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Computational Framework
• Governing equation for structural domain: principle of
virtual work (conservation of momentum, i.e.
equilibrium in a dynamic sense)
∫ ∫ ∫ ∫
ρ
xδxdV + σD( δx) dV − ρ f δxdV − tδxdS
= 0
V V V S1
ρ mass density
V current domain
current configuration
x accelerations
x
t
S 1
f
V
σ
S 2
σ Cauchy stress
D() spatial derivative operator
f
volumetric forces per unit mass
t boundary surface tractions
Must hold for all variations δ x of configuration (virtual
displacements) compatible with essential b.c.s on .
S 2
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Computational Framework (2)
• This integral form lends itself to direct application
of F.E. method. Upon spatial discretization:
Mu f ext
B T
σ dV
e e
V
= −∑ ∫
x
f
ext
S 1
e
V
S 2
M Mass matrix
u nodal displacement vector
f
ext
∑ e
discrete external forces
standard F.E. assembly operator
e
V element ( e) current volume
B matrix of shape functions derivatives
This set of discrete differential equations in time is decoupled
by diagonalization (lumping) of mass matrix M
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