A scaled boundary shell element formulation using Neumann expansion

Abstract

This paper proposes a new shell element formulation using the scaled boundary finite element (SBFE) method. A shell element is treated as a three-dimensional continuum. Its bottom surface is approximated with a quadrilateral spectral element and the shell geometry is represented through normal scaling of the bottom surface. Neumann expansion is applied to approximate the inversions of the matrix polynomials of the thickness coordinate ξ, including the Jacobian matrix and the coefficient of the second-order term in the SBFE equation. This permits the solution along the thickness to be expressed as a matrix exponential function whose exponent is a high-order matrix polynomial of ξ. After introducing the boundary conditions on the top and bottom surfaces and evaluating the resulting matrix exponential via Padé expansion, we derive the element stiffness and mass matrices. Poisson thickness locking is avoided fundamentally. Numerical examples demonstrate the applicability and efficiency of the formulation.

Appendix

In Eq. (62), ${\mathbf{E}}_{i}^{ - }$ (i = 0, 1, …, 6) are expressed as follows

$$ {\mathbf{E}}_{0}^{ - } = ({\mathbf{E}}_{0}^{0} )^{ - 1} $$

(93)

$$ {\mathbf{E}}_{1}^{ - } = {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } $$

(94)

$$ {\mathbf{E}}_{2}^{ - } = {\mathbf{E}}_{0}^{ - } \left( {{\mathbf{E}}_{2}^{0} - {\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} } \right){\mathbf{E}}_{0}^{ - } $$

(95)

$$ {\mathbf{E}}_{3}^{ - } {=} {\mathbf{E}}_{0}^{ - } \left[ {{\mathbf{E}}_{3}^{0} {-} \left( {{\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{2}^{0} {+} {\mathbf{E}}_{2}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} } \right) {+} \left( {{\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } } \right)^{2} {\mathbf{E}}_{1}^{0} } \right]{\mathbf{E}}_{0}^{ - } $$

(96)

$$ {\mathbf{E}}_{4}^{ - } = {\mathbf{E}}_{0}^{ - } \left( \begin{gathered} {\mathbf{E}}_{4}^{0} - \left( {{\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{3}^{0} + {\mathbf{E}}_{2}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{2}^{0} + {\mathbf{E}}_{3}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} } \right) \hfill \\ + \left[ {\left( {{\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } } \right)^{2} {\mathbf{E}}_{2}^{0} + {\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{2}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} + {\mathbf{E}}_{2}^{0} \left( {{\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} } \right)^{2} } \right] - \left( {{\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } } \right)^{3} {\mathbf{E}}_{1}^{0} \hfill \\ \end{gathered} \right){\mathbf{E}}_{0}^{ - } $$

(97)

$$ {\mathbf{E}}_{5}^{ - } = {\mathbf{E}}_{0}^{ - } \left( \begin{gathered} {\mathbf{E}}_{5}^{0} - \left( {{\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{4}^{0} + {\mathbf{E}}_{2}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{3}^{0} + {\mathbf{E}}_{3}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{2}^{0} + {\mathbf{E}}_{4}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} } \right) \hfill \\ + \left( \begin{gathered} \left( {{\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } } \right)^{2} {\mathbf{E}}_{3}^{0} + {\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{2}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{2}^{0} + {\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{3}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} \hfill \\ + {\mathbf{E}}_{2}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{2}^{0} + {\mathbf{E}}_{2}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{2}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} + {\mathbf{E}}_{3}^{0} \left( {{\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} } \right)^{2} \hfill \\ \end{gathered} \right) \hfill \\ - \left[ \begin{gathered} \left( {{\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } } \right)^{2} \left( {{\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{2}^{0} + {\mathbf{E}}_{2}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} } \right) \hfill \\ + \left( {{\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{2}^{0} + {\mathbf{E}}_{2}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} } \right)\left( {{\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} } \right)^{2} \hfill \\ \end{gathered} \right] \hfill \\ + \left( {{\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } } \right)^{4} {\mathbf{E}}_{1}^{0} \hfill \\ \end{gathered} \right){\mathbf{E}}_{0}^{ - } $$

