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Research article Special Issues

Sensitivity analysis and design optimization of 3T rotating thermoelastic structures using IGBEM

  • In this study, the isogeometric boundary element method (IGBEM) based on non-uniform rational basis spline (NURBS) is used to perform shape design sensitivity and optimization of rotating three-temperature (3T) thermoelastic structures. During the optimization process, the shape design sensitivity within the IGBEM formulation was derived to include precise geometries and greater continuities. It was found through the application of the IGBEM that the shape design velocity has a significant effect on accuracy of the obtained shape design sensitivity. As a result, the developed shape design sensitivity analysis (SDSA) technique based on the considered IGBEM formulation outperforms the computational solution based on the traditional SDSA method. The isogeometric shape sensitivity and optimal design for a complicated three-temperature thermoelastic problem in rotating structures are investigated. The impact of rotation on the thermal stress sensitivity, optimal three-temperature, optimal displacement and optimal three temperature thermal stress distributions are established. It is shown that the SDSA derived using IGBEM is efficient and applicable for most three-temperature thermoelastic optimization problems.

    Citation: Mohamed Abdelsabour Fahmy, Mohammed O. Alsulami, Ahmed E. Abouelregal. Sensitivity analysis and design optimization of 3T rotating thermoelastic structures using IGBEM[J]. AIMS Mathematics, 2022, 7(11): 19902-19921. doi: 10.3934/math.20221090

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  • In this study, the isogeometric boundary element method (IGBEM) based on non-uniform rational basis spline (NURBS) is used to perform shape design sensitivity and optimization of rotating three-temperature (3T) thermoelastic structures. During the optimization process, the shape design sensitivity within the IGBEM formulation was derived to include precise geometries and greater continuities. It was found through the application of the IGBEM that the shape design velocity has a significant effect on accuracy of the obtained shape design sensitivity. As a result, the developed shape design sensitivity analysis (SDSA) technique based on the considered IGBEM formulation outperforms the computational solution based on the traditional SDSA method. The isogeometric shape sensitivity and optimal design for a complicated three-temperature thermoelastic problem in rotating structures are investigated. The impact of rotation on the thermal stress sensitivity, optimal three-temperature, optimal displacement and optimal three temperature thermal stress distributions are established. It is shown that the SDSA derived using IGBEM is efficient and applicable for most three-temperature thermoelastic optimization problems.



    Various aspects of engineering are coupled or interconnected with each other, including structural response, changes of temperature, fields of electromagnetic, and interactions of fluid-structure. In thermoelasticity, the heat conductivity of an elastic body is not affected by its deformation because the statement that the distortion of an elastic body does not change the heat conductivity is true. Extensive thermoplastic design efforts are common in plant design and nuclear industries, where thermomechanical coupling by chemical and atomic responses has a significant impact on practical devise analyses. Another demand for coupled thermoelasticity is practically evaluated resources [1], of which nonhomogeneous thermal and mechanical properties are depicted by the arrange role, i.e., design parameters, to oppose thermal stacking and maintain structural intensity.

    The combined shape plan and execution evaluation of mechanical components has been a focal point in CAD and CAE businesses to get the best and optimized designing arrangement, and the recently created isogeometric analysis (IGA) system by Hughes et al. [2] encourages consistent joining between building investigation and geometric representation by utilizing the same non-uniform rational basis spline (NURBS) premise capacities to parameterize the arrangement space. In areas where modern geometry representations are required, such as shell investigation [3,4], fluid-structure interaction [5,6], and shape plan optimization [6,7], isogeometric approaches have been used. The isogeometric method was used to investigate thermoelastic behavior, the thermomechanical gun dealings problem [8], and fabric dispersion of practically evaluated structures [9,10].

    The boundary element strategy was created to decipher the supervising partial differential conditions as the boundary indispensably conditions of the relocation and footing areas over the boundary [11,12]. Several complicated thermoelastic problems have been established in the literature to solve such problems, numerical techniques, such as the boundary element method (BEM), have been investigated in the context of micropolar-thermoelasticity [13], carbon nanotube fiber reinforced composites [14], micropolar piezothermoelasticity [15], Micropolar Magne-to-thermoviscoelasticity [16], and Magneto-thermoviscoelasticity [17]. Fahmy also introduced new boundary element models for bioheat problems [18], Thermoelastic problems of Metal and Alloy Discs with Holes [19], Wave Propagation Problems of Anisotropic Fiber-Reinforced Plates [20], size-dependent thermopiezoelectric problems [21], three-temperature problems [22], and photothermal problems [23].

