Structural Optimization for Transient Response Constraints with ...

Tamkang Journal of Science and Engineering, Vol. 3, No. 3, pp. 173-186

(2000)

Structural Optimization for Transient Response Constraints with Software JIFEX* Yuanxian Gu, Biaosong Chen, Hongwu Zhang and Shutian Liu State Key Laboratory of Structural Analysis for Industrial Equipment, Dept. of Engineering Mechanics, Dalian University of Technology, Dalian 116024, China

Abstract The numerical methods of the structural design optimization with transient response constraints have been studied in the paper. The new methods of the response analysis and sensitivity analysis for the transient dynamics and the heat conduction constraints with the precise time integration have been proposed. Particularly, an efficient method of sensitivity analysis for nonlinear transient heat conduction is given. The design optimization and finite element analysis for general structures with size and shape variables and multi-type constraints are implemented in the application software JIFEX. Numerical examples have illustrated the effectiveness of the methods presented in the paper and the facility of JIFEX software. Key Words：structural optimization, sensitivity analysis, transient dynamics, heat transfer, time integration

1. Introduction The transient responses such as structural dynamics and unsteady heat transfer are important problems in structural designs and not well studied in the design optimization. These problems are not only difficult in time consuming numerical analysis, but also in the sensitivity analysis particularly. In the solution of linear and nonlinear time-dependent transient response problems, they lead to linear and nonlinear Ordinary Differential Equations (ODEs) in time domain. Among the existing methods of solving ODEs, direct time-stepping methods have been the most popular and widely advocated in most software. In particular, the time difference/ θ *-difference method[12,17,19] plays a dominant role. As for transient heat conduction analysis, whenθ, the algorithm parameter, is selected as 0, 0.5, 2/3, and 1.0, the forward, central (Crank-Nicolson), Galerkin, backward difference methods can be *

The project supported by the National Natural Science Foundation of China (19525206, 59895410) and the NKBRSF of China (No. G1999032805).

constructed respectively. With the direct iteration [12] or Newton-Raphson [17] iteration method, the above methods can be extended to nonlinear analysis. In addition to θ -difference method, many different approaches have been proposed such as hybrid transfinite element formulation[16], reduction methods[1], virtual-pulse time integral methodology[15], precise algorithm[20]. Recently, A precise time integration method was proposed to structural dynamics [21,22], and applied to other transient response problems [3,6,11,13]. The sensitivity analysis giving derivatives information of responses with respect to design variables is the basis of design optimization, and also valuable to stochastic analysis and inverse problem. The conventional methods of sensitivity analysis for transient responses have difficulties in computational efficiency and accuracy. Numerous publications [2,5,8,9,10,18] have presented the development history of research on heat transfer. Of these literatures, the direct method and the adjoint method of sensitivity analysis have been proposed based on continuum or discrete model. These authors concentrated on the sensitivity equations derivative procedure. Brief discussions on solving the equations were given in some of the

173

174

Yuanxian Gu et al.

above papers by using the time-difference integration method ( θ -difference integration method). In fact, how to solve the sensitivity equations is also important. It will make a great effect on the sensitivity precision and algorithm efficiency. In this paper, the Precise Time Integration (PTI) method [21,22] is extended to the solution of linear and nonlinear transient heat conduction problems. Particularly, new algorithms are proposed to solve the sensitivity equations of linear and nonlinear transient heat conduction, and structural transient dynamics problems with PTI method. The numerical advantages of PTI method such as unconditionally stable, high resolution, and adaptive to stiff problem have improved the sensitivity accuracy noticeably and avoided the numerical oscillations. The new methods of responses analysis and sensitivity analysis for transient constraints are implemented in the general purposed software JIFEX [4,7] for structural finite element analysis and design optimization. Numerical examples are given in the paper to illustrate the accuracy of new methods and application facilities of software JIFEX.

