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Design sensitivity analysis of nonlinear dynamic response of structural and mechanical systems

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Abstract

This paper describes a unified variational theory for design sensitivity analysis of nonlinear dynamic response of structural and mechanical systems for shape, nonshape, material and mechanical properties selection, as well as control problems. The concept of an adjoint system, the principle of virtual work and a Lagrangian-Eulerian formulation to describe the deformations and the design variations are used to develop a unified view point. A general formula for design sensitivity analysis is derived and interpreted for usual performance functionals. Analytical examples are utilized to demonstrate the use of the theory and give insights for application to more complex problems that must be treated numerically.

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Abbreviations

a :

right superscript that identifies a quantity for the adjoint structure

b:

design variable vector

B:

a strain operator

dV :

differential volume in the undeformed configuration

\(d\bar V\) :

differential volume in the fixed reference domain

f:

body force per unit undeformed volume

g :

integrand of the displacement specified boundary integral in the response functional

G :

integrand of the volume integral in the response functional

h :

integrand of the traction specified boundary integral in the response functional

J :

Jacobian of the transformation from the undeformed configuration to the reference volume

\(\bar J\) :

area metric for transformation from the undeformed configuration to the reference volume

n:

unit normal vector to the surface\(\bar \Gamma \)

r :

left superscript or subscript for quantities in the reference domain

R:

surface traction

R0 :

prescribed surface traction

R :

subscript referring to the traction specified surface

S:

second Piola-Kirchhoff stress tensor at timet

Sa :

stress tensor for the adjoint structure

t :

time

T :

total time

T :

left superscript representing the quantity at the final timeT

T :

right superscript indicating transpose of a vector or matrix

\(\bar {\rm T}\) :

Jacobian of the transformation for the time to the reference time-domain

u :

subscript referring to the displacement specified surface

u:

displacement field at timet

u0 :

prescribed displacement field

ua0 :

prescribed adjoint displacement field

ua :

displacement field for the adjoint structure

0 x i :

coordinates in the undeformed configuration

r x i :

coordinates of the particle of the body in the reference domain

\(V,\bar V\) :

volume in the undeformed configuration and the reference domain

W a :

virtual work expression in which arbitrary variations are replaced by the corresponding adjoint fields

X:

Jacobian matrix for the transformation from undeformed configuration to the reference volume

\(\bar X\) :

inverse of the Jacobian matrix X

z:

composite state vector consisting of displacement, velocity and acceleration fields

AI :

augmented ‘action’ functional defined in (9)

α :

adjoint strain operator

∇:

gradient operator

δ :

variational operator

\(\tilde \delta e^a \) :

an operator for the adjoint structure defined in (26)

\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\delta } \) :

Dirac delta function

ε :

Green-Lagrange strain tensor at timet

ε a :

strain tensor for the adjoint structure

γ :

Lagrangian multiplier for the terminal conditions

IL :

augmented Lagrangian functional defined in (9)

φ :

function that specifies the terminal conditions

Γ :

surface in the undeformed configuration

\(\bar \Gamma \) :

surface in the reference volume

Γ R ,Γ u :

traction and displacement specified surfaces in the undeformed configuration

Φ :

functional for the constitutive law

ρ :

mass density at timet

Ψ :

performance functional in the space-time domain

ψ :

integrand of the performance functionalΨ

τ :

time in the reference time-domain

ξ :

terminal time in the reference time-domain

\(\bar \delta ()\) :

total design variation of (); i.e.\(\bar \delta ( ) = \frac{{d( )}}{{db}}\delta b\)

\(\bar \bar \delta ( )\) :

explicit design variation (partial derivative) of (); i.e.\(\bar \bar \delta ( ) = \frac{{\partial ( )}}{{\partial b}}\delta b\) for which state fields are frozen

\(\tilde \delta ( )\) :

design variation of the fields that implicitly depend on the design variables, such as displacements, strains, velocities, accelerations, etc.; also design variation of functionals with respect to the implicit state fields; for this variation, the explicit dependence on the design variables is frozen

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Authors and Affiliations

  1. Instituto Superior Técnico, Departamento de Engenharia Mecânica, Universidade Técnica de Lisboa, Lisbon, Portugal

    J. B. Cardoso

  2. Optimal Design Laboratory, College of Engineering, The University of Iowa, 52242, Iowa City, IA, USA

    J. S. Arora

Authors
  1. J. B. Cardoso
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  2. J. S. Arora
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Additional information

Derivatives The ‘comma’ notation for partial derivatives is used, i.e. G,u = ∂G/∂u. An ‘upper dot’ represents material time derivative, i.e. ü = ∂2u/∂t2. A ‘prime’ implies derivative with respect to the time measured in the reference time-domain, i.e. u′ = du/dτ.

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Cardoso, J.B., Arora, J.S. Design sensitivity analysis of nonlinear dynamic response of structural and mechanical systems. Structural Optimization 4, 37–46 (1992). https://doi.org/10.1007/BF01894079

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