Frequency convergence characteristics of lumped mass Galerkin meshfree methods

Abstract

Due to the considerable complexity of meshfree approximants, there still lacks a rational theoretical estimate on the frequency accuracy of lumped mass meshfree methods. In this study, a detailed theoretical analysis is presented for the lumped mass Galerkin meshfree formulation with particular reference to the frequency accuracy. The proposed accuracy analysis is quite general and is fully based upon the reproducing or consistency conditions of meshfree shape functions and their gradients. The theoretical findings reveal that there exists an important and interesting odd/even basis degree discrepancy feature for the frequency accuracy, i.e., (p + 2)th and (p + 1)th order accurate frequencies are attained by the lumped mass Galerkin meshfree formulation using even and odd degrees of basis functions, where p is the degree of basis function employed in the meshfree shape function construction. Furthermore, it is found that interpolatory meshfree shape functions are capable of not only circumventing the non-physical modes arising from the lumped mass Galerkin meshfree formulation using the standard non-interpolatory shape functions, but also elevating the frequency accuracy for meshfree structural vibration analysis. Numerical examples well replicate the theoretical frequency accuracy analysis results for lumped mass Galerkin meshfree methods.

Appendix

(1) Proof of Eq. (42).

In Eq. (42), $\sum\limits_{{J \in S_{J} }} {\Psi_{{J,x}} (x)(x_{J} - x_{I} )^{q} }$ can be written as:

$$ \begin{aligned}\begin{aligned} & \sum\limits_{{J \in S_{J} }} {\Psi_{{J,x}} (x)(x_{J} - x_{I} )^{q} } \\ & \quad = \sum\limits_{{J \in S_{J} }} {\Psi_{{J,x}} (x)(x_{J} - x + x - x_{I} )^{q} } \\ & \quad = \sum\limits_{{J \in S_{J} }} {\Psi_{{J,x}} (x)\sum\limits_{k = 0}^{q} {C_{q}^{k} (x_{J} - x)^{q - k} (x - x_{I} )^{k} } } \end{aligned} \end{aligned} $$

(A.1)

Therefore, according to Eq. (13), Eq. (A.1) can be ultimately recast as:

$$ \begin{aligned}\begin{aligned} & \sum\limits_{{J \in S_{J} }} {\Psi_{{J,x}} (x)(x_{J} - x_{I} )^{q} } \\ & \quad = C_{q}^{q - 1} \underbrace {{\sum\limits_{{J \in S_{J} }} {\Psi_{{J,x}} (x)(x_{J} - x)} }}_{ = 1}(x - x_{I} )^{q - 1} \\ & \quad = q(x - x_{I} )^{q - 1} \\ \end{aligned}\end{aligned} $$

(A.2)

(2) Proof of Eq. (43).

In accordance with Eq. (42), Eq. (43) can be simplified as:

$$ \begin{gathered} \int_{\Omega } {\Psi_{{I,x}} (x)\underbrace {{\sum\limits_{{J \in S_{J} }} {\Psi_{{J,x}} (x)(x_{J} - x_{I} )^{q} } }}_{{ = q(x - x_{I} )^{q - 1} }}(x - x_{I} )^{2} dx} \hfill \\ = q\int_{\Omega } {\Psi_{{I,x}} (x)(x - x_{I} )^{q + 1} dx} \hfill \\ \end{gathered} $$

(A.3)

Through the integration by parts, Eq. (A.3) can be deduced as follows:

$$ \begin{aligned} & q\int_{\Omega } {\Psi_{{I,x}} (x)(x - x_{I} )^{q + 1} dx} \\ & \quad = q\int_{\Omega } {(x - x_{I} )^{q + 1} d(\Psi_{I} (x))} \\ & \quad = q\left[ {\underbrace {{\Psi_{I} (x)(x - x_{I} )^{q + 1} |_{\Omega } }}_{ = 0} - (q + 1)\int_{\Omega } {\Psi_{I} (x)(x - x_{I} )^{q} dx} } \right] \\ & \quad = - q(q + 1)\int_{\Omega } {\Psi_{I} (x)(x - x_{I} )^{q} dx} \\ \end{aligned} $$

(A.4)