2. Precise Time Integration for Transient Responses

The PTI method was proposed by Zhong [21,22] for the solution of structural dynamic equation with high precision and numerical stability. Firstly, the PTI method is presented for the problem of structural transient dynamics. The motion equation of structural transient dynamics can be expressed in the finite element discrete form as below (1)

Initial condition x (0) = x 0 and x& (0) = x& 0 are given. The M, G, K are time-invariant mass, damping and stiffness matrices with n × n dimension respectively, and f (t ) is external load

vector. In general, f (t ) ≠ {0} and Eq.(l) is non-homogeneous. Let us define

p = Mx& + Gx / 2, q = x

v& = Hv + r

⎧ q⎫ v = ⎨ ⎬, ⎩ p⎭ ⎡− M −1G / 2, GM −1G/4 − K ⎤ H =⎢ ⎥, −1 M −1 ⎣ − GM / 2, ⎦ ⎧ 0⎫ r=⎨ ⎬ ⎩ f⎭

(3a)

(3b)

where H is a matrix with 2 n × 2 n dimension. The general solution {v} of Eq.(3) is

v (t ) = exp( Ht ) ⋅ v0 + ∫ exp(H (t − s )) ⋅ r (s )ds, t

(4)

0

where v 0 are the initial conditions. Then the solution of the system at time tk is

v k = exp( Ht k )v 0 + ∫ exp[H (t k − s )] ⋅ r (s )ds tk

(5)

0

2.1 Precise Time Integration Method and Structural Transient Dynamics

M&x& + Gx& + Kx = f (t )

then, Eq.(l) becomes the following forms

(2)

Denoting time step length τ = t k +1 − tk , the solution at time tk +1 is

v k +1 = exp(Ht k +1 )v 0 +∫

t k +1

0

exp[H (t k +1 − s )] ⋅ r (s )ds

(6)

By deducing the relationship of v k and v k +1 , the integration scheme of the PTI can be obtained

v k +1 = exp( Hτ )v k + ∫ exp[H (τ − s )] ⋅ r (t k + s )ds τ

(7)

0

The matrix exponential is defined as

A = exp( H ⋅τ )

(8)

If the non-homogenous vector is linear function in the time interval (tk,tk+1), such that

Structural Optimization for Transient Response Constrains with Software JIFEX

r (t ) = r0 + r1 (t − t k )

(9)

then, the solutionνk+1 can be expressed as below

v k +1 = A[v k + H −1 (r0 + H −1r1 )] − H [r0 + H r1 + r1τ ] −1

175

−1

R = R0 + R1 (t − t k ), R0 = R(t k ),

(10)

More approximation methods to deal with other kinds of non-homogenous vectors can be found in literatures [6,13]. The calculation of exponential matrix A is a key point in PTI method. There are some algorithms of exponential matrix calculation [14]. Ｎ The 2 algorithm is used in PTI method with superposition of exponential function and second order Taylor expansion,

A = exp( H ⋅τ ) = [exp( H ⋅τ / m )] 2

(11)

for (iter = 0; iter < N ; iter + + ) (12) Aa = 2 Aa + Aa × Aa ;

When the loop is over, (13)

2.2 Linear Transient Heat Conduction Problem After space discretization by numerical techniques such as finite element, finite difference, boundary elements etc, the linear transient heat conduction equations are as follow.

MT& + KT = R

(14)

where M, K, T and R are the capacity matrix, the conduction matrix, the temperature vector and heat load vector. The PTI method can be extended to the transient heat conduction problem by adopting the following transformation,

T& = HT + r , H = -M -1 K ; r = M -1 R

(16)

r0 = M −1 R0 , r1 = M −1 R1

(17)

The integration formulations of the PTI method are

Tk +1 = A[Tk + H −1 (r0 + H −1 r1 )] − H −1 [r0 + H −1 r1 + r1τ ] ,

A = exp( Hτ )

H −1 = (− M −1 K ) = − K −1 M −1

It is noted that ∆t is a very small, for example, if Ｎ take N= 20, m =2 = 1048576, then ∆t = τ m is an extremely small time interval. However, it is same for Aa compared to the identity matrix I, and thus algorithm is particularly designed to execute the addition of matrix Aa firstly, and add them to I finally.

