Isogeometric analysis based on scaled boundary finite element method

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1 IOP Conference Series: Materials Science and Engineering Isogeometric analysis based on scaled boundary finite element method To cite this article: Y Zhang et al IOP Conf. Ser.: Mater. Sci. Eng. 37 View the article online for udates and enhancements. Related content - The scaled boundary finite element method alied to electromagnetic field roblems Jun Liu Gao Lin Fuming Wang et al. - Isogeometric finite element analysis using olynomial slines over hierarchical T- meshes Nhon Nguyen-Thanh Hung Nguyen-Xuan Stéhane P A ordas et al. - A Hamiltonian-based derivation of Scaled oundary Finite Element Method for elasticity roblems Zhiqiang Hu Gao Lin Yi Wang et al. Recent citations - Isogeometric analysis for ellitical waveguide eigenvalue roblems Yong Zhang et al - Scaled boundary finite element aroach for waveguide eigenvalue roblem G. Lin et al This content was downloaded from IP address on 3//8 at :9

2 WCCM/APCOM IOP Conf. Series: Materials Science and Engineering () 37 doi:.88/ x///37 Isogeometric analysis based on scaled boundary finite element method Y Zhang * G Lin and Z Q Hu Faculty of Infrastructure Engineering Dalian University of Technology Dalian 64 China * Corresonding author: zymarchine@gmail.com Abstract. This aer resents a new aroach which ossesses the semi-analytical feature of scaled boundary finite element method and the exact geometry feature of isogeometric analysis. NURS basis functions are emloyed to construct an exact boundary geometry. The domain boundary is discretized by NURS curves for the D case and NURS surfaces for the 3D case. Esecially the closed-form NURS curves or surfaces are needed if there are no sidefaces. The strategy of using finite elements on domain boundary with NURS shae functions for aroximation of both boundary geometry and dislacements arises from the sense of isoarametric concet. With h--k- refinement strategy imlemented the geometry is refined with maintaining exact geometry at all levels so the geometry is the same exact reresented as the initial geometry imorted from CAD system without the necessity of subsequent communication with a CAD system. Additionally numerical examle exhibits that flexible continuity within the NURS atch rather than traditional shae functions imroves continuity and accuracy of derivative stress and strain field across not only boundary elements but also domain elements as the results of the combination of the intrinsic analytical roerty along radial direction and the higher continuity roerty of NURS basis i.e. it s more owerful in accuracy of solution and less DOF-consuming than either traditional finite element method or scaled boundary finite element method.. Introduction The scaled-boundary finite element method (SFEM) is a semi-analytical numerical method develoed by Wolf and Song. The develoment of SFEM is gradual originally derived to comute the dynamic stiffness of an unbounded domain with mechanically-based derivation and then generally alied to the analysis of incomressible material and bounded domains that roved SFEM to be more general and owerful. The systematically derived weighted residual weakened version of SFEM is available from [ ] and virtual work version from [3]. SFEM combines the advantages of the finite element method and the boundary element method and ossesses unique roerties of its own such as reduction of the satial dimension by one no fundamental solution required no discretization on side face exact in the radial direction and so on. SFEM has many features in common with FEM some of them are good while others are not. Discontinuity of stress field between elements is one of the bad features. The aroximation treatment aroach is FEM-like stress recovery technique roosed by Deeks and Wolf [4] while discontinuity roblem is not fundamentally resolved. c Published under licence by Ltd