Subsequently, by employing the periodicity and symmetry of meshfree shape functions for uniform discretizations, i.e., Eq. (39), we have:

$$ \begin{gathered} \int_{\Omega } {\Psi_{I} (x)(x - x_{I} )^{q} dx}\hfill \\ \quad = \sum\limits_{{M \in S_{(I + M)I} }} {\int_{{x_{I - M} }}^{{x_{I - M + 1} }} {\Psi_{I} (x)(x - x_{I} )^{q} dx} }\hfill \\ \quad = \sum\limits_{{M \in S_{(I + M)I} }} {\int_{{x_{I - M} }}^{{x_{I - M + 1} }} {\Psi_{I + M} (x + x_{I + M} - x_{I} )(x - x_{I} )^{q} dx} } \hfill \\ \quad = \sum\limits_{{M \in S_{(I + M)I} }} {\int_{{x_{I - M} + x_{I + M} - x_{I} }}^{{x_{I - M + 1} + x_{I + M} - x_{I} }} {\Psi_{I + M} (x)(x - x_{I + M} )^{q} dx} } \hfill \\ \quad = \int_{{x_{I} }}^{{x_{I} + h}} {\sum\limits_{{(I + M) \in S_{I + M} }} {\Psi_{I + M} (x)(x - x_{I + M} )^{q} } } dx \hfill \\ \end{gathered} $$

(A.5)

Letting $q = 0$ in Eq. (A.5) gives:

$$ \begin{gathered} \int_{\Omega } {\Psi_{I} (x)(x - x_{I} )^{q} dx} \hfill \\ = \int_{{x_{I} }}^{{x_{I} + h}} {\underbrace {{\sum\limits_{{(I + M) \in S_{I + M} }} {\Psi_{I + M} (x)(x - x_{I + M} )^{q} } }}_{ = 1}} dx \hfill \\ = h \hfill \\ \end{gathered} $$

(A.6)

In case of $q \ne 0$, Eq. (A.5) becomes:

$$ \begin{gathered} \int_{\Omega } {\Psi_{I} (x)(x - x_{I} )^{q} dx} \hfill \\ = \int_{{x_{I} }}^{{x_{I} + h}} {\underbrace {{\sum\limits_{{(I + M) \in S_{I + M} }} {\Psi_{I + M} (x)(x - x_{I + M} )^{q} } }}_{ = 0}} dx \hfill \\ = 0 \hfill \\ \end{gathered} $$

(A.7)

Therefore, with the aid of Eqs. (A.4)-(A.7), Eq. (A.3) reduces to:

$$ \int_{\Omega } {\Psi_{{I,x}} (x)\sum\limits_{{J \in S_{J} }} {\Psi_{{J,x}} (x)(x_{J} - x_{I} )^{q} } (x - x_{I} )^{2} dx} = 0 $$

(A.8)

where $0 \le q \le p$.

(3) Proof of Eq. (44).

Substituting Eq. (40) into Eq. (44) gives:

$$ \begin{gathered} \int_{\Omega } {\Psi_{{I,x}} (x)(x - x_{I} )^{q} dx}\hfill\\ \quad = \sum\limits_{{N \in S_{(N + I)I} }} {\int_{{x_{I - N} }}^{{x_{I - N + 1} }} {\Psi_{{I,x}} (x)(x - x_{I} )^{q} dx} } \hfill\\ \quad = \sum\limits_{{N \in S_{(N + I)I} }} {\int_{{x_{I - N} }}^{{x_{I - N + 1} }} {\Psi_{{I + N,x}} (x + x_{I + N} - x_{I} )(x - x_{I} )^{q} dx} } \hfill\\ \quad = \sum\limits_{{N \in S_{(N + I)I} }} {\int_{{x_{I - N} + x_{I + N} - x_{I} }}^{{x_{I - N + 1} + x_{I + N} - x_{I} }} {\Psi_{{I + N,x}} (x)(x - x_{I + N} )^{q} dx} } \hfill\\ \quad = \int_{{x_{I} }}^{{x_{I} + h}} {\sum\limits_{{(N + I) \in S_{N + I} }} {\Psi_{{I + N,x}} (x)(x - x_{I + N} )^{q} } dx} \hfill\\ \end{gathered} $$

(A.9)

