Skip to Main Content
Applied Calculus of Variations for Engineers, 2nd Edition
book

Applied Calculus of Variations for Engineers, 2nd Edition

by Louis Komzsik
June 2014
Intermediate to advanced content levelIntermediate to advanced
233 pages
5h 42m
English
CRC Press
Content preview from Applied Calculus of Variations for Engineers, 2nd Edition
Computational mechanics 197
the form:
B(x, y)=
∂N
1
∂x
∂N
2
∂x
∂N
3
∂x
∂N
1
∂y
∂N
2
∂y
∂N
3
∂y
.
Since the shape functions are defined in terms of the parametric coordinates,
the derivatives of the local shape functions are computed by using the chain
rule as
∂N
i
∂v
=
∂N
i
∂x
∂x
∂v
+
∂N
i
∂y
∂y
∂v
and
∂N
i
∂w
=
∂N
i
∂x
∂x
∂w
+
∂N
i
∂y
∂y
∂w
.
These relations may be gathered as
∂N
i
∂v
∂N
i
∂w
=
∂x
∂v
∂y
∂v
∂x
∂w
∂y
∂w

∂N
i
∂x
∂N
i
∂y
.
The first term on the right-hand side is
∂x
∂v
∂y
∂v
∂x
∂w
∂y
∂w
= J,
as we found it earlier. Hence
∂N
i
∂v
∂N
i
∂w
= J
∂N
i
∂x
∂N
i
∂y
and
∂N
i
∂x
∂N
i
∂y
= J
1
∂N
i
∂v
∂N
i
∂w
.
The inverse of the Jacobian matrix may be computed by
J
1
=
adj(J)
det(J)
.
This equation clarifies the earlier warning comment about
Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.
Start your free trial

You might also like

Applied Calculus of Variations for Engineers, Third edition, 3rd Edition

Applied Calculus of Variations for Engineers, Third edition, 3rd Edition

Louis Komzsik
Algebraic and Stochastic Coding Theory

Algebraic and Stochastic Coding Theory

Dave K. Kythe, Prem K. Kythe

Publisher Resources

ISBN: 9781482253597