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Computational aspects of sensitivity calculations in linear transient structural analysis

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Abstract

A study has been performed focusing on the calculation of sensitivities of displacements, velocities, accelerations, and stresses in linear, structural, transient response problems. Several existing sensitivity calculation methods and two new methods are compared for three example problems. All of the methods considered are computationally efficient enough to be suitable for largeorder finite element models. Accordingly, approximation vectors such as vibration mode shapes are used to reduce the dimensionality of the finite element model. Much of the research focused on the convergence of both response quantities and sensitivities as a function of the number of vectors used.

Two types of sensitivity calculation techniques were considered. The first type of technique is an overall finite difference method where the analysis is repeated for perturbed designs. The second type of technique is termed semi-analytical because it involves direct analytical differentiation of the equations of motion with finite difference approximation of the coefficient matrices. To be computationally practical in large-order problems, the overall finite difference methods must use the approximation vectors from the original design in the analyses of the perturbed models. This was found to result in poor convergence of stress sensitivities in several cases. To overcome this poor convergence, two semianalytical techniques were developed. The first technique accounts for the change in eigenvectors through approximate eigenvector derivatives. The second technique applies the mode acceleration method of transient analysis to the sensitivity calculations. Both result in very good convergence of the stress sensitivities. In both techniques the computational cost is much less than would result if the vibration modes were recalculated and then used in an overall finite difference method.

A dot over a symbol indicates derivative with respect to time. A superscriptT indicates a transposed matrix.

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Abbreviations

C:

system damping matrix

\(\bar C\) :

reduced damping matrix

C x :

cost in number of floating point operations for computational taskx

E :

Youngs's modulus

f :

vector of external forces

\(\bar f\) :

vector of reduced system forces

g(t) :

scalar function representing time dependence of applied loading

h i :

thickness ini-th span or section for five-span beam and stepped beam examples

K:

system stiffness matrix

\(\bar K\) :

reduced stiffness matrix

l i :

distance from beam root to end ofi-th section in stepped beam

M:

system mass matrix

\(\bar M\) :

reduced mass matrix

M :

bending moment

m :

mass design variable in beam example

N ij :

submatrixi, j in matrix series expansion method

n c :

number of critical time points

n g :

number of degrees of freedom in finite element model

n p :

total number of physical response quantities to be calculated as function of time

n r :

number of equations in reduced system

n t :

number of time steps in numerical integration of differential equations

q:

vector of reduced system coordinates

S:

transformation matrix between element stresses and nodal displacements

t :

time

t c :

critical time in critical point constraint

t iθ :

thickness ofθ degree lamina ini-th region of skin for delta wing example

t iw :

thickness of web ini-th region for delta example

u:

vector of displacements

u i :

displacement at one-third point ini-th span of five-span beam example

u allow :

maximum allowable value of displacement

u tip :

deflection at wing or beam tip

V i :

shear force ini-th span

W ij :

submatrixi, j in matrix series expansion method

x:

vector of design variables

β :

semibandwidth (excluding diagonal) of system matrices (e.g. K, M)

Δt :

time step size in matrix series expansion integration technique

ρ :

material density

σ :

vector of elemental stresses

σ iθ :

fiber direction stress inθ degree lamina ini-th region of skin for delta wing example

σ root :

stress at beam root

τ iw :

shear stress ini-th region of web for delta wing example

Ф :

matrix with each column being basis vector

ω j :

j-th vibration frequency

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Authors and Affiliations

  1. 1411 Granada Ct., 23602, Newport News, VA, USA

    W. H. Greene

  2. Virginia Polytechnic Institute and State University, 24061, Blacksburg, VA, USA

    R. T. Haftka

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  1. W. H. Greene
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  2. R. T. Haftka
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Greene, W.H., Haftka, R.T. Computational aspects of sensitivity calculations in linear transient structural analysis. Structural Optimization 3, 176–201 (1991). https://doi.org/10.1007/BF01743074

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