1 Introduction
2 Mid-range approximation method
3 Sub-space approximations
3.1 Formulation of sub-space approximations
3.2 Sub-space approximations in trust regions
4 Demonstration on a finite element model
4.1 Beam model
4.2 Load cases
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(a) Torsional stiffness A free node on the lefthand side of the beam is constrained in all translational degrees of freedom as well as from rotation around the longitudinal axis of the beam. The node is connected to the edges of component c25 (see Fig. 8) using one dimensional elements (RBE3) that distributes forces from the free node to the structure. At the right hand side of the beam a force is applied to a lever arm, connected to the beam with the same type of connection, resulting in a moment around the longitudinal axis of the beam. The response is the resulting rotation at the point where the moment is applied. The mass of the structure is also used as a response and is extracted from this load case. The load case is analysed using a static implicit solution procedure in Altair OptiStruct (Altair Engineering, Inc., 2014a).
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(b) Cylinder impact left (CL) A heavy cylinder with an imposed velocity impacts the left section of the beam in vertical direction. The beam is supported in vertical direction by a rigid plane. At impact the beam is compressed by the heavy cylinder. The response is measured as the vertical deformation of the top part of the beam, caused by the cylinder impact. The load case, which is a dynamic explicit load case, is analysed using Altair RADIOSS (Altair Engineering, Inc., 2014b).
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(c) Cylinder impact center (CM) The load case definition is the same as for (b) except for the impact position of the cylinder which is now at the centre of the beam.
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(d) Cylinder impact right (CR) The load case definition is the same as for (b) and (c) except for the impact position of the cylinderwhich is now at the right section of the beam.
4.3 Optimisation problem set up
Response | Initial | Target | Gradients | Approximations |
---|---|---|---|---|
Mass | 1.00 |
min
| Yes | LSM |
Torsion | 0.98 | ≤1.00 | Yes | MLSM |
CL | 1.18 | ≤ 1.00 | No | MLSM |
CM | 1.18 | ≤ 1.00 | No | MLSM |
CR | 1.17 | ≤ 1.00 | No | MLSM |
Opt. | 1 | 2 | 3 | 4 | ||||
---|---|---|---|---|---|---|---|---|
ns | np | ns | np | ns | np | ns | np | |
Torsion | 26 | 39 | 26 | 39 | 26 | 39 | 26 | 39 |
CL | 26 | 39 | 7 | 39 | 7 | 10 | 7 | 10 |
CM | 26 | 39 | 12 | 39 | 12 | 18 | 11 | 18 |
CR | 26 | 39 | 7 | 39 | 7 | 10 | 7 | 10 |