A = I + Aa

R1 = (R(t k +1 ) − R(t k )) / τ

(18)

whereτ is the time step length. There are some simplifications for heat conduction, because of

m

≈ I + H ⋅ ∆t + ( H ⋅ ∆t ) / 2 = I + Aa

If matrix R varies linearly within the time interval as,

(15)

(19)

then, the PTI for linear transient heat conduction is

Tk+1 = A[Tk − K −1 (R0 − MK−1 R1 )] + K −1 [R0 − MK−1 R1 + R1τ ]

(20)

The product of K-1 and a vector can be carried out by LDLT decomposition and back substitution of the vector. The above codes are usually offered in the conventional finite element program. It is no need to program the additional code for inverse computation of matrix H. The constant, linear, and sinusoidal approximations all available to deal with the non-homogeneous vector R. 2.1. Nonlinear Transient Heat Conduction Problem In general, the heat transfer is nonlinear. The nonlinearities usually include the temperature-dependent material parameters and the radiation boundary condition. The transient nonlinear equations are

M (T )T& + K (T )T = R

(21)

where the symbolic meanings are the same as before, but they are not constant and depended on temperature. As for finite element methods, the relative matrixes are

176

Yuanxian Gu et al.

M (T ) = ∑ ∫Ω ρ (T ) ⋅ c (T )N T NdΩ

(22)

K (T ) = ∑ ∫Ω (∇ T N )k (T )(∇N )dΩ

(23)

e

The introduced matrix in the transformations can take the form as

e

e

M 0 = M (T0 ), K 0 = K (T0 )

e

where N denotes the shape function of elements, and Σ denotes the assembling of elements. It can be seen that the density ρ, specific heat c, heat conductivity k are all the functions of temperature. The right hand term R can be decomposed as two parts

R = R1 + Rr

(24)

The linear part R1 is independent of temperature. The radiation part Rr , is written as

Rr = −∑ ∫ e σεN T (T 4 − Ta4 )dS e

s

4 ⎡⎛ m ⎤ ⎞ = −∑ ∫ e σεN ⎢⎜ ∑ N iTi ⎟ − Ta4 ⎥ dS s e ⎠ ⎢⎣⎝ i =1 ⎥⎦

(25)

T

0

0

0

+ K 0 )T = R

− (K (T ) − K 0 )T

(26)

(27)

Employing the standard formula of PTI for the above equations

~ T& = HT + M 0−1 R

It means that M0, K0 are the initial heat capacity matrix, heat conduction matrix respectively. The objective of adopting the transformations is to make the matrix H constant, then the computational expense can be reduced when carrying out the precise computation of matrix exponential. These transformations don't introduce any artificial approximations and the transformed equations are identical to the original ones. In practice, because density and specific heat vary with time smoothly, they can be viewed as constant. Hence the efficiency can be improved. The analytical solution of Eq.(28) is obtained according to Eq.(7)

Tk +1 = ATk 0

Then,

M 0T& + K 0T = R − ( M (T ) − M 0 )T&

(30)

(31) τ ~ + ∫ exp( H (τ − s ))M −1 R(t k + s, T )ds

where σε is the product of Stefan-Boltzman constant and the emissivity of structural surface, Ta is the ambient temperature. The above formula can be evaluated by Guass integration so as to reduce the complexity. Some transformations are needed to solve the nonlinear equations by the PTI method. Eq.(21) is rewritten as

(M (q ) − M + M )T& + ( K (T ) − K

above

(28)

where

H = M 0−1 K 0 , (29) ~ R = R - ( M (T ) − M 0 )T& − ( K (T ) − K 0 )T

The approximations may be applied. In order to simplify the work, the following predictor-corrector algorithm is recommended. ~ Firstly the term R is divided as two parts,

~ ~ ~ R = RE + RI

(32)

where the RE is the explicit part and is only depended on time and caused by heat source , convection, heat flux, prescribed boundary temperature, and etc. RI is the implicit part related to both time and state variable caused by radiation, temperature-dependent material parameter. As for explicit part, Eq.(18) is employed, as for implicit part, the following algorithm called Predictor-Corrector is performed, (1) Predictor. Letting n the counter, when n=0, supposing ~ that R is constant within the time interval

~ ~ RI = RI (t k , Tk )

(33)

After substituting it into Eq.(31), the prediction temperature qk+1,0 at tk+1 is obtained.

Structural Optimization for Transient Response Constrains with Software JIFEX

(2) Corrector. Using the prediction temperature, letting n=n+1, linear approximation is adopted.

