Journal of Thermal Stresses ISSN: 0149-5739 (Print) 1521-074X (Online) Journal homepage: http://www.tandfonline.com/loi/uths20 Solution of steady-state thermoelastic problems using a scaled boundary representation based on nonuniform rational B-splines Peng Li, Gao Lin, Jun Liu, Yang Zhou & Bin Xu To cite this article: Peng Li, Gao Lin, Jun Liu, Yang Zhou & Bin Xu (2017): Solution of steady-state thermoelastic problems using a scaled boundary representation based on nonuniform rational Bsplines, Journal of Thermal Stresses, DOI: 10.1080/01495739.2017.1387881 To link to this article: http://dx.doi.org/10.1080/01495739.2017.1387881 Published online: 25 Oct 2017. Submit your article to this journal View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=uths20 Download by: [University of Florida] Date: 26 October 2017, At: 09:54 JOURNAL OF THERMAL STRESSES https://doi.org/10.1080/01495739.2017.1387881 Solution of steady-state thermoelastic problems using a scaled boundary representation based on nonuniform rational B-splines Peng Lia,b , Gao Lina,b , Jun Liua,b,c , Yang Zhoua,b , and Bin Xua,b a Faculty of Infrastructure Engineering, School of Hydraulic Engineering, Dalian University of Technology, Dalian, China; State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, China; c Faculty of Engineering, Centre for Offshore Foundation Systems, University of Western Australia, Perth, WA, Australia Downloaded by [University of Florida] at 09:54 26 October 2017 b ABSTRACT ARTICLE HISTORY This work explores the application of isogeometric scaled boundary method in the two-dimensional thermoelastic problems of irregular geometry. The proposed method inherits the advantages of both isogeometric analysis and scaled boundary finite element method and overcomes their respective disadvantages. In the proposed approach, the boundaries of the problem domain are discretized with nonuniform rational B-splines (NURBS) basis functions, while the temperature distributions inside the domain are represented by a sequence of power functions in terms of radial coordinate within the framework of scaled boundary finite element method. The resulting solution of the stress in radial direction can be computed analytically for the temperature changes. The construction of tensor product structure is circumvented for the two-dimensional problems as only the boundary information of the problem domain is required. Hence, the flexibility to represent the complex geometry can be significantly improved in the proposed method. Numerical examples are presented to validate the performance of the proposed method where it is seen that superior accuracy, efficiency, and convergence behavior can be achieved over the conventional scaled boundary finite element method. Received 7 July 2017 Accepted 1 October 2017 KEYWORDS Complex geometry; isogeometric analysis; nonuniform rational B-splines; scaled boundary finite element method; thermal stress; thermoelastic problems Introduction The evaluation of stress caused by thermal load is closely related to many practical engineering applications. As one of the most important research fields for solid mechanics, the thermoelastic problems have been extensively studied by many researchers. However, due to the inherent complexity of the governing equations, the closed-form analytical solutions are available for the problems with only simple geometric shapes and boundary conditions [1–3]. Hence, in most cases, the numerical techniques become the powerful alternatives to the thermal stress distributions and calculations, such as the finite element method (FEM) [4–6], boundary element method (BEM) [7–11], and meshless method (MM) [12–14]. Although all these numerical methods have been successfully applied to the solution of thermal stress due to their attractive properties, they have also exhibited some limitations of their own. As to FEM, it is easy to implement and versatile with various commercial software packages available. However, the whole computational domain in FEM is required to be discretized, which often leads to considerable workload in mesh generation for the large and complex problems. Besides, it is well known that the quality of finite element meshes can significantly influence the computational precision of this method. Compared with the FEM, the BEM is considered as an effective numerical approach for solving the thermoelastic problems based on the known fundamental solutions. In this method, only CONTACT Jun Liu liujun8128@126.com Faculty of Infrastructure Engineering, School of Hydraulic Engineering, Dalian University of Technology, No. 2 Linggong Road, Ganjingzi District, Dalian 116024, China. Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/uths. © 2017 Taylor & Francis Downloaded by [University of Florida] at 09:54 26 October 2017 2 P. LI ET AL. the boundary needs to be discretized with elements rather than the whole domain. Hence, the spatial dimension is reduced by 1. It spares computational cost in mesh generation and CPU time. But, the fundamental solutions of complex problems are available only for some special cases [15, 16]. In the other front of development, meshless methods have attracted much attention of researchers as an alternative to the mesh-based method in recent years, in which the problem domain is constructed by use of a sequence of scattering nodes instead of mesh. In addition, no information about the relationship of the nodes is required as well as the predefined nodal connectivity. Nevertheless, the high computational cost limits the application of meshless methods, therefore, any