Generalized-strain mesh-free formulation

The generalized-strain mesh-free (GSMF) formulation is a local meshfree method in the field of numerical analysis, completely integration free, working as a weighted-residual weak-form collocation. This method was first presented by Oliveira and Portela (2016),^[1] in order to further improve the computational efficiency of meshfree methods in numerical analysis. Local meshfree methods are derived through a weighted-residual formulation which leads to a local weak form that is the well known work theorem of the theory of structures. In an arbitrary local region, the work theorem establishes an energy relationship between a statically-admissible stress field and an independent kinematically-admissible strain field. Based on the independence of these two fields, this formulation results in a local form of the work theorem that is reduced to regular boundary terms only, integration-free and free of volumetric locking.

Advantages over finite element methods are that GSMF doesn't rely on a grid, and is more precise and faster when solving bi-dimensional problems. When compared to other meshless methods, such as rigid-body displacement mesh-free (RBDMF) formulation, the element-free Galerkin (EFG)^[2] and the meshless local Petrov-Galerkin finite volume method (MLPG FVM);^[3] GSMF proved to be superior not only regarding the computational efficiency, but also regarding the accuracy.^[4]

The moving least squares (MLS) approximation of the elastic field is used on this local meshless formulation.

In the local form of the work theorem, equation:

${\displaystyle \int _{\Gamma _{Q}}\mathbf {t} ^{T}\mathbf {u} ^{*}d\Gamma +\int _{\Omega _{Q}}\mathbf {b} ^{T}\mathbf {u} ^{*}d\Omega =\int _{\Omega _{Q}}{\boldsymbol {\sigma }}^{T}{\boldsymbol {\varepsilon }}^{*}d\Omega .}$

The displacement field ${\displaystyle \mathbf {u} ^{*}}$, was assumed as a continuous function leading to a regular integrable function that is the kinematically-admissible strain field ${\displaystyle {\boldsymbol {\varepsilon }}^{*}}$. However, this continuity assumption on ${\displaystyle \mathbf {u} ^{*}}$, enforced in the local form of the work theorem, is not absolutely required but can be relaxed by convenience, provided ${\displaystyle {\boldsymbol {\varepsilon }}^{*}}$ can be useful as a generalized function, in the sense of the theory of distributions, see Gelfand and Shilov.^[5] Hence, this formulation considers that the displacement field ${\displaystyle \mathbf {u} ^{*}}$, is a piecewise continuous function, defined in terms of the Heaviside step function and therefore the corresponding strain field ${\displaystyle {\boldsymbol {\varepsilon }}^{*}}$, is a generalized function defined in terms of the Dirac delta function.

For the sake of the simplicity, in dealing with Heaviside and Dirac delta functions in a two-dimensional coordinate space, consider a scalar function ${\displaystyle d}$, defined as:

${\displaystyle d=\lVert \ \mathbf {x} -\mathbf {x} _{Q}\rVert }$

which represents the absolute-value function of the distance between a field point ${\displaystyle \mathbf {x} }$ and a particular reference point ${\displaystyle \mathbf {x} _{Q}}$, in the local domain ${\displaystyle \Omega _{Q}\cup \Gamma _{Q}}$ assigned to the field node ${\displaystyle Q}$. Therefore, this definition always assumes ${\displaystyle d=d(\mathbf {x} ,\mathbf {x} _{Q})\geq 0}$, as a positive or null value, in this case whenever ${\displaystyle \mathbf {x} }$ and ${\displaystyle \mathbf {x} _{Q}}$ are coincident points.