3 WCCM/APCOM IOP Conf. Series: Materials Science and Engineering () 37 doi:.88/ x///37 Recently the concet of isogeometric analysis was introduced by Hughes et al [7]. The isogeometric analysis is based on NURS (Non-Uniform Rational -Slines) a standard technology embedded in CAD systems so the exact geometry is resented in both design and mechanics analysis rocedure the coarse mesh of NURS element constructed by exact CAD geometry is subsequently refined without communication with the CAD system but retaining exact geometry which is the reason of so-called isogeometric analysis [7]. There are h- -strategies in isogeometric analysis refinement analogues of h- -strategies in classic finite element analysis and k-refinement a new higher-order methodology [7] which has advantages of efficiency and robustness over traditional - refinement. esides the roerty of accurate reresentation of the geometry another imortant feature of the isogeometric analysis is the ability to use functions of higher order and higher continuity thus discontinuity between elements will vanish when higher continuity between elements is obtained. All subsequent meshes retain exact geometry. Throughout the isoarametric hilosohy is invoked that is the solution sace for deendent variables is reresented in terms of the same functions which reresent the geometry. For this reason we have dubbed the methodology isogeometric analysis. This aer attemts to fundamentally resolve the discontinuity shortcoming in SFEM by introducing the concet of isogeometric analysis to obtain higher continuity between elements. Thus it is with a NURS discretization on boundary only and semi-analytical roerty along radial direction is maintained. For this reason we will dub the methodology boundary isogeometric analysis or isogeometric analysis based on scaled boundary finite element method. Obviously the new aroach ossesses the semi-analytical feature of scaled boundary finite element method and the exact geometry feature of isogeometric analysis.. -slines and NURS.. -sline and NURS basis -sline is defined by -sline basis function control oints and knot vector. There are two saces in -sline: arametric sace secified by a knot vector to construct the -sline basis functions and hysical sace controlled by oints so-called control oints. The -sline arametric sace is associated with a atch in hysical sace consisted of elements. Patch lays the role of subdomain within which element tyes and material models are assumed to be uniform. A knot vector in one dimension is a non-decreasing sequence of coordinates in the arametric sace written Ξ= { ξ ξl ξ ku-} here ξi is the knot and i is the i th knot index. -sline basis functions in one dimension are defined by recursive Cox-de oor formula [9] as ξ [ ξi ξi+ ) Ni( ξ ) = elsewise (.) ξ ξ ξ i i+ + ξ Ni ( ξ) = Ni ( ξ) + Ni+ ( ξ) ξi+ ξi ξi+ + ξi+ -sline are uniform if knots are equally-saced in the arametric sace otherwise they are nonuniform. The multilicity of a knot refers to the times it aears in the knot vector. These with multilicity greater than one are referred to as reeated knots. A knot vector is oen if its first and last knots with multilicity of +. In this work only oen knot vectors are adoted as they are standard in the CAD literature. In one dimension oen -sline basis functions are interolatory at the ends of the arametric sace interval and at the corners of atches in multile dimensions but they are not in general interolatory at interior knots. This is the most distinguishing feature between knots and nodes in finite element analysis. -sline basis functions ossess basic roerties as follows Nonnegativity: basis functions are ointwise nonnegative n Partition of unity: Ni ( ξ) = ξ [ ξ ξ ku ]

4 WCCM/APCOM IOP Conf. Series: Materials Science and Engineering () 37 doi:.88/ x///37 Comact suort: N i ( ξ ) > only ξ [ ξi ξ i+ + ] Flexible continuity: a knot with multilicity of k makes the basis function with the continuity of -k at that location. n Linear indeendence: if un i i ( ξ ) = then u = i for all i in the interval. It s very useful to aly the homogeneous Dirichlet boundary condition in boundary value roblems (VP) Recursive derivation: Ni ( ξ ) = Ni ( ξ) Ni+ ( ξ) ξ ξi+ ξi ξi+ + ξi+ Non-uniform rational -sline (NURS) is built from -sline via rationalization that is ωini ( ξ ) Ri ( ξ ) = n (.) ω N ( ξ ) i i Where ωi is weight corresonding to basis Ni ( ξ ). Thus it ossesses the same roerties with - sline s such as nonnegativity Partition of unity comact suort flexible continuity and linear indeendence etc. Moreover it is more owerful to exactly construct conic curves and surfaces than -sline. And if all of the weights are equal the NURS basis degenerate to -sline basis. Figure. Quadratic NURS one-dimension shae functions. Knot sans: { }... Geometry reresentation based on NURS NURS curve can be reresented in vector form as n C( ξ ) = N ( ξ ) P (.3) i i d Where P R is control oint in control olygon. A tyical NURS curve is shown in Figure. i Figure. Quadratic NURS curve. Control oints are denoted by basis functions as in Figure. 3