improvement for computational efficiency can be considered as an important advancement in this methodology [17]. The scaled boundary finite element method (SBFEM) is a novel semianalytical numerical technique for solving partial differential equations [18]. This method combines the finite element formulations and boundary-element discretization with its own advantages. First of all, as the BEM, only the boundary requires discretization in SBFEM. But, the fundamental solution is unnecessary. Second, by applying this method, solutions can be obtained analytically in the radial direction, while solutions can be acquired numerically along the circumferential direction in finite element sense through the introduction of standard polynomial shape functions. Therefore, this methodology has shown great success on a variety of domains including elastoplasticity [19], structure analysis [20], acoustical problems [21], fracture problems [22], wave propagation [23, 24], dynamic problems [25–27], foundation engineering [28–31], sloshing problems [32–34], elastic waveguide problems [35], potential flow [36, 37], and heat transfer [38, 39]. In the case of solutions for thermal stress, Song [40] extended SBFEM to the modeling of thermal stresses at multimaterial corners under thermal load. Li et al. [41] applied SBFEM to study the fracture behaviors of piezoelectric materials and piezoelectric composites under thermal load. In the framework of scaled boundary method (SBM), the discretization method used in the circumferential direction can strongly affect the accuracy of resulting solutions [42]. Therefore, SBFEM suffers from the FEM-associated issues. The main reason of the FEM drawbacks is the use of two separate models in geometric design and numerical analysis, i.e., geometric model and analysis model. In general, the geometric model is exactly represented in CAD system by spline basis functions, while the analysis model is obtained from the approximation of geometric model by meshing, which is usually based on polynomial basis functions. Also, the discretization from the geometric model to the finite elements for numerical analysis needs extra meshing efforts and is an irreversible process, which consumes considerable manpower and time for an instant interaction that requires new and more exact analysis models. In addition, for irregular complex geometries with curvatures and torsions such as circles or cones, an exact representation of geometry can only be obtained in limit of mesh refinement, or such an inconsistency between two models leads to the loss of computational accuracy. To overcome the shortcomings of FEM mentioned above, an approach is presented [43] by introducing the concept of isogeometric analysis (IGA), where the geometric and analysis information are unified in only one model using NURBS. In contrast to the conventional isoparametric finite element analysis, the same NUBRS basis functions are used for both geometric description and approximation of filed variables in IGA. In doing so, the exact description of geometry is persevered for the analysis model and the geometric model can be directly used for analysis without additional mesh generation procedure. As a result, the mismatch between two different models is eliminated, and the communication between the two models becomes convenient and efficient. Moreover, the total number of degrees of freedom (DOF) is significantly reduced since NURBS basis functions enjoy a higher order of community across the boundary. Hence, the IGA exhibits great superiority on computational accuracy and efficiency compared to conventional FEM. This technique has been extended to various engineering fields ranging from plates [44], vibration [45], fluid mechanics [46], and fluid–structure interaction [47] to fracture mechanics [48]. Naturally, the framework of IGA has been extended to the structure shape optimization since the geometry can be exactly represented [49, 50]. Also, this approach has been combined with the extended finite element method (XFEM) [51], BEM [52], MM [53], and many fruitful applications. With the motivation of superior performance of IGA, an isogeometric scaled boundary method (IGA-SBM) is proposed combining the scaled boundary finite element method and IGA, where the NUBRS basis functions are used instead of polynomial basis functions to discrete the field variables JOURNAL OF THERMAL STRESSES 3 Downloaded by [University of Florida] at 09:54 26 October 2017 along the circumferential direction within the framework of scaled boundary method. The combination of IGA into scaled boundary method takes advantages of both methods with unique propriety of its own. No tensor product patch is required due to a reduction of spatial dimension by 1, which leads to a more flexible description for complex geometry in two-dimensional case. In this work, we further extend the proposed method to analyze the stress fields of irregular complex geometry under thermal load. The temperature changes in the domain is determined from an eigenvalue problem and expressed as a series of power functions of the radial coordinate. Then the thermal load is represented as a nonhomogenous term in the resulting ordinary differential equations. The layout of this article is as follows. First section briefs some preliminary studies including the definitions of NURBS and the basic idea of IGA. In third section, the derivation of the proposed method for the analysis of stress fields under thermal load is presented in the scaled boundary coordinates. Numerical examples are given to verify the