Novel finite element schemes for computational solid mechanics based on Bézier elements

Significance 

Recent technological advances across all fields have led to the increasing need for an optimized design that requires more sophisticated, efficient and cost-effective simulation techniques. This, in turn, has necessitated the development of several numerical simulation techniques that are majorly based on the finite element methods. Among the available finite element discretization methods, Lagrange finite elements and in particular triangular and tetrahedral elements are widely used owing to the fact that there can be easily created through mesh generation even in complex geometries. However, regardless of the remarkable improvements in numerical simulation across the industries, triangular and tetrahedral elements for accurate explicit elastodynamics simulations especially those involving incompressible models have not been fully developed. To this end, the development of robust and effective triangular and tetrahedral Lagrange elements is highly desirable.

Among the measures adopted to enhance the accuracy of modeling the incompressible materials using the triangular and tetrahedral elements include various sophisticated modifications. However, static analysis and transient problems have too posed a great threat to the achievement of the intended objectives. This has attracted significant attention from researchers who have henceforth developed several approaches. For instance, stabilized mixed formulations approaches despite being used in performing explicit elastodynamic simulations are still not favorable for large-scale applications. Consequently, mixed-stabilized formulations are difficult to be used together with the Lagrange triangular and tetrahedral elements since they do not support explicit elastodynamic simulations. Therefore, researchers have been looking for efficient alternatives and have identified isogeometric analysis in conjunction with other related techniques as a promising solution of explicit elastodynamic simulations.

In a recent paper published in the research journal, International Journal for Numerical Methods in Engineering, Dr Chennakesava Kadapa at Swansea University assessed the feasibility of using quadratic Bézier triangular and tetrahedral elements for performing elastostatic and elastodynamic simulations of both compressible and incompressible elastic models. The work entailed quadratic Bézier triangular and tetrahedral elements developed by linear mapping from the corresponding Lagrange elements. Generally, the elements were generated using the existing mesh generators. Furthermore, B-bar formulation was used to track the emerging issues related to the numerical simulation of the nearly incompressible materials, thus also reducing the dependence on the finite element simulation on the displacements.

A single element formulation was used for both elastostatic and elastodynamic simulations to obtain accurate results even for the elastic incompressible materials. Alternatively, it was noted that fewer resources were required to implement the proposed elements in the existing finite element codes. Additionally, the B-bar formulation proved to be a cost-effective approach for dealing with the incompressible model as it did not require additional independent variables as compared to the mixed-stabilized methods.

In summary, Dr. Chennakesava work demonstrated robust and cost-effective explicit elastodynamic simulation of both compressible and incompressible materials. To actualize the study, explicit dynamic simulation of a cantilever beam, two-dimensional jig, and three-dimensional connecting rod was successfully performed using the proposed elements. This was attributed to the higher-order elements employed that ensured the stability and uniform distribution of loads. Altogether, the study provides vital information that will significantly advance the field of computational mechanics.

About the author

Dr Chennakesava Kadapa is a Research Software Engineer in Swansea Academy of Advanced Computing at Swansea University, United Kingdom. Dr Kadapa obtained his PhD in Mechanical Engineering in 2014 from the Zienkiewicz Centre for Computational Engineering at Swansea University. Dr Kadapa worked as a post-doctoral research officer at the Ziekiewicz Centre from October 2013 to June 2018. The present contribution is Dr Kadapa’s independent work.

Dr Kadapa’s research interests include Finite Element Analysis, Isogeometric Analysis, Fluid-Structure Interaction, Coupled Electro-Mechanics and High-Performance Computing.

References

Kadapa, C. (2018). Novel quadratic Bézier triangular and tetrahedral elements using existing mesh generators: Applications to linear nearly incompressible elastostatics and implicit and explicit elastodynamics. International Journal for Numerical Methods in Engineering, 117(5), 543-573.

Kadapa, C. (2018). Novel quadratic Bézier triangular and tetrahedral elements using existing mesh generators: Extension to nearly incompressible implicit and explicit elastodynamics in finite strains. International Journal for Numerical Methods in Engineering, 119(2), 75-104.

Go To International Journal for Numerical Methods in Engineering

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