    The use of IGBEM in shape optimization [24] may be a natural extension of IGA-based shape optimization considerations of References [6,25], in which precise geometric data is used and shape plan factors are streamlined. Furthermore, IGBEM-based optimization has greater ideal shape plan flexibility than IGA-based optimization. [26]. Since the mid-1980s, affectability studies and plan optimization of thermally coupled frameworks have been carried out [27]. Dems and Mroz [28] performed variational thermoelasticity sensitivity analysis using sizing and shape variables. Tortorelli and Subramani [29] used the adjoint approach to analyze sensitivity for a coupled constitutive model. Optimization of shape and topology for problems of thermoelastic was introduced by Hou, Sheen [30] and Li, Steven [31]. The integral equation of boundary was used by Lee and Kwak [32] to examine optimization of shape design. The plan optimization of fabric dispersion for thermally stacked FGMs has been broadly considered owing to the design element of FGMs. Fang et al. [33] studied isogeometric boundary element analysis for two-dimensional thermoelasticity with variable temperature. Lieu and Lee [34] used the isogeometric approach to optimize practically reviewed structures while accounting for thermoelasticity. An et al. [35] investigated the implementation of isogeometric boundary element method for 2-D steady heat transfer analysis.

    In this paper, a new isogeometric boundary element method (IGBEM) based on non-uniform rational basis spline (NURBS) is developed to perform shape design sensitivity and optimization of rotating three-temperature (3T) thermoelastic structures. The isogeometric shape sensitivity and optimal design for a complicated three-temperature thermoelastic problem in rotating structures are investigated. The shape design sensitivity analysis (SDSA) derived using IGBEM is shown to be efficient and applicable to most three-temperature thermoelastic optimization problems. Using the IGBEM in the design of thermos-mechanical structures allow for accurate thermal boundary representation as well as simple design parameterization for optimization. We primarily focus on developing the continuum-based shape affectability condition of coupled thermoelastic conditions and demonstrating the optimization results of viable appropriateness utilizing the inferred equation, utilizing these benefits within the optimization strategy. Derivation and verification of thermoelastic structure sensitivity using a continuum-based coupled shape design within the IGBEM was described, the derived sensitivity formula was used to solve the coupled thermos-mechanical optimization problem. The IGBEM approach based on NURBS was discussed. The affectability of an isogeometric shape plan is then determined using a boundary fundamental condition for considered thermoelastic problem. Rotation influences on the thermal stress σ11 sensitivity, optimal 3T  distribution  θ, optimal displacement u, and optimal 3T thermal stress σ11 are presented graphically. Numerical results demonstrate the validity, accuracy, and efficiency of the proposed technique.

    The response field in the IGBEM is approximated using the unchanged premise functions that are utilized to describe the geometry within CAD. Due to the employ of NURBS premise functions, which are based on B-splines, the IGBEM has various focal points over the usual BEM, including geometric precision and ease of refinement. Let a one-dimensional tie vector Ξ which contains a collection of ties ξi can be written as

    Ξ={ξ1,ξ2,,ξn+p+1}, (1)

    where n and p are points of control number, and the function of basis order, respectively.

    A basis function of NURBS Rpi(ξ) is defined as Rpi(ξ)=Npi  (ξ)winj=1Npj(ξ)wj.

    The definition of N0i(ξ) functions is, recursively

    N0i(ξ)={1if  ξiξ<ξi+10otherwise,(p=0), (2)

    and

    Npi  (ξ)=ξξiξi+pξiNp1i(ξ)+ξi+p+1ξξi+p+1ξi+1Np1i+1(ξ),(p=1,2,3,). (3)

    In general, the IGBEM, which employs higher-order premise functions, yields higher levels of normality than the standard BEM. A NURBS bend is characterized by the summation of n sets of the p -th-order NURBS premise function Rpl(ξ) multiplied by the matching control point Bi [36,37,38]. Over each specific knot, the developed NURBS basis functions have relative invariance and (p −1) incessant differentiability, where p is the arrangement of the fundamental polynomial. If the knots are repeated k -times, the continuity of NURBS basis functions decreases by k.