effectiveness and power of the proposed method in fourth section. Finally, concluding remarks are drawn in last section. NURBS and isogeometric analysis NURBS is the standard technology for the geometric representation in CAD system since it can effectively describe the geometrical shapes of any degree of complexity and can exactly represent all kinds of conic sections. In this section, the definition of NURBS is briefly introduced. More details about NURBS can be found in Reference [54, 55]. Then, the basic concept of the NURBS-based IGA is also explained. NURBS NURBS is a generalization of B-spline, which is constructed through a weighted and rational form of B-spline. A B-spline basis function Bi,p (s) is defined by a knot vector and the polynomial order p. A knot vector 4 in one-dimensional space is a nondecreasing sequence of parametric coordinates. 4 = s1 , s2 , . . . , sn+p+1 , si ≤ si+1 , i = 1, . . . , n + p (1) where the entries si are called knots, and n is the number of basis functions. The half-open section is between two consecutive and distinct knots, is called nonzero knot span. The interval [si , si+1 ], which s1 , sn+p+1 is named as a patch. If the first and last entries in a knot vector are repeated p + 1 times, it is said to be an open knot vector. The open knot vector satisfies the Kronecker delta property at the ends of vector and it is used in the remainder of this article. The i-th B-spline basis function can be determined by the Cox-De Boor recursive formula [55] starting with p = 0: ( 1, if si ≤ s < si+1 Bi,0 (s) = (2) 0, otherwise and Bi,p (s) = si+p+1 − s s − si Bi,p−1 (s) + Bi+1,p−1 (s), si+p − si si+p+1 − si+p p≥1 (3) Figure 1 illustrates a set of linear, quadratic, cubic, and quartic B-spline basis functions constructed by the open knot vectors 41 = {0, 0, 1, 1}, 42 = {0, 0, 0, 1, 1, 1}, 43 = {0, 0, 0, 0, 1, 1, 1, 1}, and 44 = {0, 0, 0, 0, 0, 1, 1, 1, 1, 1}, respectively. Using the B-spline basis function Bi,p (s) and the weight wi , the NURBS basis function can be defined as Bi,p (s)wi Ri,p (s) = n (4) P Bj,p (s)wi j=1 Downloaded by [University of Florida] at 09:54 26 October 2017 4 P. LI ET AL. Figure 1. B-spline basis functions against various orders: (a) linear, (b) quadratic, (c) cubic, and (d) quartic. The NURBS basis function possesses several analysis-relevant properties such as nonnegatively, partition of unity, local support, and continuity easily represented with knot multiplicity. Moreover, a multidimensional NURBS basis function can be obtained using the tensor product structure of one-dimensional basis function in Eq. (4). Given the knot vectors 4 = s1 , s2 , . . . , sn+p+1 and H = t1 , t2 , . . . , tm+q+1 , a two-dimensional NURBS basis function of p × q degree is defined as Bi,p (s)Bj,q (t)wi,j p,q Ri,j (s, t) = Pm Pn i=0 j=0 Bi,p (s)Bj,q (t)wi,j (5) where Bi,p (s) and Bj,q (t) are corresponding one-dimensional B-spline basis functions of orders p and q defined on the 4 and H. wi,j denotes the related weight of geometry. In this article, since only surface is considered as analysis model, the definition is restricted to the NURBS surface only. It is defined as follows: m X n X p,q S(s, t) = Ri,j (s, t) · Pi,j (6) i=1 j=1 where Pi,j represents a bidirectional net of control points. n and m denote the number of control points in s and t direction, respectively. The patch is the domain s1 , sn+p+1 × t1 , tm+q+1 . An example of a bivariate quadratic NURBS surface is shown in Figure 2. Analysis framework based on NURBS In the present article, the analysis framework of the NURBS-based IGA is used for the circumferential approximation. The core idea of IGA is that the solution space is represented in terms of the same NUBRS basis functions as used in describing the geometry. Using the NURBS basis functions instead of polynomial basis functions as the approximating functions, the field variables in IGA can be approximated as X X u= N (s, t) ūα ; x= N (s, t) P̄α (7) α=1 α=1 JOURNAL OF THERMAL STRESSES 5 Downloaded by [University of Florida] at 09:54 26 October 2017 Figure 2. Physical mesh and control net with quadratic NURBS surface. Figure 3. The procedure of isogeometric analysis: (a) initial geometric information of analysis model, (b) global refinement by knot insertion, (c) imposing the boundary and load conditions, and (d) finite element analysis using NURBS basis functions. 6 P. LI ET AL. Downloaded by [University of Florida] at 09:54 26 October 2017 where P̄α are control points which play the identical role with nodes in FEM. ūα are the control variables related to the control points P̄α ,. s, and t are the parametric coordinates. The general procedure of IGA is summarized as follows: (1) The geometric information of analysis model such as the coordinates of control points, weights, order of basis functions, and knot vectors are obtained from the CAD files. In general, the spline data only provides the initial coarse mesh of analysis model. But, the NURBS-based IGA gives the exact description of geometry with a coarse mesh since the desirable proprieties of NURBS basis function are directly inherited from CAD system, as illustrated in Figure 3a. The parametric space of analysis model is defined by the knot vectors, while the physical space is defined as an image of the parametric space, as shown in Figure 3a and 3b. (2) For accurate analysis, the refinement of analysis model is performed by knot insertion to obtain sufficient DOF. It is worth noticing that the geometry is exactly preserved without losing any geometric information during the refinement process, as shown in Figure 