Stereometry of compressed conoids and physical adequacy of q8 element bases

The paper considers new models of bases of serendipity finite elements (FE) Q8. The standard element Q8 has been used in the finite element method (FEM) for more than 50 years despite the physical inadequacy of the spectrum of equivalent nodal loads.In recent years, the library of serendipity finite elements has been significantly replen-ished with non-standard (alternative) models. The reasons for the inadequacy of the spectrum were identified and "recipes" were proposed to eliminate this shortcoming of standard serendipity models. New approaches to modeling bases with the help of hierarchical forms force to abandon conoids - linear surfaces that are associated with intermediate nodes of standard elements. According to the authors, these Catalan surfaces (1843) are insufficiently studied and deserve the attention of modern researchers. Therefore, research is being conducted today, and it is not necessary to give up conoids. The paper shows how by compressing the surface of the conoid it is possible to obtain a mathematically sound and physically adequate spectrum of nodal loads. It is interesting that such capabilities are embedded in trigonometric functions, the popularity of which in the FEM is growing steadily.The purpose of the research is to constructively prove the existence of mathematically substantiated and (most importantly) physically adequate models of serendipity elements Q8 with the help of trigonometric bases.Trigonometric models of the finite element Q8 once again confirmed that serendipity elements are an inexhaustible source of important and interesting information. It should be noted that today it is not necessary to give up conoids for the sake of physical adequacy of the model. Conoids are also of "historical" importance to FEM. The first bases of serendipity FEs were constructed from conoids (1968).Taylor's elegant method (1972) is also based on conoids. New results show that trigo-nometric bases are able to preserve conoids and ensure the physical adequacy of the models.

A Mid-Node Mass Lumping Scheme for Accurate Structural Vibration Analysis with Serendipity Finite Elements

A mid-node mass lumping scheme is proposed to formulate the lumped mass matrices of serendipity elements for accurate structural vibration analysis. Since the row-sum technique leads to unacceptable negative lumped mass components for serendipity elements, the diagonal scaling HRZ method is frequently employed to construct lumped mass matrices of serendipity elements. In this work, through introducing a lumped mass matrix template that includes the HRZ lumped mass matrix as a special case, an analytical frequency accuracy measure is rationally derived with particular reference to the classical eight-node serendipity element. The theoretical results clearly reveal that the standard HRZ mass matrix actually does not offer the optimal frequency accuracy in accordance with the given lumped mass matrix template. On the other hand, by employing the nature of non-negative shape functions associated with the mid-nodes of serendipity elements, a mid-node lumped mass matrix (MNLM) formulation is introduced for the mass lumping of serendipity elements without corner nodal mass components, which essentially corresponds to the optimal frequency accuracy in the context of the given lumped mass matrix template. Both theoretical and numerical results demonstrate that MNLM yields better frequency accuracy than the standard HRZ lumped mass matrix formulation for structural vibration analysis.

Analysis of Distortion Parameters of Eight node Serendipity Element on the Elements Performance

Often the result produced using FEM technique on the distorted shaped element gives poor result. Moreover if the results are poor then the design will fail. Eight noded serendipity elements is the most widely used element in 2-D analysis of structures. But despite its benefits it remains distortion sensitive. In the majority of computer programs, automatic mess generation is an integral part of the program. For complex shapes, the automatic grid generation will result in distorted quadrilateral shapes. Thus the solution obtained by using these meshes will produce erroneous results so it becomes necessary to incorporate the distortion measures in to these automatic mesh generations to limit the errors. Here in this paper distortion parameters are defined in terms of the coefficient of the element’s shape polynomials and are tested for a range of distortion.

THE USE OF ENRICHED HEXAHEDRAL ELEMENTS WITH BUBBLE FUNCTIONS FOR FINITE ELEMENT ANALYSIS

In this paper, the concept that adds the interior nodes of the Lagrange elements to the serendipity elements is described and a family of enriched elements is presented to improve the accuracy of finite element analysis. By the use of the static condensation technique at the element level, the extra computation time in using these elements can be ignored. Three-dimensional elastic problems are used as examples in this paper. The numerical results show that these enriched elements are more accurate than the traditional serendipity elements. The convergence rate of the enriched elements is the same as the traditional serendipity elements. In the numerical example, the error norm of the first order enriched elements can be reduced when compared with the use of the traditional serendipity element, but the computation time is increased a little. The use of enriched second and third order hexahedral elements does not only improve accuracy, but also saves the computation time for solving the system of equations, when the precondition conjugate gradient method is used to solve the system of equations. The saving of computation time is due to the decrease in the number of iteration for the iteration method.

FINITE ELEMENT MODELING OF ACOUSTICS USING HIGHER ORDER ELEMENTS PART I: NONUNIFORM DUCT PROPAGATION

Cubic serendipity elements have been implemented into a nonuniform duct model of acoustic propagation in a moving medium. This model uses a convective potential formulation derived from the inviscid linearized mass and momentum equations. The model requires post-processing to calculate acoustic pressure. These elements outperform the quadratic serendipity elements in terms of computational efficiency based on visual observations and error norm analysis of acoustic pressure. CPU time reduction of up to 40% has been observed without sacrificing accuracy. Any penalty in numerical accuracy incurred by using serendipity elements rather than Lagrangian elements is far outweighed by the gains in dimensionality. The computational gains for calculation of acoustic potential are considerably less. Analytical expressions for the modal and convective effects on the propagating wavelength have been formulated and compared to numerical results. Preliminary assessment of alternative finite element approaches to model the convective potential formulation has been conducted. Stabilization and wave approximation methods have been implemented to solve simple one-dimensional problems.