    The three-temperature radiation heat transfer equations can be expressed as

    Cveθe(r,τ)τ1ρ[Keθe(r,τ)]=Wei(θeθi)Wer(θeθr), (4)
    Cviθi(r,τ)τ1ρ[Kiθi(r,τ)]=Wei(θeθi), (5)
    Cvrθr(r,τ)τ1ρ[Krθr(r,τ)]=Wer(θeθr). (6)

    In which

    Cvα={ceα=eciα=icrθ3rα=rand  Kα={Aeθ5/2eα=eAiθ5/2iα=iArθ3+Brα=r,

    where ρ is the material density, θ=θe+θi+θr, is the total temperature, Wei=ρAeiθ2/3e and Wer=ρAerθ1/2e are energy exchanging coefficients, and cα(α=e,i,r) are constants.

    According to Fahmy et al. [17], the three-temperature heat conduction Eqs (4)-(6) may be written as

    [Kαθα(r,τ)]+W(r,τ)=cαρδ1θα(r,τ)τ, (7)

    where

    W(r,τ)={ρWei(θeθi)ρWer(θeθr),α=e,δ1=1ρWei(θeθi),α=i,δ1=1ρWer(θeθr),α=r,δ1=T3r. (8)

    The unit mass total energy is given by

    P=Pe+Pi+Pr,Pe=ceθe,Pi=ciθi,Pr=14crθ4r. (9)

    The considered conditions can be composed as

    θα(x,y,0)=θ0α(x,y)=g1(x,τ). (10)
    Kαθαn|Γ1=0,α=e,i,θr|Γ1=g2(x,τ). (11)
    Kαθαn|Γ2=0,α=e,i,r. (12)

    Using the fundamental solution which satisfies the following equation

    D2θα+θαn=δ(rpi)δ(τr),D=Kαρc, (13)

    where pi are singular points.

    The transient heat conduction can be written as

    Cθα=DKατOS[θαqθαq]dSdτ+DKατORbθαdRdτ+Rθiαθα|τ=0dR, (14)

    which can be expressed as follows

    Cθα=S[θαqθαq]dSRKαDθατθαdR. (15)

    Let us assume that the time temperature derivative may be approximated as

    θατNj=1fj(r)jaj(τ), (16)

    where fj(r) are known functions and aj(τ) are unknown coefficients.

    Suppose that ˆθjα is a solution of

    2ˆθjα=fj. (17)

    Thus, Eq (15) can be written as

    Cθ=S[θαqθαq]dS+Nj=1aj(τ)D1(CˆθjαS[θjαqˆqjθα]dS), (18)

    where

    ˆqj=Kαˆθjαn (19)

    and

    aj(τ)=Ni=1f1jiθ(ri,τ)τ (20)

    where f1ji are defined as [33].

    {F}ji=fj(ri). (21)

    From Eqs (18) and (20), we obtain

    C˙θα+Hθα=GQ, (22)

    where

    C=[HˆθαGˆQ]F1D1, (23)

    with

    {ˆθ}ij=ˆθj(xi), (24)
    {ˆQ}ij=ˆqj(xi). (25)

    Now, we interpolate the functions θα and q as

    θα=(1T)θmα+Tθm+1α, (26)
    q=(1T)qm+Tqm+1, (27)

    where 0T=ττmτm+1τm1.

    Differentiation of (26) yields

    ˙θα=dθαdTdTdτ=θm+1αθmατm+1τm=θm+1αθmαΔτm. (28)

    Substitution of Eqs (31)-(33) into Eq (27), yields

    (CΔτm+TH)θm+1αTGQm+1=(CΔτm(1T)H)θmα+(1T)GQm, (29)

    which can be written as

    aX=b. (30)

    Because the successive over-relaxation (SOR) method requires less memory than the Jacobi and Gauss-Seidel iterative methods [38], it was efficiently implemented to solve the resulting linear algebraic systems.