3b. (3) After the imposition of boundary and load conditions, the solution is performed as in the conventional FEM, as depicted in Figure 3c and 3d. Due to the noninterpolatory nature of NURBS basis function, special treatment is required for the inhomogeneous Dirichlet conditions in IGA. Lagrange multiplier method [56] is used in this study. Methodology Governing equations for steady-state heat conduction problems The governing equation in a homogeneous and isotropic media is formulated as t T L q = 0 in (8) where represents the computational domain. q is the heat flux vector associated with the temperature T. q = − [κ] Lt T (9) h i k 0 where [κ] = 0x̂ kŷ denotes the isotropic thermal conductivity matrix. Lt is the differential operator relating the heat flux and the temperature. ∂ t x̂ L = (10) ∂ ŷ The computational domain is enclosed by the total boundaries Ŵ. The boundary conductions are written as T = T̄ on Ŵ1 ( q̄ on Ŵ2 ∂T ∂T −kx̂ nx̂ − kŷ ny = ∂ x̂ ∂ ŷ β (T − Ta ) on Ŵ3 (11) (12) P where Ŵ = 3i=1 Ŵi and Ŵi ∩ Ŵj = ∅ for i 6= j. n is the vector of the unit normal to the boundary. T̄ and q̄ are the prescribed temperature and heat flux, respectively, on the corresponding boundaries. β is the convection heat transfer coefficient, and Ta is the ambient temperature. JOURNAL OF THERMAL STRESSES 7 Downloaded by [University of Florida] at 09:54 26 October 2017 Isogeometric scaled boundary equations for heat conduction problems For the two-dimensional problem, the scaled boundary method introduces a new set of scaled boundary coordinates (ξ , s) for describing the geometry of the problem domain, as depicted in Figure 4a. One of them is the dimensionless radial coordinate ξ , ranged from the scaling center to the boundary. For a bounded domain, it has values of 0 at the scaling center and 1 on the boundary. The other one is the circumferential coordinate s (s1 ≤ s ≤ sn+p+1 ), which is the parametric coordinate of the defining boundary curve as well. In Figure 4a, the boundary Se of domain is discretized by the quadratic NURBS element using the knot vector s = {0, 0, 0, 0.5, 1, 1, 1}. In doing so, the whole computational domain is covered by scaling the boundary along the radial direction with reference to the scaling center with a scaling factor smaller than 1. The scaling center O is selected at a point from which any point on the boundary should be directly visible. This requirement can always be satisfied by dividing the total domain into subdomains where the similarity is met in each subdomain, as shown in Figure 4b. Without losing the generality, we assume that the origin of the Cartesian coordinate system is chosen at the scaling center. The Cartesian coordinates x̂, ŷ of a point inside the domain are related to the scaled boundary coordinates (ξ , s) as x̂ = x0 + ξ x (s) = ξ [N (s)] {x} (13a) ŷ = y0 + ξ y (s) = ξ [N (s)] y (13b) where x0 , y0 are the coordinates of scaling center. x (s) , y (s) denotes the corresponding point of x̂, ŷ on the boundary, which are represented by the coordinates {x} , y of control points of NURBS elements on the boundary using the NURBS basis functions [N (s)] = [N1 (s) , N1 (s) , . . . , Nn (s)]. To transform all of the derivatives in the Cartesian coordinate system to those in the scaled boundary coordinate system, the Jacobian matrix is required h i x̂ ŷ,ξ x (s) y (s) 1 0 (14) = = Ĵ (ξ , s) = ,ξ [J (s)] x̂,s ŷ,s ξ x (s),s ξ y (s),s 0 ξ where [J (s)] is the Jacobian matrix on the boundary (ξ = 1) x (s) y (s) [J (s)] = x (s),s y (s),s (15) and its determinant is equal to |J (s)| = x (s) y (s),s − y (s) x (s),s Figure 4. Discretization in IGA-SBM: (a) a bounded domain in two-dimensional space and (b) discretization by subdomains. (16) 8 P. LI ET AL. The differential operator in Eq. (10) is mapped to scaled boundary coordinate system using the following. ∂ ∂ ∂ ξ ξ i−1 t x̂ h ∂ 1 t ∂ −1 L = = [J (s)] + b2 (s) (17) = Ĵ (ξ , s) = bt1 (s) ∂ξ ξ ∂s ∂ 1 ∂ ∂ ŷ ξ s s with ( ) y (s),s 1 |J (s)| −x (s),s ( ) −y (s) t 1 b2 (s) = |J (s)| x (s) Downloaded by [University of Florida] at 09:54 26 October 2017 t b1 (s) = (18a) (18b) Using the NURBS basis functions [N (s)] in IGA, the temperature {T (ξ , s)} at a point (ξ , s) inside the domain is obtained in the form {T (ξ , s)} = [N (s)] {T (ξ )} (19) It is postulated that the same NURBS basis functions apply with the temperature functions {T (ξ )} for all lines Sξ (Figure 4a) parallel to the boundary with a constant ξ ∗ . The unknown temperature functions {T (ξ )} is analytical with respect to the radial coordinate ξ . The heat flux can be expressed in the scaled boundary coordinate system substituting Eqs. (19), (17), and (10) into Eq. (9) 1 t (20) q (ξ , s) = [−κ] Bt1 (s) {T (ξ )},ξ + B2 (s) {T (ξ )} ξ with Bt1 (s) = t B2 (s) = t b1 (s) [N (s)] t b1 (s) [N (s)],s (21a) (21b) Applying the Galerkin method [57], the isogeometric-scaled boundary equation in temperature is formulated as: t 2 T E0 ξ {T (ξ )},ξ ξ + E0t − E1t + E1t ξ {T (ξ )},ξ − E2t {T (ξ )} = 0 (22) t t t where the coefficient matrices E0 , E1 , and E2 for each boundary element e in a subdomain are given by Z t t T E0 = B1 (s) [κ] Bt1 (s) |J (s)| ds (23a) e t E1 = t E2 = Z Z (23b) e t T B2 (s) [κ] Bt1 (s) |J (s)| ds (23c) e t T B2 (s) [κ] Bt2 (s) |J (s)| ds The standard procedure for assembling the coefficient matrices can be applied as in the classic FEM. The heat flux on the boundary is derived as t t T (24) P = E0 {T} + Ē1t {T} JOURNAL OF THERMAL STRESSES where {T} denotes the boundary temperature ({T (ξ = 1)}). Z t t T Ē1 = E1 + β [N (s)]T [N (s)] ds 9 (25) Ŵ3 t P =− Z q̄ [N (s)]T ds + Ŵ2 Z βTa [N (s)]T ds (26) Ŵ3 The internal heat flux vectors on a line with a constant ξ ∗ are expressed as T Pt (ξ ) = E0t ξ {T (ξ )},ξ + Ē1t {T (ξ )} (27) Downloaded by [University of Florida] at 09:54 26 October 2017 The solution of {T (ξ )} in Eq. (22) can be obtained in the form of power function using the eigenvalue decomposition procedure in [58] Nt t X t t t t −λtN T (ξ ) = c1t ξ −λ1 φ1t + c2t ξ −λ2 φ2t + . . . + cN ξ φNt = cit ξ −λi φit (28) i=1 where N t denotes the number of DOF on the discretized boundary. The vectors {φi } and the exponents λi can be interpreted as the model temperature vector and model scaling factors, which are the eigenvectors and eigenvalues of a standard linear eigenvalue problem. The integration constants cit denote the contribution of each {φi } and are determined by the boundary conditions. The temperature for each mode is given as t {T (ξ )} = ξ −λ φ t (29) t t T P = Ē1 − λt E0t φ t (30) Substituting Eq. (29) into Eq. (27) and evaluating on the boundary (ξ = 1) lead to Substituting Eq. (29) into Eq. (22) results in the quadratic eigenvalue problem h h i i t 2 T λt E0t − λt E1t − E1t − E2t φ = {0} (31) Using Eqs. (31) and (30), the problem is written as a linear form t −1 t −1 E0 Ē1 − E0t ( φ t ) t t t −1 t − E2 + E1 E0 Ē1 + t t t −1 t t T t −1 P T −1 t t t 1 Ē − E1 E0 Ē1 − E1 E0 − Ē1 − E1 E0 ( t ) φ =λ Pt t (32) The solution of this eigenvalue problem yields 2N t modes. For the bounded domain, only half of the modes with negative real parts of eigenvalue can lead to a finite temperature at the scaling center (ξ = 0). These subsets ofnmodel and corresponding modal heat fluxes are designed by oi temperatures t h t t 81 = φ1 , φ2 , . . . , φNt t and Qt1 . The T (ξ ) in Eq. (28) is expressed as i h t T (ξ ) = 8t1 ξ −λ ct (33) 10 P. LI ET AL. t t t On the boundary (ξ = 1), Eq. (33) is written as T (ξ = 1) = 81 c . The integration constants c can be obtained by the temperature of control points on the boundary {T (ξ = 1)}. t t −1 {T (ξ = 1)} c = 81 (34) The equivalent modal heat fluxes required to cause these temperatures are equal to t t t t t −1 {T (ξ = 1)} P = Q 1 c = Q 1 81 Consequently, the stiffness matrix with respect to the boundary DOF is formulated as t t t −1 K = Q 1 81 (35) (36) Downloaded by [University of Florida] at 09:54 26 October 2017 Substituting Eq. (33) into Eq. (19) yields the temperature T (ξ , s) at any point inside the domain. Nt X t h −λt i t t T (ξ , s) = [N (s)] 81 ξ c = [N (s)] cit ξ −λi φit (37) i=1 Formulation for the thermal stress in the scaled boundary coordinates The governing equations for linear elastostatic problem with thermal load are stated as {σ } = [D] ({ε} − {ε0 }) (38) where [D] is the elasticity matrix. {ε0 } is the initial strains caused by temperature variation T. The initial strain {ε0 } for two-dimensional problems is calculated as {ε0 } == Tβ plane stress state {ε0 } == (1 + ν) Tβ plane strain state (39) (40) with β = [α α 0]T (41) where α and ν are the coefficient of thermal expansion and Poisson’s ratio, respectively. Thermal stresses are included as the initial stresses {σ0 } {σ0 } = [D] {ε0 } (42) The equations of equilibrium in the domain are written as {L}T {σ } + f = 0 (43) where f is the body load and {L}T is the differential operator relating the strains {ε} and the displacements {u} {ε} = {L}T {u} ∂ 0 x̂ ∂ {L} = 0 ŷ ∂ ∂ ŷ x̂ (44) (45) JOURNAL OF THERMAL STRESSES 11 Using Eqs. (13) and (15), {L} can be transformed into scaled boundary coordinates as {L} = [b1 (s)] ∂ 1 ∂ + [b2 (s)] ∂ξ ξ ∂s (46) with Downloaded by [University of Florida] at 09:54 26 October 2017 y (s),s 0 1 0 x (s),s [b1 (s)] = |J (s)| −x (s),s y (s),s −y (s) 0 1 0 x (s) [b2 (s)] = |J (s)| x (s) −y (s) (47a) (47b) The scaled boundary method seeks an approximate solution for the displacement field {u (ξ , s)} in the form {u (ξ , s)} = N̄ (s) {u (ξ )} (48) where {u (ξ )} is the displacement function along the radial lines and with N̄ (s) = [N1 (s) [I] , N1 (s) [I] , . . . , Nn (s) [I]] ([I] is a 2 × 2 identity matrix). Substituting Eqs. (48), (46), and (44) into (38), and using Eq. (42), the stresses are rewritten in scaled boundary coordinates as x0 , y0 1 {σ (ξ , s)} = [D] [B1 (s)] {u (ξ )},ξ + [B2 (s)] {u (ξ )} + {σ0 } (49) ξ where [B1 (s)] = [b1 (s)] N̄ (s) [B2 (s)] = [b1 (s)] N̄ (s) ,s (50a) (50b) Based on the Galerkin’s weighted residual technique [40], the isogeometric-scaled boundary finite element equation in displacement can be obtained as follows (51) [E0 ] ξ 2 {u (ξ )},ξ ξ + [E0 ] − [E1 ] + [E1 ]T ξ {u (ξ )},ξ − [E2 ] {u (ξ )} + {F (ξ )} = 0 The coefficient matrices [E0 ], [E1 ], [E2 ] and the equivalent vector load {F (ξ )} in Eq. (51) are defined as follows: Z (52a) [E0 ] = [B1 (s)]T [D] [B1 (s)] |J (s)| ds e [E1 ] = [E2 ] = Z Z [B2 (s)]T [D] [B1 (s)] |J (s)| ds (52b) [B2 (s)]T [D] [B2 (s)] |J (s)| ds (52c) e e {F (ξ )} = ξ 2 {P (ξ )} + ξ {P0 (ξ )} where {P (ξ )} and {P0 (ξ )} are the contribution of body loads and thermal stresses, respectively. Z T {P (ξ )} = N̄ (s) f |J (s)| ds (52d) (53) e {P0 (ξ )} = Z e [B1 (s)]T (ξ {σ0 }),ξ − [B2 (s)]T {σ0 } |J (s)| ds (54) 12 P. LI ET AL. The internal nodal forces q (ξ ) , which are equal to the resultants of the stresses, along the radial lines connecting the scaling center and the control points on the boundary can be expressed as q (ξ ) = [E0 ] ξ {u (ξ )},ξ + [E1 ]T {u (ξ )} + q0 (ξ ) (55) The contribution of the thermal stresses q0 (ξ ) is determined as follows: Z q0 (ξ ) = ξ [B1 (s)]T {σ0 } |J (s)| ds (56) e Downloaded by [University of Florida] at 09:54 26 October 2017 The general solution of Eq. (51) can be obtained