An energy-momentum conserving algorithm for nonlinear transient analysis within the framework of hybrid elements

2010

JoMMS

Hybrid elements, which are based on a two-field variational formulation with the displacements and stresses interpolated separately, are known to deliver very high accuracy, and to alleviate to a large extent problems of locking that plague standard displacement-based formulations. The choice of the stress interpolation functions is of course critical in ensuring the high accuracy and robustness of the method. Generally, an attempt is made to keep the stress interpolation to the minimum number of terms that will ensure that the stiffness matrix has no spurious zero-energy modes, since it is known that the stiffness increases with the increase in the number of terms. Although using such a strategy of keeping the number of interpolation terms to a minimum works very well in static problems, it results either in instabilities or fails to converge in transient problems. This is because choosing the stress interpolation functions merely on the basis of removing spurious energy modes can violate some basic principles that interpolation functions should obey. In this work, we address the issue of choosing the interpolation functions based on such basic principles of interpolation theory and mechanics. Although this procedure results in the use of more number of terms than the minimum (and hence in slightly increased stiffness) in many elements, we show that the performance continues to be far superior to displacement-based formulations, and, more importantly, that it also results in considerably increased robustness.

Section: Choice Of Stress Interpolation Functionsmentioning

confidence: 99%

Section: Choice Of Stress Interpolation Functionsmentioning

confidence: 99%

Section: Choice Of Stress Interpolation Functionsmentioning

confidence: 99%

Section: Choice Of Stress Interpolation Functionsmentioning

confidence: 99%

Section: E Elastic Wave Propagation In a Circular Diskmentioning

confidence: 99%

See 3 more Smart Citations

Improved hybrid elements for structural analysis

2010

JoMMS

A monolithic strategy for fluid–structure interaction problems

Numerical Meth Engineering

Pal

2010

Self Cite

SUMMARYIn this work, we present a new monolithic strategy for solving fluid-structure interaction problems involving incompressible fluids, within the context of the finite element method. This strategy, similar to the continuum dynamics, conserves certain properties, and thus provides a rational basis for the design of the time-stepping strategy; detailed proofs of the conservation of these properties are provided. The proposed algorithm works with displacement and velocity variables for the structure and fluid, respectively, and introduces no new variables to enforce velocity or traction continuity. Any existing structural dynamics algorithm can be used without change in the proposed method. Use of the exact tangent stiffness matrix ensures that the algorithm converges quadratically within each time step. An analytical solution is presented for one of the benchmark problems used in the literature, namely, the piston problem. A number of benchmark problems including problems involving free surfaces such as sloshing and the breaking dam problem are used to demonstrate the good performance of the proposed method.

A hybrid finite element strategy for the simulation of MEMS structures

Numerical Meth Engineering

Patil

2015

Self Cite

SUMMARYThis work describes the development of a hybrid finite element strategy for modeling the nonlinear, strongly coupled electromechanical equations governing micro-electro-mechanical systems (MEMS) structures in general and the phenomena of static and dynamic pull-in exhibited by such structures in particular. It is known that conventional finite elements 'lock' severely for thin plate or shell-type structures (which is typical of geometries that occur in microsystems). The developed hybrid finite element formulation is based on a two-field variational formulation and involves an independent interpolation of the displacement and stress fields. In a conventional structural analysis, it is known that hybrid finite elements are much more immune to volumetric, shear, and membrane locking compared with conventional elements. This work extends the hybrid finite element method to the analysis of MEMS structures. Such an extension is not straightforward due to the strong coupling with the electric field variables. The formulation is based on a total Lagrangian description and is monolithic. This combination of the hybrid strategy and the monolithic scheme results in very good coarse-mesh accuracy and quadratic convergence both within the context of the static and transient formulations that have been developed. Numerous examples from the literature are considered to demonstrate the high efficacy of the developed method.