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.
Formulation
In the local form of the work theorem, equation:
- [math]\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. }[/math]
The displacement field [math]\displaystyle{ \mathbf{u}^{*} }[/math], was assumed as a continuous function leading to a regular integrable function that is the kinematically-admissible strain field [math]\displaystyle{ \boldsymbol{\varepsilon}^{*} }[/math]. However, this continuity assumption on [math]\displaystyle{ \mathbf{u}^{*} }[/math], enforced in the local form of the work theorem, is not absolutely required but can be relaxed by convenience, provided [math]\displaystyle{ \boldsymbol{\varepsilon}^{*} }[/math] 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 [math]\displaystyle{ \mathbf{u}^{*} }[/math], is a piecewise continuous function, defined in terms of the Heaviside step function and therefore the corresponding strain field [math]\displaystyle{ \boldsymbol{\varepsilon}^{*} }[/math], 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 [math]\displaystyle{ d }[/math], defined as:
- [math]\displaystyle{ d = \lVert\ \mathbf{x}-\mathbf{x}_Q \rVert }[/math]
which represents the absolute-value function of the distance between a field point [math]\displaystyle{ \mathbf{x} }[/math] and a particular reference point [math]\displaystyle{ \mathbf{x}_Q }[/math], in the local domain [math]\displaystyle{ \Omega_Q \cup \Gamma_Q }[/math] assigned to the field node [math]\displaystyle{ Q }[/math]. Therefore, this definition always assumes [math]\displaystyle{ d=d(\mathbf{x},\mathbf{x}_Q) \geq 0 }[/math], as a positive or null value, in this case whenever [math]\displaystyle{ \mathbf{x} }[/math] and [math]\displaystyle{ \mathbf{x}_Q }[/math] are coincident points.
For a scalar coordinate [math]\displaystyle{ d\supset d(\mathbf{x},\mathbf{x}_Q) }[/math], the Heaviside step function can be defined as
- [math]\displaystyle{ H(d) = 1 \,\,\,\,\,\, if \,\,\,\,\, d\leq 0 \,\,\,\,\,\, (d=0 \,\,\, for \,\,\, \mathbf{x} \equiv \mathbf{x}_Q) }[/math]
- [math]\displaystyle{ H(d) = 0 \,\,\,\,\,\, if \,\,\,\,\, d \gt 0 \,\,\,\,\,\, (\mathbf{x} \neq \mathbf{x}_Q) }[/math]
in which the discontinuity is assumed at [math]\displaystyle{ \mathbf{x}_Q }[/math] and consequently, the Dirac delta function is defined with the following properties
- [math]\displaystyle{ \delta(d) = H'(d) = \infty \,\,\,\,\,\, if \,\,\,\,\, d=0 \,\,\, that \,\, is \,\,\, \mathbf{x} \equiv \mathbf{x}_Q }[/math]
- [math]\displaystyle{ \delta(d) = H'(d) = 0 \,\,\,\,\,\, if \,\,\,\,\, d\neq 0 \,\,\, (d\gt 0 \,\,\, for \,\,\, \mathbf{x} \neq \mathbf{x}_Q) }[/math]
and
- [math]\displaystyle{ \int\limits_{-\infty}^{+\infty} \delta(d)\,d d=1 }[/math]
in which [math]\displaystyle{ H'(d) }[/math] represents the distributional derivative of [math]\displaystyle{ H(d) }[/math]. Note that the derivative of [math]\displaystyle{ H(d) }[/math], with respect to the coordinate [math]\displaystyle{ x_i }[/math], can be defined as
- [math]\displaystyle{ H(d)_{,i}=H'(d) \,\, d_{,i}= \delta(d) \,\, d_{,i}=\delta(d) \,\, n_i }[/math]
Since the result of this equation is not affected by any particular value of the constant [math]\displaystyle{ n_i }[/math], this constant will be conveniently redefined later on.
