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Unformatted text preview: y if all the coefficients ωik are real numbers. Dependence and Independence of Variations
3.1. Computation of the metric tensor of the final object The initial metric tensor is written: Ginit = Einit ⊗ t Einit The new E final vectors represent the vectors of the final object. 27 We deduce the expression of the final tensor from the equation (Eq1) via the tensor product
G final = E final ⊗ t E final i.e. by substituting the value of E final G final = ( Einit + ∆Einit ) ⊗
then by developing the equation t ( Einit + ∆Einit ) G final = ( Einit + Ω ⋅ Einit ) ⊗ t ( Einit + Ω ⋅ Einit )
t i.e. by expressing the unit matrix of dimension n as I n :
G final = ( I n + Ω ) ⋅ ( Einit ⊗ t Einit ) ⋅ ( In + Ω)
Eq2 Finally G final = ( I n + Ω ) ⋅ Ginit ⋅ t ( In + Ω) This is the basic formula for the dimensional variations of a geometric object. This tensor equation mathematically defines a differentiable parametric manifold, the ωik parameters of which are real numbers. In this way, a topological connection is made between 3 models: the Ginit nominal model, the Ω perturbation model and the final specified model, G final .
NB. The factor ( I n + Ω ) demonstrates that a Ω perturbation has definitely been added to unit I n since, for a nil Ω perturbation, we find G final = Ginit 3.2...
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