05 | 2.354E−02 | 1.0438479315458822 |
| 9 | 2080 | 3.649E−04 | 7.207E−05 | 1.500E−02 | 1.0438478971863683 |
| 10 | 2551 | 8.706E−05 | 1.537E−05 | 4.227E−03 | 1.0438478755531098 |
| 11 | 3070 | 3.962E−05 | 6.597E−06 | 2.308E−03 | 1.0438478731313534 |
| 12 | 3637 | 1.999E−05 | 3.376E−06 | 1.242E−03 | 1.0438478735263752 |
| 13 | 4252 | 2.985E−06 | 5.006E−07 | 7.634E−04 | 1.0438478737484800 |
| 14 | 4915 | 3.022E−06 | 4.563E−07 | 9.146E−04 | 1.0438478737749743 |
| ∞ | 1.0438478737557377 |
1) The capability of more precise geometric representation of complex objects than conventional Finite Element Methods.
2) Mesh refinement without altering the geometry throughout the refinement process.
Thus, we expect that the patchwise mapping method will be effective for dealing with multiple curved [
On the other hand, the drawback of the mapping method is that it is not eligible to use control points and weights imported from Computer Aided Design (CAD) whereas the conventional IGA is available. To overcome this drawback, the approximation space of the standard IGA can be enriched by the mapping method to deal with singularities [
The author declares no conflicts of interest regarding the publication of this paper.
Kim, H. (2019) Patchwise Mapping Method for Solving Elliptic Boundary Value Problems Containing Multiple Singularities. Journal of Applied Mathematics and Physics, 7, 1572-1598. https://doi.org/10.4236/jamp.2019.77107