With the aid of Eq. (13), Eq. (A.9) becomes:

$$ \begin{gathered} \int_{\Omega } {\Psi_{{I,x}} (x)(x - x_{I} )^{q} dx}\hfill \\ \quad = \int_{{x_{I} }}^{{x_{I} + h}} {\sum\limits_{{(N + I) \in S_{N + I} }} {\Psi_{{I + N,x}} (x)(x - x_{I + N} )^{q} } dx} \hfill \\ \quad = \left\{ \begin{gathered} \int_{{x_{I} }}^{{x_{I} + h}} {\underbrace {{\sum\limits_{{(N + I) \in S_{N + I} }} {\Psi_{{I + N,x}} (x)(x - x_{I + N} )} }}_{ = - 1}dx} = - h, \, q = 1 \hfill \\ \int_{{x_{I} }}^{{x_{I} + h}} {\underbrace {{\sum\limits_{{(N + I) \in S_{N + I} }} {\Psi_{{I + N,x}} (x)(x - x_{I + N} )^{q} } }}_{ = 0}dx} = 0, \, q \ne 1 \hfill \\ \end{gathered} \right. \\ \end{gathered} $$

(A.10)

(4) Proof of Eq. (60).

In accordance with Eq. (40), ${\mathcal{C}}_{1}$ becomes:

$$ \begin{aligned} {\mathcal{C}}_{1} &= \int_{\Omega } {\Psi_{{I,x}} (x)\sum\limits_{{J \in S_{J} }} {\Psi_{{J,x}} (x)(x_{J} - x_{I} )^{2n} (x_{J} - x)^{2} } dx} \\&= \sum\limits_{{N \in S_{{(I{ + }N)I}} }} \int_{{x_{I} }}^{{x_{I} +h}} \Psi_{{I + N,x}} (x)\sum\limits_{{M \in S_{{(I{ + }M)(I{ +}N)}} }}\\&\quad \left\{\Psi_{{I + M,x}} (x) (x_{I + M} - x_{I + N})^{2n} (x_{I + M} - x)^{2} \right\}dx \\&= \int_{{x_{I} }}^{{x_{I} + h}}\!\! {\sum\limits_{{M \in S_{(I + M)(I + N)} }} \!{\left\{ \begin{gathered} \underbrace {{\sum\limits_{{I + N \in S_{(I + N)I} }} {\Psi_{{I + N,x}} (x)} (x_{I + M} - x_{I + N} )^{2n} }}_{{ = (2n)(x - x_{I + M} )^{2n - 1} }} \\ \times \Psi_{{I + M,x}} (x)(x_{I + M} - x)^{2} \\ \end{gathered} \right\}dx} }\; \\&= (2n)\int_{{x_{I} }}^{{x_{I} + h}} {\underbrace {{\sum\limits_{{(I + M) \in S_{{I{ + }M}} }} {\Psi_{{I + M,x}} (x)(x - x_{I + M} )^{2n + 1} } }}_{ = 0}dx} \\ &= 0 \\\end{aligned} $$

(A.11)

(5) Proof of Eq. (62).

Based upon the binomial expansion approach and the completeness conditions of the derivatives for meshfree shape functions as indicated in Eq. (13), we have:

$$ \begin{aligned} {\mathcal{C}}_{3} &= 2\int_{\Omega } \Psi_{{I,x}} (x)\sum\limits_{{J \in S_{J} }} \Psi_{{J,x}} (x)(x_{J} - x_{I} )^{2n}\\&\quad \times \left[ {(x_{J} - x)(x - x_{I} )} \right]dx \\&= 2\int_{\Omega } \Psi_{{I,x}} (x)\sum\limits_{{J \in S_{J} }}\Psi_{{J,x}} (x)(x_{J} - x + x - x_{I} )^{2n}\\&\quad \times \left[ {(x_{J} - x)(x - x_{I} )} \right]dx \\&= 2\sum\limits_{k = 0}^{2n} {C_{2n}^{k} } \int_{\Omega }\Psi_{{I,x}} (x)\sum\limits_{{J \in S_{J} }} \Psi_{{J,x}}(x)\\&\quad \times (x_{J} - x)^{2n + 1 - k} (x - x_{I} )^{k + 1} dx \\&= 2C_{2n}^{0} \int_{\Omega } \Psi_{{I,x}}(x)\underbrace{{\sum\limits_{{J \in S_{J} }} \Psi_{{J,x}} (x)(x_{J}- x)^{2n + 1} }}_{ = 0}\\&\quad \times (x - x_{I} )dx + \cdots \\&\quad + 2C_{2n}^{2n}\int_{\Omega } {\Psi_{{I,x}} (x)\underbrace {{\sum\limits_{{J \in S_{J} }} {\Psi_{{J,x}} (x)(x_{J} - x)^{1} } }}_{ = 1}(x - x_{I})^{2n + 1} dx} \\&= 2C_{2n}^{2n} \underbrace {{\int_{\Omega }{\Psi_{{I,x}} (x)(x - x_{I} )^{2n + 1} dx} }}_{ = 0} = 0 \\\end{aligned} $$

(A.12)

(6) Proof of Eq. (66).

