Abstract
In this paper, methods for developing isoparametric tetrahedral finite elements (FE) based on the absolute nodal coordinate formulation (ANCF) are presented. The proposed ANCF tetrahedral elements have twelve coordinates per node that include three position and nine gradient coordinates. The fundamental differences between the coordinate parametrizations used for conventional finite elements and the coordinate parametrizations employed for the proposed ANCF tetrahedral elements are discussed. Two different parametric definitions are introduced: a volume parametrization based on coordinate lines along the sides of the tetrahedral element in the straight (un-deformed) configuration and a Cartesian parametrization based on coordinate lines directed along the global axes. The volume parametrization facilitates the development of a concise set of shape functions in a closed form, and the Cartesian parametrization serves as a unique standard for the element assembly. A linear mapping based on the Bezier geometry is used to systematically define the cubic position fields of ANCF tetrahedral elements: the complete polynomial-based eight-node mixed-coordinate and the incomplete polynomial-based four-node ANCF tetrahedral elements. An element transformation matrix that defines the relationship between the volume and Cartesian parametrizations is developed and used to convert the parametric gradients to structure gradients, thereby allowing for the use of a standard FE assembly procedure. A general computational approach is employed to formulate the generalized inertia, external, and elastic forces. The performance of the proposed ANCF tetrahedral elements is evaluated by comparison with the conventional linear and quadratic tetrahedral elements and also with the ANCF brick element. In the case of small deformations, the numerical results obtained show that all the tetrahedral elements considered can correctly produce rigid body motion. In the case of large deformations, on the other hand, the solutions of all the elements considered are in good agreement, provided that appropriate mesh sizes are used.
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This research was supported, in part, by the National Science Foundation (Project # 1632302).
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Appendix A
Appendix A
The linear transformations between the Bezier tetrahedral patch basis functions \(g_{k} ,k=1,2,\ldots ,20\), and the shape functions of the ANCF/ENMC tetrahedral element \(\bar{{s}}_{k} ,k=1,2,\ldots ,20\), are given by
The linear transformations between the shape functions of the ANCF/ENMC tetrahedral element \(\bar{{s}}_{k} ,k=1,2,\ldots ,20\), and the shape functions of the ANCF four-node (FN) tetrahedral element \(s_{k} ,k=1,2,\ldots ,16\), are given by:
The shape functions of the ENMC and FN elements associated with the Cartesian gradient vectors \(\mathbf{r}_x^k \), \(\mathbf{r}_y^k \), and \(\mathbf{r}_z^k \) can be systematically evaluated. The ENMC shape functions \(\bar{{s}}_k^{*} ,k=1,2,\ldots ,20\), associated with the Cartesian gradients can be explicitly written in terms of the tetrahedral volume parameters as
The ANCF/FN shape functions \(s_k^{*} ,k=1,2,\ldots ,16\), associated with the Cartesian gradients can be written as
The above FN shape functions can be written in terms of the ENMC shape functions as
To obtain the above equations, the following constant coefficients need to be defined:
where \(x_{k} \), \(y_{k} \), and \(z_{k} \) represent the Cartesian coordinates of the tetrahedron vertex k, and \(V=V_1 +V_2 +V_3 +V_4 \). The tetrahedron volume coordinates \(\xi ,\eta ,\zeta \), and \(\chi \) can be expressed as a linear combination of the tetrahedral Cartesian coordinates x, y, and z as:
It can also be shown that the ENMC and FN shape functions written in terms of the Cartesian gradients \(\bar{{s}}_k^{*} \) and \(s_k^{*} \) can be expressed in terms of the ENMC and FN shape functions associated with the volume gradients \(\bar{{s}}_{k} \) and \(s_{k} \) as \({\bar{\mathbf{S}}}^{{*}}={\bar{\mathbf{S}}\bar{\mathbf{T}}}\) and \(\mathbf{S}^{{*}}=\mathbf{ST}\), where the transformation matrices \({\bar{\mathbf{T}}}\) and \(\mathbf{T}\) are previously defined in the paper.
In developing new ANCF finite elements with certain number and type of nodal coordinates, the use of incomplete polynomials may be necessary. In order to avoid trials and errors in identifying such incomplete polynomials and obtain symmetric structure in x, y, and z, the method of algebraic constraint equations used in developing the FN element presented in this paper can be systematically used. For the FN element, one can use from the outset, the algebraic constraint equations to reduce the number of each polynomial basis function from 20 to 16 and to systematically define the incomplete polynomial which has the following basis expressed in terms of the volume coordinates:
Using the relationship between the Cartesian and volume coordinates (Eq. A.13), the basis functions can be written in terms of the Cartesian coordinates x, y, and z as \(h_i =\sum _{j=1}^{20} {a_{i,j} b_j } \), where
and the coefficients \(a_{i,j} ,i=1,2,\ldots ,16,j=1,2,\ldots ,20\), can be systematically defined. One can show that the incomplete polynomial defined by the basis functions \(h_i =\sum _{j=1}^{20} {a_{i,j} b_j } \) has the linear terms 1, x, y, and z that ensure that the rigid body motion can be correctly described, and such a polynomial will have a symmetric structure in x, y, and z.
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Pappalardo, C.M., Wang, T. & Shabana, A.A. Development of ANCF tetrahedral finite elements for the nonlinear dynamics of flexible structures. Nonlinear Dyn 89, 2905–2932 (2017). https://doi.org/10.1007/s11071-017-3635-6
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DOI: https://doi.org/10.1007/s11071-017-3635-6