Locking-free finite elements for large deformations

Research project

Development of adaptive, deformation-dependent finite element formulations for stable and locking-free analysis of large deformation problems.


  • Stability of finite elements in finite deformation problems
  • Locking in the nonlinear regime
  • Capturing physical instabilities

Project description

The finite element method is a well-established numerical method for the solution of various problems in mechanics, as for instance to determine the response of an elastic body due to external loads. A common problem is the appearance of locking effects, which often results in an overly stiff structural response. While for linear problems many well performing elements have been developed, there are still open questions concerning nonlinear elements for finite deformation problems. The aim of this project is the analysis of the locking phenomena in the nonlinear regime and the development of novel nonlinear finite element formulations.  

Deformation numerical and physical instabilities
Deformed finite element mesh for different formulations with the same load case (LTR: locking – too stiff response, numerical instability, physical instability).
Numerical instability
Numerical instability
Physical instability
Physical instability

Stability of finite elements

Many locking-free elements which are known to be stable for linear problems show undesired artificial numerical instabilities (“hourglassing”) in the finite deformation range. In addition, (real) physical instabilities can occur in nonlinear problems. For the development of new finite elements it is crucial to eliminate the numerical instabilities but simultaneously be able to capture the physical instabilities accurately.

Locking in the nonlinear regime

Another focus of the project is a systematic investigation of the influence of the geometrically nonlinear parts of the tangent stiffness matrix, namely the initial displacement stiffness and the geometric stiffness on the locking behaviour. Furthermore, the behaviour of elements, which perform well in the linear regime is investigated in the nonlinear context.

Project data

Project title:
Adaptive, deformation-dependent finite element formulations for stable and locking-free analysis of large deformation problems

German Research Foundation (DFG), Research Grant BI 722/11-1, GEPRIS project number 299369509



Simon Bieber

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