by a standard linear eigenvalue problem in the form " #( ) ( ) {φ} {φ} − [E0 ]−1 [E0 ]−1 [E1 ]T =λ (57) q q − [E2 ] + [E1 ] [E0 ]−1 [E1 ]T − [E1 ] [E0 ]−1 For N DOF on the discretized boundary, the solution of Eq. (57) yields 2N modes. Considering the finiteness of displacement at the scaling center, only half of eigenvalues with negative real parts are selected for a bounded domain. The subset of N modal displacements is designated by [81 ] = [{φ1 } , {φ2 } , . . . , {φN }] and the corresponding modal forces by [Q1 ]. Solution procedure When the temperature changes are represented in terms of a series of power functions of the radial coordinate, the solutions of stresses can be obtained analytically in the radial direction [40]. Substituting Eq. (39) into Eq. (42) and utilizing one term of temperature field specified in Eq. (37) leads to the initial stresses cause by the temperature changes as follows: {σ0 } = [D] Td (s) ξ d β (58) with d = −λti (59) Td (s) = [N (s)] 8ti cit (60) {P0 (ξ )} = P̄0 ξ d (61) Substituting Eq. (58) into Eq. (54) yields where P̄0 = Z e [B1 (s)]T (d + 1) − [B2 (s)]T [D] Td (s) β |J (s)| ds (62) By substituting Eq. (61) in Eq. (52d), the nonhomogeneous term {F (ξ )} for vanishing body loads can be denoted as {F (ξ )} = P̄0 ξ b (63) with the constant b=d+1 (64) Considering the finiteness of the integral in particular solutions, b > 0 is required. This requirement can always be satisfied since all eigenvalues λti are negative in a bounded domain. By utilizing Eq. (61), the thermal load mode displacement can be obtained in the form [58]: {u0 (ξ )} = ξ b {u∗ } (65) JOURNAL OF THERMAL STRESSES Substituting Eq. (65) into Eq. (51) (in the absence of body loads) yields b2 [E0 ] + b [E1 ]T − [E1 ] − [E2 ] {u∗ } + P̄0 = 0 and the nodal displacement for the thermal load mode is derived as follows: −1 {u∗ } = − b2 [E0 ] + b [E1 ]T − [E1 ] − [E2 ] P̄0 13 (66) (67) The equivalent nodal forces caused by these displacements can be obtained by the substitution of Eq. (67) into Eq. (55) as q∗ (ξ ) = b [E0 ] + [E1 ]T ξ b {u∗ } + q0 (ξ ) (68) Downloaded by [University of Florida] at 09:54 26 October 2017 The complete solution of Eq. (51) for vanishing body loads becomes {u (ξ )} = [81 ] ξ −λ {c} + ξ b {u∗ } (69) Formulating Eq. (69) on the boundary (ξ = 1), the displacements at the boundary control points {ub } = {u (ξ = 1)} can be expressed as {ub } = [81 ] {c} + {u∗ } (70) To evaluate the integration constants {c}, Eq. (70) can be rewritten as {c} = [81 ]−1 ({ub } − {u∗ }) The equivalent boundary nodal forces required to cause these displacements are {P} = [Q1 ] {c} + q∗ (ξ = 1) (71) (72) Substituting Eq. (71) into Eq. (72) and using Eqs. (68) and (56) yields [K] {ub } = {P} + [K] {u∗ } − q∗ (ξ = 1) (73) [K] = [Q1 ] [81 ]−1 (74) where q∗ (ξ = 1) = b [E0 ] + [E1 ]T {u∗ } + Z [B1 (s)]T {σ0 } |J (s)| ds (75) e The boundary conductions place constraints on subsets of {ub } and {P}, and solution proceeds as in the standard finite element method. Once the solutions of boundary displacements {ub } are determined from Eq. (73), Eq. (71) can be utilized to obtain the integration constants {c}. Using Eq. (48), the displacement field in the domain is then recovered as {u (ξ , s)} = N̄ (s) [81 ] ξ −λ {c} + ξ b {u∗ } (76) Substituting Eq. (69) and its derivative into Eq. (49), the stress field is obtained as {σ (ξ , s)} = [D] ξ b−1 (b [B1 (s)] + [B2 (s)]) {u∗ } + [D] N X ci ξ −λi −1 (−λi [B1 (s)] + [B2 (s)]) {φi } + {σ0 } (77) i=1 Here the stress distribution in Eq. (77) represents the contribution of one term of temperature field specified in Eq. (37), and the resulting thermal stress filed is then obtained by the superposition of the solutions for the individual terms of the power series. 14 P. LI ET AL. Results and discussion Downloaded by [University of Florida] at 09:54 26 October 2017 In this section, some numerical examples are investigated to verify the performance of proposed method for solving two-dimensional thermoelasticity problems with complex geometry. For comparison, the computer codes of IGA-SBM and SBFEM in Mathematica are developed to evaluate the same problems through the same quadratic basis functions. To obtain more information, it is necessary to compare the resulting results obtained from IGA-SBM and SBFEM with the exact solutions or other finite element software results. The following relative L2 error norm is used for the convergence study: R 1/2 T (u − uh ) (u − uh ) d eL 2 = (78) R 1/2 T u ud where u and uh represent the exact and numerical solution, respectively. When the exact solutions are not available for the presented numerical examples, the solutions obtained using the finite element commercial software ANSYS with a fine refinement are used as the reference solutions. In the following analysis, the international standard unit system is utilized for the thermoelastic parameters and thus no units are indicated. Circular plate To investigate the influence of selection of the scaling center and domain partition on the proposed method, a circular plate with analytical solution in the polar coordinate is modeled as the first example, as shown in Figures 5 and 7. In this example, the prescribed heat flux is ∂T/∂n = Cosθ around the boundary, and the temperatures are T (r = 3, θ = 0) = 3 and T (r = 3, θ = π ) = −3 at the ends of the horizontal diameter. The coefficient of thermal conductivity is the unit. The known analytical solution for this problem is T (r, θ ) = r cos θ . As shown in Figure 5, different scaling centers C0 = (0, 0), C1 = (0.5, 0), C2 = (1, 0), C3 = (1.5, 0), C4 = (2, 0), C5 = (2.5, 0) are selected to model this example. Figure 6 compares the numerical results obtained using different scaling centers with analytical solutions. It can be observed that there is excellent agreement between the numerical results against different scaling centers and the analytical solutions. Domain partitions in various cases are shown in Figure 7. The computational results against different Figure 5. The computational model of circular plate with different scaling centers. Downloaded by [University of Florida] at 09:54 26 October 2017 JOURNAL OF THERMAL STRESSES Figure 6. Comparison of temperatures using different scaling centers. Figure 7. The computational model of circular plate with different domain partitions. Figure 8. Comparison of temperatures using different domain partitions. 