    Figure 1 illustrates the case of considered model in an open domain Ω that is bounded by a closed surface Γ, The boundaries existing of the mechanical and thermal are independently on Ω. Temperature Γ0θ, flux Γ1θ, and convection Γ2θ. boundaries are thermal boundaries. Displacement ΓD and traction ΓN. boundaries are mechanical boundaries. In addition, the boundaries are independent. The properties of considered model material in domain Ω are supposed to be isotropic and elastic, as well as temperature independent.

    Figure 1.  Boundary problem of thermoelasticity.

    The internal heat production rate Q and the next thermal boundary conditions apply to the body: A endorsed temperature θ0 on Γ0θ, a endorsed heat flux q on Γ1θ, and an surrounding temperature θ on the convection boundary Γ2θ. In addition, a body power strength b as well as the next mechanical boundary conditions apply to the body: A endorsed displacement u on ΓD and a endorsed traction t on ΓN.

    The equilibrium equation is

    σij,j+bi=0  in  Ω. (31)

    The boundary conditions are

    ui=ui  on  ΓD, (32)

    and

    ti=σijnj=ti  on  ΓN. (33)

    Suppose that the stress-strain relation in thermoelasticity is weakly coupled and is stated as

    σij=2μ[εij+112vεkkδij]E12ναθδij. (34)

    With θ (θ=θe+θi+θr) is the total temperature calculated from section 3, μ, εij, δij, ν, E and α are shear modulus, strain tensor, Kronecker delta, Poisson's ratio, Young's modulus, and thermal expansion coefficient, respectively.

    The solution u* should satisfy the following:

    σij,j(u)=δ(xˆx)ei, (35)
    ui=Uijej, (36)
    σij(u)nj=Tijei, (37)

    here ei represents a unit vector.

    Uij(x,ˆx)=18πμ(1ν)[(34v)ln1rδij+r,ir,j], (38)

    and

    Tij(x,ˆx)=14π(1ν)r[rn[(12ν)δij+2r,irj]+(12v)(nir,jnjr,i)], (39)

    with r=∥ˆxx is the distance function. The mechanical problem boundary integral equations can be written accordingly using the fundamental solution:

    uj(ˆx)+ΓTij(x,ˆx)uidΓ=ΓtiUij(x,ˆx)dΓ+Eα12vΩUij,i(x,ˆx)θdΩ+ΩbiUijdΩ. (40)

    Using a Galerkin vector, Eq (40) can be rewritten as

    uj(ˆx)+ΓTij(x,ˆx)uidΓΓPj(x,ˆx)θ(x)dΓ=ΓtiUij(x,ˆx)dΓ+ΓqnQj(x,ˆx)dΓ+ΩbiUijdΩ. (41)

    A two-dimensional problem has the following fundamental solutions:

    Pi(x,ˆx)=α(1+ν)4π(1v){[ln(1r12)nir,ir,knk]} (42)

    and

    Qi(x,ˆx)=(1+v)4πk(1ν)αrr,i[ln(1r)12]. (43)

    Integral characters [39] for fundamental solutions of the singular integral at were utilized, as well as the transformation method. In Eq (41), we can write the regularized boundary integral equations as follows:

    ΓTij(x,ˆx){ui(x)ui(ˆx)}dΓΓPj(x,ˆx)θ(x)dΓ=ΓtiUij(x,ˆx)dΓ+ΓqnQj(x,ˆx)dΓ+ΩbiUijdΩ. (44)

    Isoparametric mapping, based on pth order NURBS basis functions, is a technique for expressing geometric point and response as follows:

    qn(Ξ)=CPTI=1RpI(Ξ)wI, (45)
    u(ξ)=CPI=1RpI(ξ)yI (46)

    and

    t(ξ)=CPTI=1RpI(ξ)zI, (47)

    with vI is a coefficient of temperature, wI is a coefficient of normal flux, yI is a coefficient of displacement, zI is a coefficient of traction, CP is control points number, and CPT is normal fluxes number. Finally, for simplicity, Eqs (27) and (28) are discretized in a form of matrix using NURBS based basis functions.

