Hierarchic sets of shape functions constructed from enriched Fourier series

Abstract

Based on the concept of improving the convergence of Fourier series by elimination of the Gibbs phenomenon, this work proposes a rational approach to construct hierarchic sets of shape functions of any desired degree of continuity from suitably modified (enriched) trigonometric series. Hierarchic sets of either enriched cosines or enriched sines are then used in Ritz solutions of beam and plate bending problems to illustrate the benefits provided by the enrichment in the quality of approximations. For a given degree of continuity, it is observed that a set of enriched cosines yields, in general, more accurate results and faster convergence than a set of enriched sines. To the authors’ knowledge, there is no published information yet on hierarchic sets of enriched cosines. Moreover, it is shown that some hierarchic sets, which are already well-established in the literature, may lead not only to spurious oscillations but also to erroneous results.

Appendices

Appendix A

For $C^{0}$ continuity, the components of the set ${\mathbf{F}}_{c}$ of enriched cosines read

$$\begin{aligned} F_{c1} & = \frac{1}{4}\left( {3\xi + 1} \right)\left( {\xi - 1} \right)\;\;\;\;F_{c2} = \frac{1}{4}\left( {3\xi - 1} \right)\left( {\xi + 1} \right) \\ F_{cj} & = \cos \frac{\pi }{2}\left( {j - 3} \right)\left( {\xi + 1} \right) - F_{c1} + \left( { - 1} \right)^{j} F_{c2} \;\;\;\;j = 3, 4, \ldots \\ \end{aligned}$$

(A.1)

with properties

$$\begin{aligned} F_{c1} & = 1\;\;\;\;F_{c2} = F_{c3} = \ldots = 0 \;\;\;\;{\text{at}}\; \xi = - 1 \\ F_{c2} & = 1\;\;\;\;F_{c1} = F_{c3} = \ldots = 0\;\;\;\;{\text{at}}\; \xi = 1. \\ \end{aligned}$$

(A.2)

For numerical convenience, one replaces $x$ by the nondimensional coordinate $\xi = 2x/L - 1$ ($- 1 \le \xi \le 1$). The first two functions of ${\mathbf{F}}_{c}$ are the nodal components collected in ${\varvec{\psi}}$, whereas the remaining ones are the hierarchic components collected in ${\mathbf{H}}$. The components of the set ${\mathbf{F}}_{s}$ of enriched sines are

$$\begin{aligned} F_{s1} & = - \frac{1}{2}\left( {\xi - 1} \right)\;\;\;\;F_{s2} = \frac{1}{2}\left( {\xi + 1} \right) \\ F_{sj} & = \sin \frac{\pi }{2}\left( {j - 2} \right)\left( {\xi + 1} \right)\;\;\;\;j = 3, 4, \ldots \\ \end{aligned}$$

(A.3)

with the same properties (A.2).

Similarly, the components of ${\mathbf{F}}_{c}$ for $C^{1}$ continuity are given by

$$\begin{aligned} F_{c1} & = - \frac{1}{16}\left( {3\xi + 1} \right)\left( {5\xi + 7} \right)\left( {\xi - 1} \right)^{2} \;\;\;\;F_{c2} = - \frac{L}{32}\left( {\xi + 1} \right)\left( {5\xi + 1} \right)\left( {\xi - 1} \right)^{2} \\ F_{c3} & = - \frac{1}{16}\left( {3\xi - 1} \right)\left( {5\xi - 7} \right)\left( {\xi + 1} \right)^{2} \;\;\;\; F_{c4} = \frac{L}{32}\left( {\xi - 1} \right)\left( {5\xi - 1} \right)\left( {\xi + 1} \right)^{2} \\ F_{cj} & = \cos \frac{\pi }{2}\left( {j - 5} \right)\left( {\xi + 1} \right) - F_{c1} + \left( { - 1} \right)^{j} F_{c3} \;\;\;\;j = 5, 6, \ldots \\ \end{aligned}$$

(A.4)