15 Downloaded by [University of Florida] at 09:54 26 October 2017 16 P. LI ET AL. Figure 9. The problem definition of thick-walled hollow cylinder. Figure 10. Boundary discretization for thick-walled hollow cylinder using IGA-SBM: (a) coarse model and (b) refined model. Figure 11. Boundary discretization for thick-walled hollow cylinder using SBFEM: (a) coarse model and (b) refined model. JOURNAL OF THERMAL STRESSES 17 cases are shown in Figure 8. As can be seen, the results obtained from different domain partitions are in very good agreement with analytical solutions. Thick-walled hollow cylinder Downloaded by [University of Florida] at 09:54 26 October 2017 The second test for IGA-SBM on the thermal stress is a thick-walled hollow cylinder (a plane stress problem) subjected to a steady thermal load [59], as shown in Figure 9. For the annular region, the inner boundary has a radius of 0.25 and the outer boundary has a radius of 1. The specified temperatures 3 and 1 are applied to the inner and outer boundary, respectively. In addition, the inner boundary of the thick-walled hollow cylinder is subject to the displacement 0.25 in the radial direction while the outer boundary is fixed. In the computation, the material properties are set as follows: E = 1.0, ν = 0.0, α = 1.0, kx = ky = 1, and ρc = 1. The exact solutions of the temperatures T (r), displacements ur (r), Table 1. Comparison of computed results using SBFEM and IGA-SBM for the thick-walled hollow cylinder model. Temperature, displacement, and stress at r = 0.7 Method Number of elements Degrees of freedom Tr ur σr SBFEM 12 24 48 96 12 24 48 96 20 44 92 188 20 32 56 104 1.404 1.458 1.493 1.510 1.487 1.502 1.510 1.513 1.515 −1.717 −1.759 −1.775 −1.792 −1.780 −1.788 −1.794 −1.799 −1.801 −1.854 −1.896 −1.935 −1.966 −1.915 −1.961 −1.973 −1.975 −1.979 IGA-SBM Exact solutions Figure 12. Contour plots of IGA-SBM solutions in the radial direction for thick-walled hollow cylinder: (a) temperature, (b) displacement, and (c) stress. 18 P. LI ET AL. and stresses σr (r) for this problem in cylindrical coordinates are expressed as ln (r) ln (2) (79) r ln (r) 2 ln (2) (80) Tr (r) = 1 − ur (r) = − Downloaded by [University of Florida] at 09:54 26 October 2017 σr (r) = −1 + ln (r) − 1 2 ln (2) (81) Figure 13. Contour plot of exact solutions in the radial direction for thick-walled hollow cylinder: (a) temperature, (b) displacement, and (c) stress. Figure 14. Comparison of CPU time between IGA-SBM and SBFEM for thick-walled hollow cylinder. JOURNAL OF THERMAL STRESSES 19 Downloaded by [University of Florida] at 09:54 26 October 2017 To have a good visibility, the whole computational domain is divided into four subdomains, where each subdomain is visible from its scaling center. As shown in Figure 10a and 10b, an initial coarse isogeometric model is defined and then the h-refinement is performed to obtain the refined model by knot insertion algorithm. Figure 11 shows the quadratic computational models for conventional SBFEM. From Figures 10a and 11a, it is seen that the complex boundary can be exactly represented by IGA-SBM even in a coarse discretization. The results of radial temperature, displacement, and stress obtained using IGA-SBM and SBFEM at the same number of elements are listed in Table 1 and are compared to the exact solutions. It is seen that the computed values obtained using the proposed method agree with the exact solutions up to 3 digits with less number of DOFs when compared to the conventional SBFEM. It is also seen that less number of DOFs is required in IGA-SBM than in the conventional SBFEM at the same number of elements since the NURBS basis functions possess higher degree of continuity across the element boundaries. In addition, a comparison of contour plots inside the domain Figure 15. Convergence comparison between the IGA-SBM and conventional SBFEM for thick-walled hollow cylinder. Figure 16. A multiconnected square domain with nine complex voids. Downloaded by [University of Florida] at 09:54 26 October 2017 20 P. LI ET AL. Figure 17. Initial computational model of a square domain with nine complex voids: (a) IGA-SBM discretization and (b) SBFEM discretization. Figure 18. Contour plots of IGA-SBM solutions in the radial direction for the multiconnected domain: (a) temperature, (b) displacement, and (c) stress. JOURNAL OF THERMAL STRESSES 21 is made in Figures 12 and 13, which are computed by IGA-SBM in 56 control points and exact solutions, respectively. Figures 12 and 13 show that the resulting results of IGA-SBM are in excellent agreement with the exact results, where the analytical solutions are almost reproduced even using 56 DOFs. To evaluate the numerical efficiency of SBFEM and IGA-SBM, the runtime of CPU for both techniques at the same mesh refinement is compared in