5 WCCM/APCOM IOP Conf. Series: Materials Science and Engineering () 37 doi:.88/ x///37 3. Linear elasticity and NURS based scaled boundary finite element method 3.. Linear elasticity Linear elasticity roblem is a classic boundary value roblem (VP) in mechanical analysis governed by three grous of equations in domain: equilibrium equations (Equation 3.a) strain-dislacement equations (Equation 3.b) constitutive equations (Equation 3.c) and two sets of boundary conditions on boundary: Dirichlet boundary condition Neumann boundary condition. Generally as it s difficult to get analytical solution numerical method needs be exlored to obtain aroximate solution varying from finite difference method to finite element method boundary element method and scaled boundary finite element method. T [L] { σ }+{f}= (3.a) { ε } = [ L]{ u} (3.b) { σ} = [ D]{ ε} (3.c) 3.. NURS based scaled boundary finite element method Numerical method exlored to solve linear elasticity roblem in this work is NURS based scaled boundary finite element method in which only boundary needs be discretized by NURS and analytical solution along radial direction will be attained Coordinate system transformation. There are two coordinate systems: Cartesian coordinate system and scaled boundary coordinate system. In Cartesian coordinate system for hysics domain ( ˆx ŷ ẑ ) denotes coordinate of the arbitary oint in hysical domain ( x yz) denotes the coordinate on boundary of hysical domain and ( ˆx ŷ ẑ ) denotes the scaling center. Note that in scaled boundary coordinate system for arameter sace (ξ η ζ ) denotes corresonding coordinate in arameter sace where ξ reresents the dimensionless radial coordinate running from zero at scaling center to unity at domain boundary and others accord with boundary. The formula of transformation from Cartesian coordinate system to scaled boundary coordinate system is written as xˆ( ξ η ζ ) = ξx( η ζ ) + xˆ (3.a) yˆ( ξ η ζ ) = ξy( η ζ ) + yˆ (3.b) zˆ( ξ η ζ ) = ξz( η ζ ) + zˆ (3.c) This is a general reresentation ζ vanishes if boundary is a curve and z disaears when domain is two-dimension. The gradient oerators in both hysics domain and arameter sace are related by so called Jacobi matrix as xˆ ˆ ξ y ξ zˆ ξ ˆ [ J ( ξ η ζ )] = x ˆ ˆ η y η zˆ η = diag( ξξ )[ J ( ηζ )] (3.3) xˆ ˆ ˆ y z ζ ζ ζ The latter item is called boundary Jacobi matrix for its indeendency on radial coordinate ξ x y z [ J ( ηζ )] = x η y η z η (3.4) x y z ζ ζ ζ Equation (3.3) and (3.4) result in ˆ J = [ J] diag( ) (3.5) ξ ξ So the gradient oerators in both hysics domain and arameter sace related as 4

