The basis of each structural optimization is a precise structural analysis on well founded structural mechanics. In the field of structural mechanics our team can refer to long term experience and many new developments. In the following section the most important components of a FEM analysis should be introduced:

  • Element formulations
  • Boundary conditions
  • Material formulations
  • Solution strategies

Element formulations

Element formulations are used to model the specific structural response based on kinematic assumptions. These kinematic assumptions specify the relation between structural deformations (displacements, rotations) and strains. During the analysis process the element formulations are necessary to compute stiffness and mass matrices, stresses, and element loads. Usually, the different element formulations are separated in 1-D, 2-D and 3-D elements.

1-D Elements

If one dimension of the structure is much larger than the other two dimensions these structure can be modelled by 1-D elements, e.g. beam elements or truss elements. Beam elements can be formulated by Euler Bernoulli or Timoshenko theory according to their suitability to thin or compact beam structures. The basic difference of both formulations is the incorporation of specific shear deflections that is provided by the Timoshenko theory. It is well known that Timoshenko beams may suffer from locking problems. EAS (Enhanced Assumed Strain) or reduced integration can overcome this problem.

truss element

 

2-D Elements

This group of finite elements consists of plate, wall, membrane and shell elements. Wall elements are not discussed here because of their low practical relevance. Plate elements can be considered as simplified shell elements. The basic difference between these formulations are the missing surface curvatures of plate elements. Membrane elements can consider curvature but they do not produce any bending stiffness. Such elements are used to simulate the behavior of textiles, e.g. tents. Shell elements are very powerful and complex finite elements used to model the behavior of curved surface structures like roofs, car body components or air plane components. There exist specific shell formulations for thin and thicker shells. Thick shells can be modelled with elements that are based on the Reissner-Mindlin theory. Pure displacement based Reissner-Mindlin shells suffer from locking phenomena. Remedies are EAS (Enhanced Assumed Strain) methods, ANS (Assumed Natural Strain) methods or application of higher order elements with quadratic shape functions.

References:

  • M. Bischoff, F. Koschnick, K.-U. Bletzinger, Stabilized DSG elements – a new paradigm in finite element technology, in Proceedings of the 4th European LS-DYNA Users Conference, Ulm.
Shell Elements

 

3-D Elements

The well known solid element belong to the group of 3-D Elements. Possible discretization types are Tetras, Hexas and Pentas. Locking problems are also present for pure displacement based solid elements. Possible remedies are EAS or ANS enhancements as well as application of higher order elements using quadratic shape functions. In this context 10-noded Tetras are very important. They are suitable for automatic meshing and also produce accurate analysis results. It has to be noted that the application of quadratic solids elements results in a large number of degrees of freedom an time consuming system evaluations. Solid elements are often applied to discretize casted or milled parts. They are also very important in topology optimization.

References:

  • M. Fischer,Geometrische und volumetrische Lockingeffekte bei kontinuumsbasierten finiten Elementen und ihre Vermeidung durch die EAS-Methode, Master Thesis, EADS MAS Ottobrunn, 2007.
  • H. Masching, Implementierung mehrerer Kontinuumselementformulierungen in einem objektorientierten Finite-Elemente-Programm unter Anwendung der EAS-Methode, Master Thesis, Lehstuhl für Statik, Technische Universität München, 2009.
Solid Elemente

 

Boundary conditions

Boundary conditions are used to model external actions on structures. In general, it is distinguished between structural loads and support conditions.

External loads

External loads can be formulated via node loads or element loads. The most important type of node loads are nodal forces that are applied directly on the displacement DOFs of the node. Shell elements usually incorporate nodal rotations as additional degrees of freedom. This allows specification of moment loads. Element loads are applied to elements because the specific element formulation is necessary to compute the consistent nodal forces. Well known element loads are surface loads, pressure loads or dead load.

Reference:

  • Jrusjrungkiat A.,  Wüchner R.,  Bletzinger K.-U., Aspects of nonlinear analysis for an inflatable membrane coupled with enclosed fluids, In proceeding of the Structural Membranes 2009, Stuttgart, Germany, October 7-9, 2009

 

Prescribed displacements / rotations

Prescribing displacements or rotations can be used to model the action external deformations. The most important special case is the modelling of supports. In this case the prescribed displacement or rotation takes the value '0'.

Material formulations

Structures may consist of many different material types. The specific properties of materials are modelled by suitable material laws which relate structural strains and stresses. The most simple material law is Hookes law used to model linear elasticity for small displacements. In case of finite displacements the linear elasticity is described by the St. Venant Kirchhoff material law. There exists a huge variety of different materials suitable to model plasticity of metals, creeping of polymers or concrete, distribution of cracks, etc. Orthotropic materials laws are often used to model the behavior of composites. Fibre reinforced materials like composites need specific material laws to consider the material orientation.

Reference:

  • H. Masching, M. Fischer, M. Firl, K.-U. Bletzinger, Parameter Free Structural Optimization of Large Lightweight Composite Structures, 3rd ECCOMAS Thematic Conference on the Mechanical Response of Composites, 21st - 23rd September 2011, Hannover, Germany

 

Solution strategies

Different types of analysis methods require different solution strategies. In general, one distinguishes linear and nonlinear solutions and eigenvalue problems. Linear solution methods are applied if the structural deformations are small and if the deformations do not change the load carrying behavior significantly. Such strategies formulate structural equilibrium on the reference model and result in linear equation systems. If the assumptions of linear analysis are not fulfilled it is necessary to apply the more precise nonlinear solution strategies. These methods solve a nonlinear problem by iterative solutions of linear subproblems formulated by specific path following methods. Well known path following strategies are:

  • load control
  • displacement control
  • arc-length control

Another class of solution strategies are eigenvalue problems. Solving eigen systems allows for computation of eigenfrequencies, linear buckling loads and nonlinear buckling loads.

References:

  • Reiner Reitinger, Ekkehard Ramm, Kai-Uwe Bletzinger, Shape Optimization of Buckling Sensitive Structures Computing Systems in Engineering 5, 1994, pp. 65-75
  • Ekkehard Ramm, Kai-Uwe Bletzinger, Reiner Reitinger, Shape optimization of shell structures. In : Seiken - IASS Symposium on 'Nonlinear Analysis and Design of Shell and Spatial Structures', Tokyo, Japan, Oktober 1993, IASS - Bulletin 34, pp 103-121