~ ~ RI = RI (t k , Tk ) ~ ~ + RI (t k +1 , Tk +1,n−1 ) − RI (t k , Tk ) s / τ ,

(

s ∈ [0,τ ]

)

(34)

After substituting it into Eq.(31) again, the new prediction temperature Tk+1,n is obtained. The above formula can be executed consecutively until the two predictions satisfy the norm condition

Tk +1,n − Tk +1,n −1 < δ

(35)

177

dv& dv =H + R0 + R1 (t − t k ) dα dα dr (t k ) dH (t k ) + R0 = vk dα dα dH (t k ) ⎡ v k +1 − v k ⎤ R1 = ⎢ ⎥ ; when t = tk dα ⎣ ∆t k ⎦

(37a)

(37b)

The general solution of Eq.(37) similar to that of Eq.(10) can be obtained as below

dv k +1 dv =A k dα dα + (T − I )H −1 R0 + H −1 R1 − H −1 R1τ

(

)

(38) where δ is the given error tolerance. Because the linear approximation is employed in the above formula, this predictor-corrector can achieve second order precision. The algorithm is easy to implemented in program. Of course, higher precision algorithm can be construct by using higher precision approximations and the algorithm is still stable because of high precise computation of matrix exponential which guarantees the stability.

3. Sensitivity Analysis for Transient Responses with PTI Method The sensitivity analysis methods for structural transient responses, i.e. the linear dynamics, the linear and the nonlinear heat conduction, have been proposed on the basis of precise time integration. The new methods are more accurate compared to conventional time difference methods such as Newmark and 6-difference methods. 3.1 Sensitivity Analysis of Structural Transient Dynamics Differentiating Eq.(3a) with respect to design variable α , we have the sensitivity equation of general vector ν as following

dv& dv dH dr =H + R, R = v+ dα dα dα dα

(36)

It is noted that Eq.(36) is same as Eq.(3a).Making the linear approximation to the right-hand vector in Eq.(36) same as that in Eq.(3a),then, the Eq.(36) is transformed as below

With the initial condition of original dynamics problem

dx0 dx& 0 = 0, =0 dα dα

(39)

the initial condition and first step of Eq.(38) is obtained as

dv 0 = 0, dα dv1 = ( A − I )H −1 R0 (t 0 ) + H −1 R1 (t 0 ) dα − H −1 R1 (t 0 )(t1 − t 0 )

(

)

(40)

Now, we have the precise time integration formulations Eq.s(38) and Eq.(40) to calculate the derivatives of vector v k +1 (k=0,1,......,n). It is clear from the definitions of Eq.(2) and (3) that the derivatives of dynamic displacements x just is the first half part of derivatives of general vector v .Same as the general procedures of sensitivity analysis for transient responses, the Eq.(38) is computed simultaneously with Eq.(10). With computing the matrix exponential T firstly, Eq.(10)and (38) are efficient explicit multiply calculation. This is a feature of the new sensitivity analysis algorithm with the PTI method. Since the matrix H is composed of M, K and G and these matrices are time independent, The derivatives of H can be computed once before the calculation of Eq.s(10) and (38). In the above presentations, the external load vector r is assumed to be the linear

178

Yuanxian Gu et al.

non-homogenous. For other kinds of non-homogenous external load vector of Eq.(l), some other approximation treatment methods have been proposed in literatures[6,13]. Similarly, the right-hand vector R in the sensitivity equation Eq.(36) is approximated linearly as Eq.(37). The other approximation procedures available for the new sensitivity analysis algorithm with the PTI method can be derived easily. According to the definitions in Eq.(3), the second half part of dv / dα is the derivative of generalized vector p, dp / dα . Therefore, the derivatives of velocity vector x& can be computed with Eq.(2) as below

⎛ dp dM dx& dx ⎞ ⎞ ⎛ dG x& − 0.5⎜ x +G = − M −1 ⎜⎜ − ⎟⎟ dα dα ⎠ ⎟⎠ ⎝ dα ⎝ dα dα (41)

The formulation calculating the derivatives of acceleration vector &x& can be obtaion form the motion question of structural dynamics Eq.(l) as below

d&x& dG ⎛ df dM = M − 1⎜ − x&& − x& dα dα ⎝ dα dα dx& dK dx ⎞ x−K −G − ⎟ dα dα dα ⎠

(42)

The sensitivity equation of linear heat conduction can be obtained differentiating Eq.(14) as

T'=

dT dM dK dR , M'= , K'= , R' = dα dα dα dα

(44)

It shows that if PTI is applied to solve the sensitivity equations, the exponential matrix needn't to compute again. Furthermore, it is assumed that the temperature and its rate in time vary linearly within the time interval.