For a scalar coordinate ${\displaystyle d\supset d(\mathbf {x} ,\mathbf {x} _{Q})}$, the Heaviside step function can be defined as

${\displaystyle H(d)=1\,\,\,\,\,\,if\,\,\,\,\,d\leq 0\,\,\,\,\,\,(d=0\,\,\,for\,\,\,\mathbf {x} \equiv \mathbf {x} _{Q})}$

${\displaystyle H(d)=0\,\,\,\,\,\,if\,\,\,\,\,d>0\,\,\,\,\,\,(\mathbf {x} \neq \mathbf {x} _{Q})}$

in which the discontinuity is assumed at ${\displaystyle \mathbf {x} _{Q}}$ and consequently, the Dirac delta function is defined with the following properties

${\displaystyle \delta (d)=H'(d)=\infty \,\,\,\,\,\,if\,\,\,\,\,d=0\,\,\,that\,\,is\,\,\,\mathbf {x} \equiv \mathbf {x} _{Q}}$

${\displaystyle \delta (d)=H'(d)=0\,\,\,\,\,\,if\,\,\,\,\,d\neq 0\,\,\,(d>0\,\,\,for\,\,\,\mathbf {x} \neq \mathbf {x} _{Q})}$

and

${\displaystyle \int \limits _{-\infty }^{+\infty }\delta (d)\,dd=1}$

in which ${\displaystyle H'(d)}$ represents the distributional derivative of ${\displaystyle H(d)}$. Note that the derivative of ${\displaystyle H(d)}$, with respect to the coordinate ${\displaystyle x_{i}}$, can be defined as

${\displaystyle H(d)_{,i}=H'(d)\,\,d_{,i}=\delta (d)\,\,d_{,i}=\delta (d)\,\,n_{i}}$

Since the result of this equation is not affected by any particular value of the constant ${\displaystyle n_{i}}$, this constant will be conveniently redefined later on.

Consider that ${\displaystyle d_{l}}$, ${\displaystyle d_{j}}$ and ${\displaystyle d_{k}}$ represent the distance function ${\displaystyle d}$, for corresponding collocation points ${\displaystyle \mathbf {x} _{l}}$, ${\displaystyle \mathbf {x} _{j}}$ and ${\displaystyle \mathbf {x} _{k}}$. The displacement field ${\displaystyle \mathbf {u} ^{*}(\mathbf {x} )}$, can be conveniently defined as

${\displaystyle \mathbf {u} ^{*}(\mathbf {x} )={\Bigg [}{\frac {L_{i}}{n_{i}}}\,\sum _{l=1}^{n_{i}}H(d_{l})+{\frac {L_{t}}{n_{t}}}\,\sum _{j=1}^{n_{t}}H(d_{j})+{\frac {S}{n_{\Omega }}}\,\sum _{k=1}^{n_{\Omega }}H(d_{k}){\Bigg ]}\mathbf {e} }$

in which ${\displaystyle \mathbf {e} =[1\,\,\,\,1]^{T}}$ represents the metric of the orthogonal directions and ${\displaystyle n_{i}}$, ${\displaystyle n_{t}}$ and ${\displaystyle n_{\Omega }}$ represent the number of collocation points, respectively on the local interior boundary ${\displaystyle \Gamma _{Qi}=\Gamma _{Q}-\Gamma _{Qt}-\Gamma _{Qu}}$ with length ${\displaystyle L_{i}}$, on the local static boundary ${\displaystyle \Gamma _{Qt}}$ with length ${\displaystyle L_{t}}$ and in the local domain ${\displaystyle \Omega _{Q}}$ with area ${\displaystyle S}$. This assumed displacement field ${\displaystyle \mathbf {u} ^{*}(\mathbf {x} )}$, a discrete rigid-body unit displacement defined at collocation points. The strain field ${\displaystyle {\boldsymbol {\varepsilon }}^{*}(\mathbf {x} )}$, is given by

${\displaystyle {\boldsymbol {\varepsilon }}^{*}(\mathbf {x} )=\mathbf {L} \,\mathbf {u} ^{*}(\mathbf {x} )={\Bigg [}{\frac {L_{i}}{n_{i}}}\,\sum _{l=1}^{n_{i}}\mathbf {L} \,H(d_{l})+{\frac {L_{t}}{n_{t}}}\,\sum _{j=1}^{n_{t}}\mathbf {L} \,H(d_{j})+{\frac {S}{n_{\Omega }}}\,\sum _{k=1}^{n_{\Omega }}\mathbf {L} \,H(d_{k}){\Bigg ]}\mathbf {e} ={\Bigg [}{\frac {L_{i}}{n_{i}}}\,\sum _{l=1}^{n_{i}}\,\delta (d_{l})\,\mathbf {n} ^{T}\,+{\frac {L_{t}}{n_{t}}}\,\sum _{j=1}^{n_{t}}\,\delta (d_{j})\,\mathbf {n} ^{T}\,+{\frac {S}{n_{\Omega }}}\,\sum _{k=1}^{n_{\Omega }}\,\delta (d_{k})\,\mathbf {n} ^{T}{\Bigg ]}\mathbf {e} }$