6 WCCM/APCOM IOP Conf. Series: Materials Science and Engineering () 37 doi:.88/ x///37 When [ ] J xˆ ξ ξ ˆ () = = [ J] = [ J] yˆ η ξ η zˆ ζ ξ ζ is divided by column that is [ J ] { j} { j }{ j } = Then the gradient oerators in Cartesian system can be rewritten as () = { j} + { j} + { j3} ξ ξ η ξ ζ (3.6) 3 (3.7) We define a ermutation oerator A () which mas a three-comonent vector 3 X ={x x x 3 } to a matrix by 6 3 x x x 3 A(X)= 3 (3.9) x 3 x x3 x x x Then the differential oerator [L] in Equation (3.a) equals 3 [ L] = [ b ] + [ b ] + [ b ] (3.) ξ ξ η ξ ζ with 3 [ b ( )] A ( j ) [ b ( )] A ( j ) [ b ( ηζ )] = A ( j ) (3.) ηζ = { } ηζ = { } { } oundary isogeometric discretization. Domain boundary is discretized by NURS the knot vector on two directions of domain boundary are denoted by Η= { η ηl η kv-} Ζ= { ζ ζl ζ kw-} resectively. q r denote the degree of NURS basis function. Knot interval [ ηk ηk+ ) [ ζl ζ l+ ) in arameter sace mas into an element V on boundary of hysical domain the corresonding nonzero NURS basis indices ( k l ) the range k= q - k q - k + L q l=r-lr-l+ L r. We define the total number of nonzero basis in that element by cnum = (q+)(r +) as the same number of control oints of the element V. The coordinate of an arbitrary oint on boundary element V kl is written as 3 { X ( cnum )} = { xi} Ni ( ) = [ N ( )]{ x} 3 3 (3.8) ηζ ηζ ηζ (3.) Where { X( ηζ )} = ( X( ηζ ) Y( ηζ ) Z( ηζ ) ) denotes arbitrary oint coordinate on boundary element kl V { xi} = ( xi yi zi ) denotes the corresonding control oint of boundary element V and {} x kl is the total control oint coordinate of V i is univariate index corresonding to bivariate indices ( k l ) Ni is aforementioned basis function either -sline basis or NURS basis and [ N ( ηζ )] is the matrix form. Consequentially coordinate in domain is reresented by 5

7 WCCM/APCOM IOP Conf. Series: Materials Science and Engineering () 37 doi:.88/ x///37 = + cnum = i i + { Xˆ ( ξηζ )} ξ{ X( ηζ )} { xˆ } ξ { x} N ( ηζ ) { xˆ } (3.3) To simlify subsequent writing the suerscrit k l will be omitted. A tyical domain discretized by NURS is shown in Figure 3. Figure 3. A tyical domain discretization by NURS. Control oints are denoted by control net is denoted by black olylines and exact geometry is green-colored. Noninterolation of control oints is shown Isoarametric concet. The isoarametric concet is adoted to aroximate dislacement field on domain boundary so the dislacement of an arbitrary oint on the boundary surface has the similar form to Equation (3.) cnum { u( ξ η ζ )} = { ui( ξ)}* Ni( η ζ ) = [ N( η ζ )]{ u( ξ )} (3.4) Generally { ui( ξ )} = ( ui( ξ) vi( ξ) wi( ξ)) denotes the corresonding dislacement field variable on control oint of similar scaled boundary. The vector { ui} = ( ui vi wi) denotes the corresonding dislacement on control oint of boundary element V kl where ξ =. Consequentially aroximated strain field is obtained from strain-dislacement equations: { ε ( ξηζ )} = [ L]{ u( ξηζ )} = [ ]{ u( ξ)} ξ + [ ]{ u( ξ)} (3.5) ξ where [ ( ηζ )][ ( ηζ )] reresent the strain-dislacement relationshi [ ( ηζ )] = [ b ][ N( η ζ )] (3.6a) 3 [ ( ηζ )] = [ b ][ N( ηζ )] η + [ b ][ N( ηζ )] ζ (3.6b) and then substituting (3.5) into (3.c) leads to the stress vector { σ ( ξηζ )} = [ D ]{ εξηζ ( )}=[ D ]([ ]{u( ξ)} ξ + [ ]{u( ξ)}) (3.7) ξ Scaled boundary finite element equations. The governing differential equations Equation (3.a) is weakened by Galerkin s weighted residual method [] or virtual work rincile [3]. Scaled boundary finite element equations system is exressed as: {[ ] { ( )} (( )[ ] [ ] T T E ξ u ξ ξξ + s E + E [ E ]) ξ{ u( ξ)} ξ + (( s )[ E ] [ E ]){ u( ξ)} + { F( ξ)} = (3.8) where s (= or 3) denotes the satial dimension. [ E ] = [ E ] e [ E ] = [ E ] e [ E ] = [ E ] e are the global coefficient matrices assembled from boundary element coefficient matrices in the finite element manner. e e e 6

