which is the best for modelling different continuum and structural elements; updated Lagrangian formulation or total Lagrangian formulation.
Thanks
Look at the problem that you are going to solve and define what kind of the displacements are involved.
Total Lagragian formulations analyse what is the present configuration in relationship to the first / initial one (in general, fixed in time), therefore it means the total displacement. Usually this is the case when you deal with big displacements and/or velocities. Therefore, it is recommended for fluids, for total or potential energy aproaches, for other cases as in Mechanical Engineering: the space-ship flights, the motion of navy ships, trucks and the aircrafts etc. Understanding that any formulation that you adopt has an error, use the one that
corresponds to the needs that you have. (e.g. there is no sense to worry about differences of order of tenth millimeters when your displacements ranges in meters or more).
The Updated Lagrangian formulations tries to capture what happen in the updtated configuration in relationship to the previous known configuration.
This comes to be important when the displacements becomes near or even
represent also the deformation of the body (e.g. the displacements becomes
of order of tenths of millimeter and the errors are reduced to one thousandth
of it). Therefore many times it is recommended for the Structural Engineering
use.
The order of displacements is a good parameter to choose the kind of
formulation, but it doesn't mean that you cannot use a TL for a structural
analysis, but can make you worry about the precision of the involved answers.
Some energy applications are verywell defined, suited and recommended for
use with TL.
Maybe, some part of the question must be also addressed to the deformation
which is involved. And sometimes, you must remove the effects of the rigid-body
motion from the analysis. In this case, it is common to use the co-rotational system, just to guarantee that you will not get espurious forces or strains, which
belong to rotations of the body (among others), which in fact don't provide
deformations. I adopted Updated Lagrangian co-rotational system in my works,
so I'm a bit suspect by push the answer for this side (lgs).
I recommend you to look at the old (but always good) Theory and Analysis of
Nonlinear Framed Structures, from Yeong-Bin YANG and Shyh-Rong KUO, printed
by Prentice Hall, 1994, ISBN 0-13-109224-3. Good luck!
https://www.amazon.com/Theory-Analysis-Nonlinear-Framed-Structures/dp/0131092243
Dear@ Khaled Megahed
- Total Lagrangian (T.L.) Formulation:
It is also called Lagrangian formulation, in this solution scheme all static and kinematic variables are referred to the initial configuration at time 0.
- Updated Lagrangian (U.L.) Formulation:
This formulation is quite analogous to the total Lagrangian formulation, except that in this solution scheme all static and kinematic variables are referred to the configuration at time t.
- T.L. Formulation versus U.L. Formulation:
Upon comparing U.L. and T.L. we can observe that they are quite analogous and the only theoretical difference between the two formulations lies in the choice of different reference configurations for the kinematic and static variables. Both the T.L. and U.L. formulations include all kinematic nonlinear effects due to large displacements, large rotations, and large strains, but the modeling of large strains depends on the constitutive relations.
I hope that the answer is clear.
Kind Regards
Look at the problem that you are going to solve and define what kind of the displacements are involved.
Total Lagragian formulations analyse what is the present configuration in relationship to the first / initial one (in general, fixed in time), therefore it means the total displacement. Usually this is the case when you deal with big displacements and/or velocities. Therefore, it is recommended for fluids, for total or potential energy aproaches, for other cases as in Mechanical Engineering: the space-ship flights, the motion of navy ships, trucks and the aircrafts etc. Understanding that any formulation that you adopt has an error, use the one that
corresponds to the needs that you have. (e.g. there is no sense to worry about differences of order of tenth millimeters when your displacements ranges in meters or more).
The Updated Lagrangian formulations tries to capture what happen in the updtated configuration in relationship to the previous known configuration.
This comes to be important when the displacements becomes near or even
represent also the deformation of the body (e.g. the displacements becomes
of order of tenths of millimeter and the errors are reduced to one thousandth
of it). Therefore many times it is recommended for the Structural Engineering
use.
The order of displacements is a good parameter to choose the kind of
formulation, but it doesn't mean that you cannot use a TL for a structural
analysis, but can make you worry about the precision of the involved answers.
Some energy applications are verywell defined, suited and recommended for
use with TL.
Maybe, some part of the question must be also addressed to the deformation
which is involved. And sometimes, you must remove the effects of the rigid-body
motion from the analysis. In this case, it is common to use the co-rotational system, just to guarantee that you will not get espurious forces or strains, which
belong to rotations of the body (among others), which in fact don't provide
deformations. I adopted Updated Lagrangian co-rotational system in my works,
so I'm a bit suspect by push the answer for this side (lgs).
I recommend you to look at the old (but always good) Theory and Analysis of
Nonlinear Framed Structures, from Yeong-Bin YANG and Shyh-Rong KUO, printed
by Prentice Hall, 1994, ISBN 0-13-109224-3. Good luck!
https://www.amazon.com/Theory-Analysis-Nonlinear-Framed-Structures/dp/0131092243
It depends on the problem you are solving. for example, the ULD has the following advantages: • Effective treatment of large rotations; • Easily adapt with finite elements that have rotational d.o.f like beams and shells; • Decoupling of material nonlinearity and the geometric nonlinearity; • Automatic reorientation of material by removal of the rigid body motion, avoiding the complexity of using invariants of the continuum mechanics; • And finally, it has the advantages of exploiting the library programs in small displacements.
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