f ( t + h ) = f ( t k ) + ( f ( t k +1 ) − f ( t k ))h / τ

[ ]

h ∈ 0 ,τ

f 0 + f1 h ,

(45)

f (t k ) = R' (t k ) − M' T&k − K' Tk

3.2 Sensitivity Analysis of Linear Transient Heat Conduction

where

T&' = HT' + M −1 f , H = − M −1 K , f = R' − M' T& − K' T

=

In summary, the new sensitivity analysis algorithm with the PTI method is first to calculate the matrices A and H-1 and the derivatives of M, K, G, f and H, r, then calculate the derivatives of generalized state vector v and displacements by Eq.(38), derivatives of velocity &x& by Eq.(41), and derivatives of acceleration x by Eq.(42). To general case, the derivatives of the load vector f(t) has been considered here although loads are normally variable independent.

MT& '+ KT ' = R'− M 'T& − K 'T

Eq.(43) is very similar to the original one Eq.(14) except for the right hand terms, so the same solution algorithm can be applied. The M ' , K ' , R' are derivatives of capacity matrix, conduction matrix, heat load vector with respect to design variable which are time- and temperatureindependent. Thus, they are computed only once with analytical or finite difference method. Before solving the above equations, the temperature and the temperature rate in time have been obtained. Then the sensitivity equations is transformed as

(43)

The integration formulation for sensitivity analysis is

[

(

T'k+1 = A Tk' − K −1 f0 − MK−1 f1

[

+ K −1 f0 − MK−1 f1 + f1τ

]

)] (46)

3.3 Sensitivity Analysis of Nonlinear Transient Heat Conduction By differentiating nonlinear heat conduction, the sensitivity equations are

~ M 0 T&' + K 0 T' = f , (47) ~ ~ f = R' − M' (T ) − M 0' T& − K' (T ) − K 0' T

(

) (

)

~

Since the right-hand term f depends on temperature T, the sensitivity equation is nonlinear

~

too. The problem is how to evaluate f . As the same before, it can be decomposed into two parts: the explicit part and the implicit part.

Structural Optimization for Transient Response Constrains with Software JIFEX

~ ~ ~ f = fE + fI

(48)

179

4. Optimization Modeling and Solution Algorithm Implemented in JIFEX

~

where f E is the explicit part which is independent of temperature caused by the sensitivity of heat source, convection, heat flux, prescribed boundary temperature with respect to design parameter. The explicit part can be

~

calculated by Eq.(18) firstly. f I is the implicit part which is depend on temperature, and deduce with finite difference method at time tk,

~ ~ ~ k RI (t k , α + ∆α , TK + Tk' ⋅ ∆α ) − RI (t k , Tk ) fI = ∆α (49) where ∆α is the perturbation of the design variable. The predictor-corrector algorithm is recommended. 1.

Predictor. Let n is the counter of iteration, when n=0,

~ ~ f I = f (t k , Tk )

(50)

~

After substituting f I , into Eq.(47), we get the first prediction of Tk' +1,0 by the PTI method. 2.

Corrector. Let n = n + 1 ,

(

)

~ ~ ~ ~ f = f (t k , Tk ) + f (t k +1 , Tk +1,n−1 ) − f (t k , Tk ) h τ ~ ~ = f 0 + f1h, h ∈ [0, τ ] (51)

~ Also substituting f , into Eq.(47), we get the corrective value of Tk' +1,n . The above corrector

formulas can be also executed by iteration. The convergence condition can be

Tk′+1,n − Tk′+1,n−1 < δ

(52)

It can be realized that the process of sensitivity analysis is very similar to the one of solving heat transfer equations. This is very benefit for implementation.