Having defined the displacement and the strain components of the kinematically-admissible field, the local work theorem can be written as

${\displaystyle {\frac {L_{i}}{n_{i}}}\sum _{l=1}^{n_{i}}\,\int \limits _{\Gamma _{Q}-\Gamma _{Qt}}\!\!\!\!\!\!\mathbf {t} ^{T}H(d_{l})\mathbf {e} \,d\Gamma +{\frac {L_{t}}{n_{t}}}\sum _{j=1}^{n_{t}}\,\int \limits _{\Gamma _{Qt}}\!{\overline {\mathbf {t} }}^{T}H(d_{j})\mathbf {e} \,d\Gamma +{\frac {S}{n_{\Omega }}}\sum _{k=1}^{n_{\Omega }}\,\int \limits _{\Omega _{Q}}\mathbf {b} ^{T}H(d_{k})\mathbf {e} \,d\Omega ={\frac {S}{n_{\Omega }}}\sum _{k=1}^{n_{\Omega }}\,\int \limits _{\Omega _{Q}}{\boldsymbol {\sigma }}^{T}\delta (d_{k})\,\mathbf {n} ^{T}\mathbf {e} \,d\Omega .}$

Taking into account the properties of the Heaviside step function and Dirac delta function, this equation simply leads to

${\displaystyle {\frac {L_{i}}{n_{i}}}\sum _{l=1}^{n_{i}}\,\mathbf {t} _{\mathbf {x} _{l}}=-\,{\frac {L_{t}}{n_{t}}}\sum _{j=1}^{n_{t}}\,{\overline {\mathbf {t} }}_{\mathbf {x} _{j}}-\,{\frac {S}{n_{\Omega }}}\sum _{k=1}^{n_{\Omega }}\,\mathbf {b} _{\mathbf {x} _{k}}}$

Discretization of this equations can be carried out with the MLS approximation, for the local domain ${\displaystyle \Omega _{Q}}$, in terms of the nodal unknowns ${\displaystyle {\hat {\mathbf {u} }}}$, thus leading to the system of linear algebraic equations that can be written as

${\displaystyle {\frac {L_{i}}{n_{i}}}\sum _{l=1}^{n_{i}}\,\mathbf {n} _{\mathbf {x} _{l}}\mathbf {D} \mathbf {B} _{\mathbf {x} _{l}}{\hat {\mathbf {u} }}=-\,{\frac {L_{t}}{n_{t}}}\sum _{j=1}^{n_{t}}\,{\overline {\mathbf {t} }}_{\mathbf {x} _{j}}-\,{\frac {S}{n_{\Omega }}}\sum _{k=1}^{n_{\Omega }}\,\mathbf {b} _{\mathbf {x} _{k}}}$

or simply

${\displaystyle \mathbf {K} _{Q}\,{\hat {\mathbf {u} }}=\mathbf {F} _{Q}}$

This formulation states the equilibrium of tractions and body forces, pointwisely defined at collocation points, obviously, it is the pointwise version of the Euler-Cauchy stress principle. This is the equation used in the Generalized-Strain Mesh-Free (GSMF) formulation which, therefore, is free of integration. Since the work theorem is a weighted-residual weak form, it can be easily seen that this integration-free formulation is nothing else other than a weighted-residual weak-form collocation. The weighted-residual weak-form collocation readily overcomes the well-known difficulties posed by the weighted-residual strong-form collocation,^[6] regarding accuracy and stability of the solution.

• Moving least squares
• Finite element method
• Boundary element method
• Meshfree methods
• Numerical analysis
• Computational Solid Mechanics