(98)

$$ {\mathbf{E}}_{6}^{ - } = {\mathbf{E}}_{0}^{ - } \left( \begin{gathered} {\mathbf{E}}_{6}^{0} - \left( {{\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{5}^{0} + {\mathbf{E}}_{2}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{4}^{0} + {\mathbf{E}}_{3}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{3}^{0} + {\mathbf{E}}_{4}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{2}^{0} + {\mathbf{E}}_{5}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} } \right) \hfill \\ + \left[ \begin{gathered} \left( {{\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } } \right)^{2} {\mathbf{E}}_{4}^{0} + {\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{2}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{3}^{0} + {\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{3}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{2}^{0} + {\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{4}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} \hfill \\ + {\mathbf{E}}_{2}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{3}^{0} + {\mathbf{E}}_{2}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{2}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{2}^{0} + {\mathbf{E}}_{2}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{3}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} \hfill \\ + {\mathbf{E}}_{3}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{2}^{0} + {\mathbf{E}}_{3}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{2}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} + {\mathbf{E}}_{4}^{0} \left( {{\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} } \right)^{2} \hfill \\ \end{gathered} \right] \hfill \\ - \left[ \begin{gathered} \left( {{\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } } \right)^{2} \left( {{\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{3}^{0} + {\mathbf{E}}_{2}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{2}^{0} + {\mathbf{E}}_{3}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} } \right) \hfill \\ + \left( {{\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{2}^{0} + {\mathbf{E}}_{2}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} } \right){\mathbf{E}}_{0}^{ - } \left( {{\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{2}^{0} + {\mathbf{E}}_{2}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} } \right) \hfill \\ + \left( {{\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{3}^{0} + {\mathbf{E}}_{2}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{2}^{0} + {\mathbf{E}}_{3}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} } \right)\left( {{\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} } \right)^{2} \hfill \\ \end{gathered} \right] \hfill \\ + \left[ \begin{gathered} \left( {{\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } } \right)^{3} \left( {{\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{2}^{0} + {\mathbf{E}}_{2}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} } \right) + \left( {{\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } } \right)^{2} {\mathbf{E}}_{2}^{0} \left( {{\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} } \right)^{2} \hfill \\ + \left( {{\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{2}^{0} + {\mathbf{E}}_{2}^{0} {\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} } \right)\left( {{\mathbf{E}}_{0}^{ - } {\mathbf{E}}_{1}^{0} } \right)^{3} \hfill \\ \end{gathered} \right] \hfill \\ - \left( {{\mathbf{E}}_{1}^{0} {\mathbf{E}}_{0}^{ - } } \right)^{5} {\mathbf{E}}_{1}^{0} \hfill \\ \end{gathered} \right)^{\vphantom{\dfrac{\sum_{n}^{18}}{18}}}{\mathbf{E}}_{0}^{ - } $$

(99)

In Eq. (63), ${\mathbf{Z}}_{i}^{jk}$ (j, k = 1, 2; i = 0, 1, …, 6) are expressed as follows

$$ {\mathbf{Z}}_{i}^{11} = {\mathbf{E}}_{0}^{ - } ({\mathbf{E}}_{i}^{1} )^{{\text{T}}} - \sum\limits_{s = 1}^{i} {\left[ {{\mathbf{E}}_{s}^{ - } ({\mathbf{E}}_{i - s}^{1} )^{{\text{T}}} } \right]} \, (i = 0,1,...,6) $$

(100)

$$ {\mathbf{Z}}_{i}^{12} = {\mathbf{E}}_{i}^{ - } \, (i = 0,1,...,6) $$

(101)

$$\begin{aligned} {\mathbf{Z}}_{i}^{21} &= \omega^{2} {\mathbf{M}}_{i}^{0} - {\mathbf{E}}_{i}^{2} + \sum\limits_{t = 0}^{i} \\&\quad\times{\left( {\left[ {{\mathbf{E}}_{t}^{1} {\mathbf{E}}_{0}^{ - } - \sum\limits_{s = 1}^{t} {\left( {{\mathbf{E}}_{t - s}^{1} {\mathbf{E}}_{s}^{ - } } \right)} } \right]({\mathbf{E}}_{i - t}^{1} )^{{\text{T}}} } \right)} \, (i = 0,1,2) \end{aligned}$$

(102)

$$ \begin{aligned}{\mathbf{Z}}_{i}^{21} &= - {\mathbf{E}}_{i}^{2} + \sum\limits_{t = 0}^{i} {\left( {\left[ {{\mathbf{E}}_{t}^{1} {\mathbf{E}}_{0}^{ - } - \sum\limits_{s = 1}^{t} }\right.}\right.}\\&\quad\times {\left.{\left.{{\left( {{\mathbf{E}}_{t - s}^{1} {\mathbf{E}}_{s}^{ - } } \right)} } \right]({\mathbf{E}}_{i - t}^{1} )^{{\text{T}}} } \right)} \, (i = 3,4,5,6) \end{aligned}$$

(103)

$$ {\mathbf{Z}}_{i}^{22} = - {\mathbf{E}}_{i}^{1} {\mathbf{E}}_{0}^{ - } + \sum\limits_{s = 1}^{i} {\left( {{\mathbf{E}}_{i - s}^{1} {\mathbf{E}}_{s}^{ - } } \right)} \, (i = 0,1,...,6) $$

(104)