with properties

$$\begin{aligned} F_{c1} & = 1\;\;\;\;F_{c2} = F_{c3} = F_{c4} = F_{c5} = \cdots = 0 \\ \frac{{{\text{d}}F_{c2} }}{{{\text{d}}x}} & = \frac{2}{L}\frac{{{\text{d}}F_{c2} }}{{{\text{d}}\xi }} = 1\;\;\;\;\frac{{{\text{d}}F_{c1} }}{{{\text{d}}\xi }} = \frac{{{\text{d}}F_{c3} }}{{{\text{d}}\xi }} = \frac{{{\text{d}}F_{c4} }}{{{\text{d}}\xi }} = \frac{{{\text{d}}F_{c5} }}{{{\text{d}}\xi }} = \cdots = 0\;\;\;\;{\text{at}}\;{ }\xi = - 1 \\ F_{c3} & = 1\;\;\;\;F_{c1} = F_{c2} = F_{c4} = F_{c5} = \cdots = 0 \\ \frac{{{\text{d}}F_{c4} }}{{{\text{d}}x}} & = \frac{2}{L}\frac{{{\text{d}}F_{c4} }}{{{\text{d}}\xi }} = 1\;\;\;\;\frac{{{\text{d}}F_{c1} }}{{{\text{d}}\xi }} = \frac{{{\text{d}}F_{c2} }}{{{\text{d}}\xi }} = \frac{{{\text{d}}F_{c3} }}{{{\text{d}}\xi }} = \frac{{{\text{d}}F_{c5} }}{{{\text{d}}\xi }} = \cdots = 0\;\;\;\;{\text{at}}\;\xi = 1. \\ \end{aligned}$$

(A.5)

The components of ${\mathbf{F}}_{s}$ are

$$\begin{aligned} F_{s1} & = \frac{1}{4}\left( {\xi + 2} \right)\left( {\xi - 1} \right)^{2} \;\;\;\;F_{s2} = \frac{L}{8}\left( {\xi + 1} \right)\left( {\xi - 1} \right)^{2} \\ F_{s3} & = - \frac{1}{4}\left( {\xi - 2} \right)\left( {\xi + 1} \right)^{2} \;\;\;\;F_{s4} = \frac{L}{8}\left( {\xi - 1} \right)\left( {\xi + 1} \right)^{2} \\ F_{sj} & = \sin \frac{\pi }{2}\left( {j - 4} \right)\left( {\xi + 1} \right) - \frac{\pi }{L}\left( {j - 4} \right)\left[ {F_{s2} + \left( { - 1} \right)^{j} F_{s4} } \right]\;\;\;\; j = 5, 6, \ldots \\ \end{aligned}$$

(A.6)

with the same properties (A.5).

For $C^{2}$ continuity, the components of ${\mathbf{F}}_{c}$ change to

$$\begin{aligned} F_{c1} & = \frac{1}{32}\left( {\xi - 1} \right)^{3} \left( {35\xi^{3} + 99\xi^{2} + 87\xi + 19} \right) \\ F_{c2} & = \frac{L}{32}\left( {\xi + 1} \right)\left( {\xi - 1} \right)^{3} \left( {7\xi^{2} + 11\xi + 2} \right) \\ F_{c3} & = \frac{{L^{2} }}{384}\left( {\xi + 1} \right)^{2} \left( {\xi - 1} \right)^{3} \left( {7\xi + 1} \right) \\ F_{c4} & = \frac{1}{32}\left( {\xi + 1} \right)^{3} \left( {35\xi^{3} - 99\xi^{2} + 87\xi - 19} \right) \\ F_{c5} & = - \frac{L}{32}\left( {\xi - 1} \right)\left( {\xi + 1} \right)^{3} \left( {7\xi^{2} - 11\xi + 2} \right) \\ F_{c6} & = \frac{{L^{2} }}{384}\left( {\xi - 1} \right)^{2} \left( {\xi + 1} \right)^{3} \left( {7\xi - 1} \right) \\ F_{cj} & = \cos \frac{\pi }{2}\left( {j - 7} \right)\left( {\xi + 1} \right) - F_{c1} + \left( { - 1} \right)^{j} F_{c4} + \left( {\frac{\pi }{L}} \right)^{2} \left( {j - 7} \right)^{2} \left[ {F_{c3} - \left( { - 1} \right)^{j} F_{c6} } \right] \;\;\;\;j = 7, 8, \ldots \\ \end{aligned}$$

(A.7)