8 WCCM/APCOM IOP Conf. Series: Materials Science and Engineering () 37 doi:.88/ x///37 ζ η l+ k+ e T [ ] [ E ] = [ ( ηζ )] D [ ( η ζ )] J dηdζ (3.9a) ζl ηk ζl+ ηk+ e T [ ] [ E ] = [ ( ηζ )] D [ ( η ζ )] J dηdζ (3.9b) ζl ηk ζl+ ηk+ e T [ ] [ E ] = [ ( ηζ )] D [ ( η ζ )] J dηdζ (3.9c) ζ l η k Note that the integration interval in Equation (3.9) is the interval in arameter sace of NURS element which varies from element to element. { F( ξ )} = { F( ξ )} e { P} = { P} e is the force vector resulted from the body force traction on the e e side-faces or boundary resectively. Generally the above equation set is a nonhomogeneous second-order Euler-Cauchy equation and degenerated to a homogeneous equation when{ F( ξ )} vanishes that is both body force and traction on side-face force disaear. The homogeneous equation is solved by eigenvalue method [5] or Matrix function method [6]. Introducing the variable +.5( s ) ξ {()} u ξ { X ( ξ )} =.5( s ) (3.) ξ { q( ξ)} The homogeneous second-order ODEs (3.8) are exressed as a first-order ODEs ξ{ X( ξ)} ξ = [ Z]{ X( ξ)} (3.) With the coefficient Hamiltonian matrix T [ E ] [ E ].5( s )[ I] [ E ] [ Z] = T (3.) [ E ][ E ] [ E ] [ E ].5( s )[ I] [ E ][ E ] Solution of Scaled boundary finite element equations. In the eigenvalue method [5] conducting eigen-decomosition of coefficient Hamiltonian matrix [ Z ] the eigenvalues[ Λ ] and eigenvector [ Φ] are artitioned conformably as [ Φu] [ Φu] [ λn ] [ Z][ Φ ] = [ Φ][ Λ ] = [ q] [ q] (3.3) Φ Φ λ The general solutions and internal forces is.5( s ) λ λ n u( ξ) ξ ([ ] ξ c [ ] ξ = Φ + Φ c ) (3.4a) { } { }.5( s ) { c } = is set in unbounded domain and { } [ ] { } u { } [ λ ] n { } λ { } u + q q q( ξ) = ξ ([ Φ ] ξ c + [ Φ ] ξ c ) (3.4b) c = is set in bounded domain the stiffness matrix on the boundary is obtained by eliminating integration constant as[ K] =Φ [ ][ Φ ] for bounded domain q u and[ K] = Φ [ ][ Φ ] for unbounded domain. oundary dislacement is solved from[ K]{ u} = { P} q u followed by the determination of the integration constants { c }. Then dislacement strain and stress at any oint in domain can be obtained by (3.4) (3.5) (3.7) resectively. 7

9 WCCM/APCOM IOP Conf. Series: Materials Science and Engineering () 37 doi:.88/ x///37 4. Numerical Examles The Lame s roblem [8] is illustrated to verify the effectiveness of isogeometric analysis based on SFEM. The Lame s roblem is a lane stress roblem of a circular cavity in a full-lane subjected to a uniform ressure on the cavity wall. The exact solution for stress comonents in Cartesian coordinate system is R σ x ( r θ) = (sinθ cos θ) (4.a) r R σ y ( r θ) = (cosθ sin θ) (4.b) r R σ xy ( r θ) = cosθsinθ (4.c) r The scaling centre O is chosen at the centre of the circular cavity. Taking advantage of the symmetry of the roblem only a quarter of the unbounded domain is modeled. The domain boundary is dicretized by both traditional Lagrange and NURS element shown in Figure 4. Comarison the solution of the NURS SFEM and traditional SFEM with the analytical solution is given in following subsections. (a) -order Lagrange element (b) -order NURS element Figure 4. Discretization of unbounded domain. 4.. Exact geometry reresentation oundary geometry is resented by Lagrange basis in traditional SFEM while NURS basis in NURS SFEM. Comarison of the distance between the oints on the boundary wall and the scaling center is showed in Figure 5. It s observed that Lagrange element reresents boundary geometry aroximately excet for the nodal location while NURS element can exactly reresent it ointwisely. 8

