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In the generation of element geometries, if dx dy represents an area element in the real element and dεdη represents the corresponding area element in the master element, then what is the expression for Jacobian j^e?

(a) (dx dy) (dεdη)

(b) \(\frac{dx}{d\epsilon} \frac{dy}{d\eta}\)

(c) \(\frac{d\epsilon}{dx} \frac{d\eta}{dy}\)

(d) \(\frac{1}{(dx dy) (d\epsilon d\eta)}\)

I had been asked this question in unit test.

This interesting question is from Modelling Considerations topic in division Interpolation Functions, Numerical Integration and Modelling Considerations of Finite Element Method

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The correct option is (b) \(\frac{dx}{d\epsilon} \frac{dy}{d\eta}\)

Explanation: The numerical evaluation of integrals over actual elements involves a one-to-one mapping between the actual element and the master element. This requirement can be expressed as j^e>0, where j^e is the Jacobian matrix. J^e represents the ratio of an area element in the real element to the corresponding area element in the master element.

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