with properties

$$\begin{aligned} F_{c1} & = 1\;\;\;\;F_{c2} = F_{c3} = F_{c4} = F_{c5} = F_{c6} = F_{c7} = \cdots = 0 \\ \frac{{{\text{d}}F_{c2} }}{{{\text{d}}x}} & = \frac{2}{L}\frac{{{\text{d}}F_{c2} }}{{{\text{d}}\xi }} = 1\;\;\;\;\frac{{{\text{d}}F_{c1} }}{{{\text{d}}\xi }} = \frac{{{\text{d}}F_{c3} }}{{{\text{d}}\xi }} = \frac{{{\text{d}}F_{c4} }}{d\xi } = \frac{{{\text{d}}F_{c5} }}{{{\text{d}}\xi }} = \frac{{{\text{d}}F_{c6} }}{{{\text{d}}\xi }} = \frac{{{\text{d}}F_{c7} }}{{{\text{d}}\xi }} = \cdots = 0 \\ \frac{{{\text{d}}^{2} F_{c3} }}{{{\text{d}}x^{2} }} & = \frac{4}{{L^{2} }}\frac{{{\text{d}}^{2} F_{c3} }}{{{\text{d}}\xi^{2} }} = 1\;\;\;\;\frac{{{\text{d}}^{2} F_{c1} }}{{{\text{d}}\xi^{2} }} = \frac{{{\text{d}}^{2} F_{c2} }}{{{\text{d}}\xi^{2} }} = \frac{{{\text{d}}^{2} F_{c4} }}{{{\text{d}}\xi^{2} }} = \frac{{{\text{d}}^{2} F_{c5} }}{{{\text{d}}\xi^{2} }} = \frac{{{\text{d}}^{2} F_{c6} }}{{{\text{d}}\xi^{2} }} = \frac{{{\text{d}}^{2} F_{c7} }}{{{\text{d}}\xi^{2} }} = \cdots = 0\;\;\;\;{\text{at}} \;\xi = - 1 \\ F_{c4} & = 1\;\;\;\;F_{c1} = F_{c2} = F_{c3} = F_{c5} = F_{c6} = F_{c7} = \cdots = 0 \\ \frac{{{\text{d}}F_{c5} }}{{{\text{d}}x}} & = \frac{2}{L}\frac{{{\text{d}}F_{c5} }}{{{\text{d}}\xi }} = 1\;\;\;\; \frac{{{\text{d}}F_{c1} }}{{{\text{d}}\xi }} = \frac{{{\text{d}}F_{c2} }}{{{\text{d}}\xi }} = \frac{{{\text{d}}F_{c3} }}{{{\text{d}}\xi }} = \frac{{{\text{d}}F_{c4} }}{{{\text{d}}\xi }} = \frac{{{\text{d}}F_{c6} }}{{{\text{d}}\xi }} = \frac{{{\text{d}}F_{c7} }}{{{\text{d}}\xi }} = \cdots = 0 \\ \frac{{{\text{d}}^{2} F_{c6} }}{{{\text{d}}x^{2} }} & = \frac{4}{{L^{2} }}\frac{{{\text{d}}^{2} F_{c6} }}{{{\text{d}}\xi^{2} }} = 1\;\;\;\;\frac{{{\text{d}}^{2} F_{c1} }}{{{\text{d}}\xi^{2} }} = \frac{{{\text{d}}^{2} F_{c2} }}{{{\text{d}}\xi^{2} }} = \frac{{{\text{d}}^{2} F_{c3} }}{{{\text{d}}\xi^{2} }} = \frac{{{\text{d}}^{2} F_{c4} }}{{{\text{d}}\xi^{2} }} = \frac{{{\text{d}}^{2} F_{c5} }}{{{\text{d}}\xi^{2} }} = \frac{{{\text{d}}^{2} F_{c7} }}{{{\text{d}}\xi^{2} }} = \cdots = 0\;\;\;\;{\text{at}}\; \xi = 1. \\ \end{aligned}$$

(A.8)

In turn, the components of ${\mathbf{F}}_{s}$ are given by

$$\begin{aligned} F_{s1} & = - \frac{1}{16}\left( {\xi - 1} \right)^{3} \left( {3\xi^{2} + 9\xi + 8} \right)\;\;\;\;F_{s2} = - \frac{L}{32}\left( {\xi + 1} \right)\left( {3\xi + 5} \right)\left( {\xi - 1} \right)^{3} \\ F_{s3} & = - \frac{{L^{2} }}{64}\left( {\xi + 1} \right)^{2} \left( {\xi - 1} \right)^{3} \;\;\;\;F_{s4} = \frac{1}{16}\left( {\xi + 1} \right)^{3} \left( {3\xi^{2} - 9\xi + 8} \right) \\ F_{s5} & = - \frac{L}{32}\left( {\xi - 1} \right)\left( {3\xi - 5} \right)\left( {\xi + 1} \right)^{3} \;\;\;\;F_{s6} = \frac{{L^{2} }}{64}\left( {\xi - 1} \right)^{2} \left( {\xi + 1} \right)^{3} \\ F_{sj} & = \sin \frac{\pi }{2}\left( {j - 6} \right)\left( {\xi + 1} \right) - \frac{\pi }{L}\left( {j - 6} \right)\left[ {F_{s2} + \left( { - 1} \right)^{j} F_{s5} } \right] \;\;\;\;j = 7, 8, \ldots \\ \end{aligned}$$

(A.9)

with the same properties (A.8).

The sets (A.3), (A.6) and (A.9) are also developed in [21, 23] and [22], respectively, but without worrying about convergence aspects of the sine series expansion. The hierarchic functions $F_{sj}$ ($j = 5, 6, \ldots$) of the set (A.6) were first proposed by Iguchi [6].