Following the similar routine used in the proof of Eq. (60), ${\mathcal{C}}_{1}$ in Eq. (66) can be obtained as follows:

$$ \begin{aligned} {\mathcal{C}}_{1} &=\int_{\Omega } {\Psi_{{I,x}} (x)\sum\limits_{{J \in S_{J} }}{\Psi_{{J,x}} (x)(x_{J} - x_{I} )^{2n} (x_{J} - x)^{2} } dx}\\&=\sum\limits_{{N \in S_{(I + N)I} }} \int_{{x_{I - N} }}^{{x_{I -N +1} }} \Psi_{{I,x}} (x)\sum\limits_{{M \in S_{(I - N + M)I} }}\Big\{\Psi_{{I - N + M,x}} (x)\\&\quad \times (x_{I - N + M}-x_{I} )^{2n} (x_{I - N + M} - x)^{2}\Big\}dx \\&= \sum\limits_{{N \in S_{(I + N)I} }} \int_{{x_{I - N} }}^{{x_{I -N + 1} }}\sum\limits_{{M \in S_{(I + M)(I + N)} }} \\&\quad \Big\{\Psi_{{I + N,x}} (x + x_{I + N} - x_{I} )\Psi_{{I + M,x}} (x + x_{I + N} - x_{I} )\\&\quad \times (x_{I - N + M} - x_{I})^{2n} (x_{I - N + M} - x)^{2}\Big\}dx \\&= \sum\limits_{{N \in S_{(I + N)I} }}\int_{{x_{I - N} + x_{I + N}- x_{I} }}^{{x_{I - N + 1} + x_{I + N} -x_{I} }} \sum\limits_{{M \in S_{(I + M)(I + N)} }} \\&\quad \Big\{ \Psi_{{I + N,x}} (x)\Psi_{{I + M,x}} (x) (x_{I + M} - x_{I +N} )^{2n} (x_{I - N + M} - x\\&\quad + x_{I + N} - x_{I})^{2}\Big\}dx \\ &= \sum\limits_{{N \in S_{(I + N)I} }} \int_{{x_{I}}}^{{x_{I} + h}} \Psi_{{I + N,x}}(x)\sum\limits_{{M \in S_{(I + M)(I + N)} }}\\&\quad \Big\{ \Psi_{{I + M,x}} (x) (x_{I + M} -x_{I + N})^{2n} (x_{I + M} - x)^{2} \Big\}dx \\ &= \int_{{x_{I}}}^{{x_{I}+ h}} \sum\limits_{{M \in S_{(I + M)(I + N)} }}\Bigg\{ \sum\limits_{{I + N \in S_{(I + N)I} }}{\Psi_{{I +N,x}} (x)} (x_{I + M} - x_{I + N} )^{2n}\\&\quad \times \Psi_{{I +M,x}} (x)(x_{I + M} - x)^{2}\Bigg\}dx \\\end{aligned}$$

(A.13)

In the meantime, based upon Eq. (42), Eq. (A.13) can be recast as:

$$ \begin{aligned} {\mathcal{C}}_{1} & = \int_{{x_{I} }}^{{x_{I} + h}}\!\! {\sum\limits_{{M \in S_{(I + M)(I + N)} }} \!{\left\{ \begin{gathered} \underbrace {{\sum\limits_{{I + N \in S_{(I + N)I} }} {\Psi_{{I + N,x}} (x)} (x_{I + M} - x_{I + N} )^{2n} }}_{{ = 2n(x - x_{I + M} )^{2n - 1} }} \\ \times \Psi_{{I + M,x}} (x)(x_{I + M} - x)^{2} \\ \end{gathered} \right\}dx} } \\ & = 2n\int_{{x_{I} }}^{{x_{I} + h}} {\sum\limits_{{(I + M) \in S_{I + M} }} {\Psi_{{I + M,x}} (x)(x - x_{I + M} )^{2n + 1} } dx} \\ \end{aligned} $$

(A.14)

(7) Proof of Eq. (71).