    HNIyIHθI¯HθIvI=GDIzI+GθIwI. (48)

    Therefore, the algebraic equation can be solved by combining Eqs (34) and (35):

    HIuI=GItI, (49)

    where

    HI=[HNIHθI0HθI], (50)
    GI=[GDI¯GθI0GI], (51)
    uI=[yIvI], (52)
    tI=[zIwI], (53)

    where

    ˆξ'i=ξi+1+ξi+2+ξi+pp,i=1,2,,n1. (54)

    Recall that we can simplify the thermoelasticity problem by not including the heat generator and the body intensity force, thereby revealing the fundamental solution in Eqs (42) and (43).

    ΓTij(x,ˆx){ui(x)ui(ˆx)}dΓΓPj(x,ˆx)θ(x)dΓ=ΓtiUij(x,ˆx)dΓ+ΓqnQj(x,ˆx)dΓ, (55)

    where

    ˙u=u'+uTV, (56)
    ˙t=t'+tTV (57)

    and

    ˙r=r'+rTV. (58)

    Now, we can write Eq (55) as

    [ΓTij(x,ˆx){ui(x)ui(ˆx)}dΓ]'[ΓPi(x,ˆx)θ(x)dΓ]'=[ΓUij(x,ˆx)ti(x)dΓ]'+[ΓQi(x,ˆx)qn(x)dΓ]' (59)

    where

    [ΓTij(x,ˆx){ui(x)ui(ˆx)}dΓ]'=ΓTij(x,ˆx){ui(x)ui(ˆx)}Vk,sskdΓ+Γ[˙Tij(x,ˆx){ui(x)ui(ˆx)}]dΓ+ΓTij(x,ˆx)˙ui(x)dΓΓTij(x,ˆx)˙ui(ˆx)dΓ, (60)
    [ΓPj(x,ˆx)θ(x)dΓ]'=Γ{˙Pj(x,ˆx)θ(x)+Pj(x,ˆx)˙θ(x)}dΓ+ΓPj(x,ˆx)θ(x)Vk,sskdΓ, (61)
    [ΓtiUij(x,ˆx)dΓ]'=Γ{tiUij(x,ˆx)Vk,ssk+˙tiUij(x,ˆx)+ti˙Uij(x,ˆx)}dΓ, (62)
    [ΓQj(x,ˆx)qn(x)dΓ]'=Γ{˙qnQj(x,ˆx)+qn˙Qj(x,ˆx)+qnQj(x,ˆx)Vk,ssk}dΓ. (63)

    Appendix and References [40,41] provide more information on the material derivative formulae. Using Eqs (55)-(58), we can rewrite (54) as follows:

    ΓTij(x,ˆx){˙ui(x)˙ui(ˆx)}dΓΓ˙tiUij(x,ˆx)dΓΓPi(x,ˆx)˙θ(x)dΓΓ˙qnQi(x,ˆx)dΓ=Γ˙Tij(x,ˆx){ui(x)ui(ˆx)}dΓΓTij(x,ˆx){ui(x)ui(ˆx)}Vk,sskdΓ+Γ˙Pi(x,ˆx)θ(x)dΓ+ΓPi(x,ˆx)θ(x)Vk,sskdΓ+Γti˙Uij(x,ˆx)dΓ+ΓtiUij(x,ˆx)Vk,sskdΓ+Γqn˙Qi(x,ˆx)dΓ+ΓqnQi(x,ˆx)Vk,sskdΓ. (64)

    By using isogeometric discretization, we can write Eqs (59) and (60) in a matrix form as

    HI˙uIGI˙tI=HIuIHVIuI+˙GItI+GVItI. (65)

    The h-refinement technique is used for adding control points by using the following formula:

    Bi=(xi,yi,zi;wi),i=1,,n (66)

    and its proposed control points are stated as

    Bwi=(wixi,wiyi,wizi,wi). (67)

    If a new knot ˜ξ[ξk,ξk+1) is inserted as

    ˜Ξ={ξ1,ξ2,,ξk,˜ξ,ξk+1,,ξn+p,ξn+p+1}, (68)

    the position of fresh control points is upgraded by the next equation:

     Bwi=αiBwi+(1αi)Bwi1, (69)

    where

    αi={1ikp ξξiξi+pξikp+1ik0k+1i. (70)

    The parallelepiped shape radiator for a loop heat pipe (LHP) of a titanium Ti6Al4V as shown in Figure 2 is optimized for space nuclear power system like the one considered in Hartenstine et al. [25], L=305mm and h=26.2mm, where two pipes enter the radiator symmetrically and exit after three folds from the same side. Also, the order of basis functions considered in the calculation is p=3.