Figure 14. It is observed that the computational efficiency is improved significantly by introducing NURBS basis functions to the framework of scaled boundary method. A convergence study is performed in Figure 15 using h-refinement in both IGASBM and SBFEM. As shown in Figure 15, a better convergence performance is observed as the number of DOFs is increased in proposed method. Downloaded by [University of Florida] at 09:54 26 October 2017 Multiconnected domain In the third numerical example, the thermoelastic analysis by IGA-SBM and SBFEM is tested on a square domain with nine complex voids under the plane strain condition, as shown in Figure 16. The top side of the analyzed domain is being at the prescribed temperature of T̄ = 100 and the bottom side undergoes the heat flux of 2, while the left and right sides of the square are in a convection environment with β = 100 and Ta = 20. The boundaries of the voids are insulated. Additionally, the left, bottom, and right boundaries of the problem domain are restrained in the normal direction, while the top boundary is free to displacement. The material properties used in this case are k = 200, E = 210GPa, ρc = 1, α = 0.02, ν = 0.3. To satisfy the visible condition from the scaling center, the whole problem domain is divided into 16 subdomains and the corresponding scaling centers are selected for each of the subdomains. Figure 19. Contour plots of FEM solutions in the radial direction for the multiconnected domain: (a) temperature, (b) displacement, and (c) stress. Downloaded by [University of Florida] at 09:54 26 October 2017 22 P. LI ET AL. The initial boundary discretization using IGA-SBM and SBFEM for the problem is depicted in Figure 17a and 17b, respectively. For the purpose of error analysis and convergence study, a corresponding finite element analysis is performed for the reference solution obtained from ANSYS software with sufficient large number of nodes. Figure 18 represents contour plot of temperature, displacement, and stress obtained using the IGA-SBM discretization shown in Figure 17a with 272 control points, while Figure 19 depicts the finite element solutions with 4429 nodes. From Figures 18 and 19, it is seen that the contours are in an excellent agreement with each other. To test the numerical accuracy and efficiency in calculating this problem, the relative L2 error norm in stress is computed for each mesh using h-refinement in both IGA-SBM and SBFEM. Figure 20 shows the efficiency analysis results of IGA-SBM and SBFEM. It is observed that less computing time is required in IGA-SBM under the condition of same error, and the error of proposed method is less than that of SBFEM when the same computational time is used. Hence, the proposed method has the advantage in both numerical efficiency and accuracy. A convergence performance comparison is made between IGASBM and SBFEM in Figure 21. As can be seen, a higher rate of convergence is obtained using IGA-SBM as the number of DOFs increases. Figure 20. Comparison of computational efficiency for the multiconnected domain against the L2 error. Figure 21. Comparison of convergence performance for the multiconnected domain at a range of meshes. JOURNAL OF THERMAL STRESSES 23 Downloaded by [University of Florida] at 09:54 26 October 2017 Conclusion Taking into account the combined formulations, this work expounds the application of the isogeometric scaled boundary method (IGA-SBM) for the two-dimensional isotropic thermoelastic problems. The proposed method incorporates the IGA within the framework of the conventional scaled boundary finite element method (SBFEM). This is achieved using the NURBS basis functions to represent the boundary and field variables in the circumferential direction instead of standard Lagrange basis functions. Such a combination provides a powerful numerical method. First, the boundary geometry of the problem domain can be exactly described with IGA-SBM rather than the approximation in finite element sense. Moreover, by demonstrating numerical examples, IGA-SBM exhibits the improved numerical accuracy, efficiency, and convergence performance for thermoelastic analysis in comparison to the conventional SBFEM. Third, the use of NURBS in scaled boundary method enhances the ability for describing the complex geometry since no tensor product structure is required. Thus, each numerical technique in IGA-SBM is used to handle the particularities of the problem that better fit its positive features. The present method can be further extended for the analysis of unbounded domain, which is being performed. Funding This research was supported by the National Natural Science Foundation of China (grant Nos. 51779033, 51409038), the National Major Scientific Research Program in 13th 5-Year Plan (grant No. NMSRP 2016 YFB0201001) and the National Natural Science Foundation of China (grant No. 51421064). These supports are gratefully acknowledged. References 1. M. A. Kouchakzadeh and A. Entezari, “Analytical solution of classic coupled thermoelasticity problem in a rotating disk,” J. Therm. Stresses, vol. 38, pp. 1267–1289, 2015. 2. A. Entezari and M. A. Kouchakzadeh, “Analytical solution of generalized coupled thermoelasticity problem in a rotating disk subjected to thermal and mechanical shock loads,” J. Therm. Stresses, vol. 39, pp. 1588–1609, 2016. 3. E. K. Kakhki, S. M. Hosseini, and M. Tahani, “An analytical solution for thermoelastic damping in a micro-beam based on generalized theory of thermoelasticity and modified couple stress theory,” Appl. Math. 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