Consider that [math]\displaystyle{ d_l }[/math], [math]\displaystyle{ d_j }[/math] and [math]\displaystyle{ d_k }[/math] represent the distance function [math]\displaystyle{ d }[/math], for corresponding collocation points [math]\displaystyle{ \mathbf{x}_l }[/math], [math]\displaystyle{ \mathbf{x}_j }[/math] and [math]\displaystyle{ \mathbf{x}_k }[/math]. The displacement field [math]\displaystyle{ \mathbf{u}^{*}(\mathbf{x}) }[/math], can be conveniently defined as
- [math]\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} }[/math]
in which [math]\displaystyle{ \mathbf{e}=[1\,\,\,\, 1]^T }[/math] represents the metric of the orthogonal directions and [math]\displaystyle{ n_i }[/math], [math]\displaystyle{ n_t }[/math] and [math]\displaystyle{ n_\Omega }[/math] represent the number of collocation points, respectively on the local interior boundary [math]\displaystyle{ \Gamma_{Qi}=\Gamma_Q-\Gamma_{Qt}-\Gamma_{Qu} }[/math] with length [math]\displaystyle{ L_i }[/math], on the local static boundary [math]\displaystyle{ \Gamma_{Qt} }[/math] with length [math]\displaystyle{ L_t }[/math] and in the local domain [math]\displaystyle{ \Omega_Q }[/math] with area [math]\displaystyle{ S }[/math]. This assumed displacement field [math]\displaystyle{ \mathbf{u}^{*}(\mathbf{x}) }[/math], a discrete rigid-body unit displacement defined at collocation points. The strain field [math]\displaystyle{ \boldsymbol{\varepsilon}^{*}(\mathbf{x}) }[/math], is given by
- [math]\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} }[/math]
Having defined the displacement and the strain components of the kinematically-admissible field, the local work theorem can be written as
- [math]\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. }[/math]
Taking into account the properties of the Heaviside step function and Dirac delta function, this equation simply leads to
- [math]\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} }[/math]
Discretization of this equations can be carried out with the MLS approximation, for the local domain [math]\displaystyle{ \Omega_Q }[/math], in terms of the nodal unknowns [math]\displaystyle{ \hat{\mathbf{u}} }[/math], thus leading to the system of linear algebraic equations that can be written as
- [math]\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} }[/math]
or simply
- [math]\displaystyle{ \mathbf{K}_Q\,\hat{\mathbf{u}}=\mathbf{F}_Q }[/math]
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.
See also
- Moving least squares
- Finite element method
- Boundary element method
- Meshfree methods
- Numerical analysis
- Computational Solid Mechanics
References
- ↑ Oliveira, T. and A. Portela (2016). "Weak-Form Collocation – a Local Meshless Method in Linear Elasticity". Engineering Analysis with Boundary Elements.
- ↑ Belytschko, T., Y. Y. Lu, and L. Gu (1994). "Element-free Galerkin methods". International Journal for Numerical Methods in Engineering. 37.2, pp. 229–256.
- ↑ Atluri, S.N., Z.D. Han, and A.M. Rajendran (2004). "A New Implementation of the Meshless Finite Volume Method Through the MLPG Mixed Approach". CMES: Computer Modeling in Engineering and Sciences. 6, pp. 491–513.
- ↑ Oliveira, T. and A. Portela (2016). "Comparative study of the weak-form collocation meshless formulation and other meshless methods". Proceedings of the XXXVII Iberian Latin-American Congress on Computational Methods in Engineering. ABMEC, Brazil
- ↑ Gelfand, I.M., Shilov, G.E. (1964). Generalized Functions. Volume I, Academic Press, New York.
- ↑ Kansa, E.J.,(1990) "Multiquadrics: A Scattered Data Approximation Scheme with Applications to Computational Fluid Dynamics", Computers and Mathematics with Applications, 19(8-9), 127--145.
Original source: https://en.wikipedia.org/wiki/Generalized-strain mesh-free formulation.
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