    Figure 2.  Geometry of the considered radiator model.

    We selected a quarter of the cross-section A for the radiator considered model. Thermal boundary conditions made of incoming heat Q on the upper side and fixed temperature θ inside the pipes.

    By considering the relationship between the solution of one-temperature heat conduction model (θ) and the solution of three-temperature heat conduction model (θα) [14], the 3T distribution θ becomes

    θ=θαSummationθαAverage

    where

    θαSummation=θe+θi+θr  and  θαAverage=θe+θi+θr3.

    Now, we can write the considered optimization problem as follows:

    (I) According to the objective function θ based on θα subjected to (10)-(12), we obtain the optimal 3T distribution θ as in Figures 3 and 4 below.

    (II) According to the objective function u* subjected to (35)-(37), we obtain the optimal 3T displacement u as in Figures 5 and 6 below.

    (III) According to (I) and (II), we can calculate the optimal 3T thermal stress σ11 as in Figures 7 and 8 below.

    The design and nondesign domains must be defined first in the optimization process. In almost all applications, the heat pipes are installed in a sandwich panel with a honeycomb structure at the core. Using the proposed coupled topology optimization algorithm, the honeycomb structure is filled with a single isotropic material for the radiator redesign (design domain). However, the dimensions of the cross section, radiator, and heat pipes remain unchanged. A titanium alloy Ti6Al4V with the thermoelastic properties listed in Table 1 was considered for the redesign and production of the component. It is possible to see that there are eight holes in each section. They are symmetric about both axes of the rectangular section. The radius R=9mm of the pipe is constant all over the radiator. Each fold has radius r=12, resulting in a distance between the pipes within the domain of d=24mm.

    Table 1.  Considered material Ti6Al4V properties.
    Material E ν ρ k α
    Ti6Al4V 113.8 GPa 0.342 4430 kg/m3 6.7 W/mK 9e-6 1/K

     | Show Table
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    Figures 9 and 10 show the effect of rotation on the 3T thermal stress σ11 sensitivity distribution along x -axis

    Figure 9.  Variation of the 3T thermal stress σ11 sensitivity along x -axis in the non-rotating case.
    Figure 10.  Variation of the 3T thermal stress σ11 sensitivity along x -axis in the rotating case.

    Figures 3 and 4 show the effect of rotation on the optimal 3T  distribution  θ along x -axis.

    Figure 3.  Optimal 3Tdistributionθ along x -axis in the non-rotating case.
    Figure 4.  Optimal 3T distribution θ along x -axis in the rotating case.

    Figures 5 and 6 show the effect of rotation on the optimal displacement u distribution along x -axis.

    Figure 5.  Optimal displacement u distribution along x -axis in the non-rotating case.
    Figure 6.  Optimal displacement u distribution along x -axis in the rotating case.

    Figures 7 and 8 show the effect of rotation on the optimal thermal stress σ11 distribution along x -axis.

    Figure 7.  Optimal 3T thermal stress σ11 distribution along x -axis in the non-rotating case.
    Figure 8.  Optimal 3T thermal stress σ11 distribution along x -axis in the rotating case.

    Figures 11 shows the variations of the special case 3T thermal stress sensitivity along x -axis for BEM [42,43,44,45], lattice Boltzmann method (LBM) of Yin and Zhang [46] and finite element method (FEM) of Soliman and Fahmy [47]. It is clear from this figure that the BEM results of the proposed technique are in excellent agreement with the LBM and FEM, thus confirming the validity and accuracy of our proposed technique.

    Figure 11.  Variation of the 3T thermal stress σ11 sensitivity along x -axis in the rotating case.