JIFEX is a new finite element software for structural analysis and design optimization on the MS Windows95/NT platforms. JIFEX possesses more powerful capabilities of finite element analysis design optimization, structural finite element modeling based on the CAD package AutoCAD, and graphics visualization. The design optimization of JIFEX offers the users many facilities to define their optimal model. In the optimal model, the objective function and the constraint functions can be the structural weight, the strength, the stiffness, the vibration behavior, the buckling stability. The design variables considered in the design optimization are size variables and shape variables. The size variables include the thickness of shell, plate and membrane elements, cross sectional area of the bar element, and cross sectional geometric parameters of the beam element. The variables of composite elements, i.e. layer thickness of laminate plate, ply orientation, core height of honeycomb sandwich plate, are also classified as size variables. Shape variables are the natural variables, i.e. geometric parameters of the spline and quadratic curves and the surface of the structural boundary, as well as node coordinates. Furthermore, JIFEX can conduct efficient structural sensitivity analysis which provide the users with important information for optimal design, reliability analysis and inverse problem research. With the help of JIFEX, the users can complete the optimal design of complex built-up structure. JIFEX has been enhanced recently. The new sensitivity analysis techniques introduced above for transient structure, linear and nonlinear heat conduction and thermal structural responses are implemented. The optimal model can include the transient responses (such as the transient displacement, the velocity, the acceleration), the temperature and thermal responses (such as the thermal stress and thermal displacement). So the software can fulfil more complicated problem of structural optimization. The optimization algorithms of the program JIFEX are improved sequential linear programming (SLP) and sequential quadratic programming (SQP). The general mathematical modeling of structural design optimization problems is as follow

180

Yuanxian Gu et al.

min f ( X ) ⎧ ⎪ ⎨ s.t. g j ( X ) ≤ 0 j = (1,2.....m ) (53) ⎪ x ≤ x ≤ x i = (1,2.....n ) iL i iU ⎩ where f (X) is the objective function, n is the number of design variables xi , m is the number of constrains functions gj(X), xiU, and xiL are upper bounds and lower bounds of the variables respectively. The objective function and constraint functions may be weight, displacements, stresses, vibration frequencies, dynamic responses, buckling loads, and other behaviors of structures. In the SLP algorithm, the objective and constraints functions are approximated with linear extensions at the current design point of the optimization iteration. Then the original problem (53) is transformed into the following linear programming problem.

⎧ min f ( X 0 ) + ∇T f ( X 0 )∆X ⎪ T ⎨s.t. g j ( X 0 ) + ∇ g j ( X 0 )∆X ≤ 0 ⎪ xiL ≤ xi ≤ xiU ⎩ where

the

∇T f ( X 0 ) and

∇T g j ( X 0 )

(54)

are

derivative gradients of the objective function and constraint functions respectively. The linear programming (54) is solved with the Lamke pivot algorithm to find a new design. These approximation and solution procedure are repeated until the convergence is reached. In the SQP algorithm, the constraints are the same as in Eq.(54), and the objective function is approximated with second order extension as follows

f ( X 0 ) + ∇ T f ( X 0 )∆X +

1 T ∆ X∇ 2 f ( X 0 )∆X 2

Then the problem is transformed into a quadratic programming problem, and is also solved by the Lamke pivot algorithm. The SQP algorithm is used when the objective function is structural weight since its second order derivatives are easy to compute. While the SLP algorithm is used for other cases with objectives of buckling load, stress, vibration frequency and so on. To overcome the error problem caused in the linear approximation of constraint functions and to ensure iteration convergence, the approximate line search and adaptive move limit approaches have been employed. The basic SLP and SQP

algorithms mentioned above are improved to be a two-stage solution procedure for each iteration: (a) solving linear or quadratic programming problem to obtain a new design X, and (b) finding the accepted new design X new = X 0 + βd by the approximate line search along the direction of d = X − X 0 The approximate line search is based on the following Goldstein criterion.

β (1 − η )∇ T F ( X 0 )d ≤ F ( X 0 + βd ) − F ( X 0 ) ≤ βη∇ T F ( X 0 )d 0 < η < 0.5 (55) where β is the step length of the line search with initial value 1.0, and η is a prescribed constant factor. F(Ｘ) is the original objective function or a modified multi-criterion objective function. In the later case, the algorithm is extended to deal with the multiple criterion design optimizations and solve the unfeasible design problems. The Goldstein criterion means that the design satisfying condition (55) is better in some degree than the old design X0, though it may not be the optimum. This way, it makes the objective function F(X) decrease step by step and the iteration convergence is stable. Starting from X and with the initial step length β =l.0, the approximate line search is carried out with the following three conditions and finished within a few steps. (a) If the Goldstein criterion (55) is satisfied, then the X new = X 0 + βd is accepted as a new design and the line search stops. (b) If F(X o + βd)-F(X 0 ) > βη∇ T F ( X 0 )d , then reduce the step length β to continue the line search, and reduce the move limit for next iteration; (c) If F(X o + βd)-F(X 0 )