Appendix B

2.1 Beam bending

It is convenient to rewrite the potential energy (48) as follows

$${\Pi }\left( v \right) = \frac{L}{2}\mathop \int \limits_{ - 1}^{1} \left[ {\frac{8EI}{{L^{4} }}v_{,\xi \xi }^{2} + q_{0} e^{\xi + 1} v} \right]{\text{d}}\xi ,$$

(B.1)

where a comma followed by $\xi$ indicates differentiation with respect to $\xi$. Substitution of the approximation $v\left( \xi \right) \approx {\mathbf{H}}^{T} {\mathbf{h}}$ into (B.1) yields

$${\Pi }\left( {\mathbf{h}} \right) = {\mathbf{h}}^{T} \left( {\frac{1}{2}{\mathbf{kh}} - {\mathbf{f}}} \right)$$

(B.2)

with

$${\mathbf{k}} = \frac{8EI}{{L^{3} }}\mathop \int \limits_{ - 1}^{1} {\mathbf{H}}_{,\xi \xi } {\mathbf{H}}_{,\xi \xi }^{T} {\text{d}}\xi \;\;\;\;{\mathbf{f}} = - \frac{{q_{0} L}}{2}\mathop \int \limits_{ - 1}^{1} e^{\xi + 1} {\mathbf{H}}{\text{d}}\xi .$$

(B.3)

The discrete form of the equations that govern the response of the beam bending is given by the stationary condition of ${\Pi }$ with respect to ${\mathbf{h}}$:

$$\frac{{\partial {\Pi }}}{{\partial {\mathbf{h}}}} = \textbf{0} \;\; \Rightarrow \;\; {\mathbf{kh}} = {\mathbf{f}}.$$

(B.4)

2.2 Plate bending

In the nondimensional coordinates $\xi = 2x/a - 1$ and $\eta = 2y/b - 1$, the potential energy (52) reads

$${\Pi }\left( w \right) = \mathop \int \limits_{ - 1}^{1} \mathop \int \limits_{ - 1}^{1} \left\{ {\frac{2D}{{ab}}\left[ {\frac{{b^{2} }}{{a^{2} }}w_{,\xi \xi }^{2} + 2\nu w_{,\xi \xi } w_{,\eta \eta } + \frac{{a^{2} }}{{b^{2} }}w_{,\eta \eta }^{2} + 2\left( {1 - \nu } \right)w_{,\xi \eta }^{2} } \right] + \frac{ab}{4}q_{0} w} \right\}{\text{d}}\xi {\text{d}}\eta .$$

(B.5)

Substitution of the approximation (55), now written as $w\left( {\xi ,\eta } \right) \approx {\mathbf{R}}^{{\text{T}}} {\mathbf{c}}$ in which ${\mathbf{R}}$ and ${\mathbf{c}}$ collect the Ritz basis functions and the unknown coefficients, into (B.5) gives

$${\Pi }\left( {\mathbf{c}} \right) = {\mathbf{c}}^{T} \left( {\frac{1}{2}{\mathbf{kc}} - {\mathbf{f}}} \right)$$

(B.6)

with

$$\begin{aligned} {\mathbf{k}} & = \frac{4D}{{ab}}\mathop \int \limits_{ - 1}^{1} \mathop \int \limits_{ - 1}^{1} \left[ {\frac{{b^{2} }}{{a^{2} }}{\mathbf{R}}_{,\xi \xi } {\mathbf{R}}_{,\xi \xi }^{T} + 2\nu {\mathbf{R}}_{,\xi \xi } {\mathbf{R}}_{,\eta \eta }^{T} + \frac{{a^{2} }}{{b^{2} }}{\mathbf{R}}_{,\eta \eta } {\mathbf{R}}_{,\eta \eta }^{T} + 2\left( {1 - \nu } \right){\mathbf{R}}_{,\xi \eta } {\mathbf{R}}_{,\xi \eta }^{T} } \right]d\xi {\text{d}}\eta \\ {\mathbf{f}} & = - \frac{ab}{4}\mathop \int \limits_{ - 1}^{1} \mathop \int \limits_{ - 1}^{1} q_{0} {\mathbf{R}}d\xi {\text{d}}\eta . \\ \end{aligned}$$

(B.7)

The discrete form of the equations that govern the response of the plate bending is

$$\frac{{\partial {\Pi }}}{{\partial {\mathbf{c}}}} = \textbf{0}\;\; \Rightarrow \;\; {\mathbf{kc}} = {\mathbf{f}}.$$

(B.8)

The Ritz basis can be expressed by means of the Kronecker product [38] as

$${\mathbf{R}}\left( {\xi ,\eta } \right) = {\mathbf{X}}\left( \xi \right) \otimes {\mathbf{Y}}\left( \eta \right),$$

(B.9)

retaining $M$ and $N$ shape functions in the sets ${\mathbf{X}}$ and ${\mathbf{Y}}$, respectively.