Similar to the proof of Eq. (66), substituting Eqs. (40) and (42) into Eq. (71) gives:

$$ \begin{aligned}\begin{aligned} {\mathcal{D}}_{1} & = \int_{\Omega } {\Psi_{{I,x}} (x)\sum\limits_{{J \in S_{J} }} {\Psi_{{J,x}} (x)(x_{J} - x_{I} )^{2n - 1} (x_{J} - x)^{2} } (x - x_{I} )dx} \\ & = \sum\limits_{{N \in S_{(I + N)I} }} \int_{{x_{I} }}^{{x_{I} + h}} {\Psi_{{I + N,x}} (x)\sum\limits_{{M \in S_{(I + M)(I + N)} }} {\left\{ \begin{gathered} \Psi_{{I + M,x}} (x)(x_{I + M} - x_{I + N} )^{2n - 1} \\ \times (x - x_{I + N} )(x_{I + M} - x)^{2} \\ \end{gathered} \right\}dx} } \hfill \\ & = \int_{{x_{I} }}^{{x_{I} + h}} {\sum\limits_{{M \in S_{(I + M)(I + N)} }} {\left\{ \begin{gathered} \underbrace {{\sum\limits_{{I + N \in S_{(I + N)} }} {\Psi_{{I + N,x}} (x)} (x_{I + M} - x_{I + N} )^{2n - 1} \times (x - x_{I + N} )}}_{{ = (x - x_{I + M} )^{2n - 1} }} \\ \times \Psi_{{I + M,x}} (x)(x_{I + M} - x)^{2} \\ \end{gathered} \right\}dx} } \\ & = \int_{{x_{I} }}^{{x_{I} + h}} {\sum\limits_{{I + M \in S_{(I + M)} }} {\Psi_{{I + M,x}} (x)(x - x_{I + M} )^{2n + 1} } dx} \\ \end{aligned}\end{aligned} $$

(A.15)

According to Eqs. (67) and (68), Eq. (A.15) becomes:

$$ {\mathcal{D}}_{1} = \int_{{x_{I} }}^{{x_{I} + h}} {\underbrace {{\sum\limits_{{I + M \in S_{(I + M)} }} {\Psi_{{I + M,x}} (x)(x - x_{I + M} )^{2n + 1} } }}_{ = 0}dx} \; = 0 $$

(A.16)

(8) Proof of Eq. (82).

For convenience of development, Eqs. (82) can be expressed as follows:

$$ \begin{aligned} & \sum\limits_{{J \in S_{JI} }} {\int_{\Omega } {\nabla \Psi_{I} ({\varvec{x}}) \cdot \nabla \Psi_{J} ({\varvec{x}})(x_{J} - x_{I} )^{2n + 2} d\Omega } } \\ & \quad = \int_{\Omega } {\nabla \Psi_{I} ({\varvec{x}}) \cdot \sum\limits_{{J \in S_{J} }} {\nabla \Psi_{J} ({\varvec{x}})(x_{J} - x_{I} )^{2n + 2} } d\Omega } \\ & \quad = \int_{{\Omega_{y} }} {\int_{{\Omega_{x} }} {\nabla \Psi_{I} ({\varvec{x}}) \cdot \sum\limits_{{J \in S_{J} }} {\nabla \Psi_{J} ({\varvec{x}})(x_{J} - x_{I} )^{2n + 2} } dxdy} } \\ \end{aligned} $$

(A.17)

in which $\Omega_{x}$ and $\Omega_{y}$ represent the directional problem domains along the $x$ and $y$ directions, respectively.

In accordance with Eqs. (63) and (74), we have:

$$ \begin{aligned} &\sum\limits_{{J \in S_{JI} }} {\int_{\Omega } {\nabla \Psi_{I} ({\varvec{x}}) \cdot \nabla \Psi_{J} ({\varvec{x}})(x_{J} - x_{I} )^{2n + 2} d\Omega } } \\&\quad = \int_{{\Omega_{y} }} {\underbrace {{\int_{{\Omega_{x} }} {\nabla \Psi_{I} ({\varvec{x}}) \cdot \sum\limits_{{J \in S_{J} }} {\nabla \Psi_{J} ({\varvec{x}})(x_{J} - x_{I} )^{2n + 2} } dx} }}_{ = 0}dy} = 0 \\ \end{aligned} $$

(A.18)