10 WCCM/APCOM IOP Conf. Series: Materials Science and Engineering () 37 doi:.88/ x///37. exact traditional NURS radius Figure 5. Comarison of the exactness of geometric reresentation there are two elements in both traditional and NURS SFEM and the basis functions are second-order in both traditional and NURS SFEM 4.. Less total DOF-consuming It s shown in Figure 4 that there is the same number of elements in traditional and NURS SFEM but NURS method only consumes 8 DOF less than DOF in traditional method. It s not accident for the NURS element each other shares more DOF. In Figure 4b element and share control oint and 3 generally -order NURS elements share -k control oints each other k the common arameter multilicity. Thus less total DOF-consuming is obvious Flexible continuity solution along boundary The solution of Cartesian stress comonents and their comarison are given in Figure 6..5 traditional exact.5 NURS exact Stress σx/ Stress σx/ (a) - σ x by traditional SFEM (b) σ x by NURS SFEM trditional exact.5 NURS exact Stress σy/ Stress σy/ (c) - σ y by traditional SFEM (d) σ y by NURS SFEM Figure 6. Comarison of stress comonents on boundary oundary discretization of two second-order elements. 9

11 WCCM/APCOM IOP Conf. Series: Materials Science and Engineering () 37 doi:.88/ x///37 The traditional SFEM solution for stress comonents is discontinuous across element as shown in Figure 6(a c) while the NURS SFEM solution is continuous across element as shown in Figure 6(b d). And the relative error of stress solution is demonstrated in Figure 7. The accuracy of solution by NURS SFEM is almost 5 times of traditional SFEM at the same number of elements of two. stress relative error 8% Stress σx Stress σy 4% Stress σxy % % stress relative error.5%.%.5%.% % -.% Stress σx Stress σy Stress σxy -8% -.5% (a) Stress error by traditional SFEM (b) Stress error by NURS SFEM Figure 7. Relative error of stress comonents on boundary oundary discretization of two second-order elements. 5. Conclusions NURS based scaled boundary finite element method (NSFEM) is resented by introducing the concet of isogeometric analysis to SFEM. The advanced method ossesses semi-analytical and flexible continuity roerty at the same time. In comarison with the traditional SFEM the advanced method has the following advantages. Firstly the exact domain geometry is resented. Secondly it consumes less DOF. Thirdly solution owns flexible continuity across elements. Finally higher accuracy solution is obtained. The numerical examle demonstrates the above advantages. Reference [] Wolf J P and Song C The scaled boundary finite-element method-a rimer: derivations Comut. Struct [] Song C and Wolf J P The scaled boundary finite-element method-a rimer: solution rocedures Comut. Struct [3] Deeks A J and Wolf J P A virtual work derivation of the scaled boundary finite-element method for elastostatics Comut. Mech [4] Deeks A J and Wolf J P Stress recovery and error estimation for the scaled boundary finite-element method Int. J. Num. Meth. Eng [5] Song C and Wolf J P 999 ody loads in scaled boundary finite-element method Com. Meth. Al. Mech. Eng [6] Song C 4 A matrix function solution for the scaled boundary finite-element equation in statics Comut. Methods Al. Mech. Engrg [7] Hughes T J R Cottrell J A and azilevs Y 5 Isogeometric analysis CAD finite elements NURSexact geometry and mesh refinement Comut. Methods Al. Mech. Engrg [8] Gonld P L 999 Introduction to Linear Elasticity nd edition (erlin: Sringer Verlag) [9] Piegl L and Tiller W 997 The NURS ook (erlin: Sringer Verlag)