    Table 2 shows a comparison of the computer resources needed to perform special case of sensitivity analysis and optimization of rotating three-temperature thermoelastic structures using BEM [42,43,44,45], lattice Boltzmann method (LBM) of Yin and Zhang [46] and finite element method (FEM) of Soliman and Fahmy [47]. It can be seen from this table that the proposed BEM is more accurate and efficient than the LBM and FEM.

    Table 2.  Comparison of computer resources required for BEM, LBM and FEM.
    FEM [47] LBM [46] BEM [42,43,44,45]
    CPU time (min) 24 20 2
    Memory (MB) 22 18 1
    Disc space (MB) 34 28 0
    Accuracy of results (%) 2.1 1.8 1.1

     | Show Table
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    In this study, the isogeometric boundary element method (IGBEM) based on non-uniform rational basis spline (NURBS) is used to perform sensitivity analysis and optimization of rotating three-temperature thermoelastic structures. To include precise geometries and greater continuities, we derive a shape design sensitivity equation within the isogeometric boundary element method formulation. In the considered boundary element technique, the shape design velocity field is divided into normal and tangential components, which has a significant effect on the accuracy of shape design sensitivity. As a result, the developed isogeometric SDSA technique based on the considered boundary element formulation outperforms the traditional DSA method's computational solution. In rotating structures, the isogeometric shape sensitivity and the optimal design for a complex 3T thermoelastic problem are established. The impact of rotation on thermal stress sensitivity, optimal three-temperature, optimal displacement, and optimal thermal stress distributions is investigated. The SDSA derived from IGBEM is shown to be efficient and applicable for most three-temperature thermoelastic optimization problems.

    The accuracy of the proposed method has been confirmed by comparing the obtained results with the lattice Boltzmann method (LBM) results and finite element method (FEM) results. The performance of the proposed method has been confirmed.

    This research was funded by [Deanship of Scientific Research at Umm Al‐Qura University] grant number [22UQU4340548DSR09] And the APC was funded by [Deanship of Scientific Research at Umm Al‐Qura University].

    The authors would like to thank the Deanship of Scientific Research at Umm Al‐Qura University for supporting this work by Grant Code: (22UQU4340548DSR09).

    The authors declare no conflicts of interest.

    The fundamental solutions material and derivatives are derived, as follows:

    ˙η(x,ˆx)=1γ12πr˙r=1γ12πrr,k(Vk(x)Vk(ˆx)), (A1)
    ˙wn(x,ˆx)=˙r2πr2rn12πr(rn)=12πr2nr,k(Vk(x)Vk(ˆx))
    12πr1r(δikr,ir,k){Vk(x)Vk(ˆx)}ni12πrr,i(Vk,snk)si, (A2)
    ˙Uij=18πμ(1v)r{δjkri+δikr,j(34v)δijr,k2rir,jr,k}{Vk(x)Vk(ˆx)}, (A3)
    ˙Tij=(12v)4π(1v)r2{(2rinjr,kδiknjδjkniδijr,kr,mnm)}{Vk(x)Vk(ˆx)}14π(1v)r2{r(12v)(δijnk+2r,irjnk)+2rn(δikrj3r,ir,krj+δjkri)}{Vk(x)Vk(ˆx)}
    +(12v)4π(1v)r(r,iVl,snlsj+r,jVl,snlsi)14π(1v)r{rkVl,snlsk((12v)δij+2r,ir,j)}, (A4)
    ˙Pi=α(1+v)4π(1v)r,kr(Vk(x)Vk(ˆx))α(1+v)4π(1ν)ln(1r12)Vl,snlSi
    α(1+v)4π(1ν)1r(δikr,ir,k)(Vk(x)Vk(ˆx))r,lnl
    α(1+v)4π(1v)r,i1r(δlkr,lr,k)(Vk(x)Vk(ˆx))nl+α(1+v)4π(1v)r,ir,kVl,snlSk,˙Qi=(1+v)4πk(1v)αr,k(Vk(x)Vk(ˆx))r,i[ln(1r)12] (A5)
    (1+v)4πk(1ν)αr1r(δikr,ir,k)(Vk(x)Vk(ˆx))[ln(1r)12]
    +(1+ν)4πk(1ν)αrr,ir,kr(Vk(x)Vk(ˆx)). (A6)


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