1 Introduction

Numerical methods based on differential governing equations have been extensively applied for the simulation of a broad range of engineering problems, especially after the advent of high-speed computers. Tremendous efforts have been made to develop the so-called meshless approaches with considerable superiority over the conventional mesh-based methods because of the elimination of the process of computational mesh establishment (Monaghan 1994; Koshizuka et al. 1998; Liu et al. 2006). The main advantage of these schemes is their ability to approximate functions by only several scattered calculation nodes without any predefined connectivity. Therefore, the meshless methods have been successfully applied to study practical fluid and solid mechanics problems with complex computational domains.

Smoothed particle hydrodynamics (SPH) is the first meshless method introduced for the simulation of the collision of galaxies (Gingold and Monaghan 1977). The method has been successfully applied in diverse areas such as free surface flows (Shao and Lo 2003, Shimizu et al. 2020), solid mechanic problems (Chen et al. 2013, Khayyer et al. 2021), multiphase flows (Rezavand et al. 2018), fluid–structure interactions (Khayyer et al. 2018; Gotoh et al. 2021), and Industrial fluid flows (Cleary et al. 2021). In this method, function approximation can be conducted through several calculation nodes with arbitrary configurations for the interpolation process based on the theory of integral interpolants and weight or kernel functions. Discrete element method (DEM) (Nayroles et al. 1991), element-free Galerkin (EFG) (Belytschko et al. 1994), material point method (MPM) (Sulsky et al. 1994), particle-in cell (PIC) (Sulsky et al. 1995), moving particle semi-implicit (MPS) (Koshizuka and Oka 1996), meshless local Petrov–Galerkin (MLPG) (Atluri and Zhu 1998), and discrete least squares meshless (DLSM) (Firoozjaee and Afshar 2009) are among the well-known meshless methods employed to study of a wide range of engineering problems.

Despite attractive characteristics and promising features of SPH, it suffers from low accuracy, especially near computational domain boundaries. Various modifications have been introduced to ameliorate this fundamental drawback. A high-order Laplacian model using a particle-averaged spatial derivative was proposed by Hu and Adams (2007). Oger et al. (2007) proposed a normalization technique to obtain a promising higher-order Laplacian model. Another higher-order Laplacian model was introduced by Schwaiger (2008) using a gradient approximation derived from Taylor series expansion (TSE). A higher-order particle interpolation method was proposed for first-order derivatives in the context of SPH using the TSE method (Zheng et al. 2010). A more accurate Laplacian model was introduced by Huang et al. (2016) which does not require the kernel gradient. A new Laplacian model was also developed through coupling SPH, TSE, and moving least squares (MLS) methods which offered higher accuracy compared to several higher-order models (Shobeyri 2019; Heydari et al. 2019). Recently, the Voronoi diagram was proposed to improve the accuracy of the standard SPH Laplacian model (Shobeyri and Ardakani 2017; Shobeyri and Yourdkhani 2017). A comprehensive study dealing with error estimation in SPH method for second derivatives was also conducted by Fatehi and Manzari (2011). Some improved models were also proposed to amend deficits of the SPH method for solid mechanics applications (Vidal et al. 2007; Batra and Zhang 2008; Wang et al. 2014; Xiao et al. 2020).

In the mixed formulation, the second-order partial differential equations (PDEs) representing elasticity problems are rewritten in the form of first-order differential equations including the displacements and stresses. This formalism has been widely used in the finite element method to solve contact problems (Papadopoulos and Taylor 1992). A mixed least squares method was also proposed to study linear elasticity problems (Tchonkova and Sture 1997). Some problems in linear elasticity were studied using the MLPG mixed method (Atluri et al. 2006). A mixed discrete least squares meshless method was introduced for an efficient solution of planer elasticity problems (Amani et al. 2012). A mixed finite element method was presented using quadrilateral and hexahedral meshes to solve nearly incompressible elasticity or Stokes equations (Lamichhane 2014). Two different approaches were examined to stabilize mixed finite element methods for linear elasticity using simplicial grids (Chen et al. 2017). This mixed approach has been recently employed to facilitate the solution process of elasticity problems using a linear gradient elasticity theory of the Helmholtz type (Jalusic et al. 2020).

In this study, a novel mixed SPH method was introduced to solve two well-known benchmark static elasticity problems. In this approach, the quadratic governing equations are recast in a system of first-order differential equations based on displacements and stresses parameters. For more accurate simulation, the first-order derivatives are approximated using a higher-order gradient model in the context of TSE-improved SPH. The proposed model showed the significant advantage of the elimination of costly calculation of the second-order derivatives leading to higher accuracy. The classic SPH Laplacian model (Shao and Lo 2003) and the newly proposed higher-order SPH Laplacian model (Shobeyri 2019) were extended to solve the original quadratic PDEs representing the governing equations of the test problems at higher accuracy. Regarding the high accuracy of the MLS method, an MLS-based mixed model is also included to compare its performance with the mixed SPH model. A comparison of the accuracy of the models suggests the smallest error for the mixed models, especially the SPH model.

2 Governing Equations of Elasticity Problems

The general PDEs describing governing equations of linear static elasticity problems are given by Afshar et al. 2012.

$$ (\lambda + 2\mu )\frac{{\partial^{2} u}}{{\partial x^{2} }} + \mu \frac{{\partial^{2} u}}{{\partial y^{2} }} + (\lambda + \mu )\frac{{\partial^{2} v}}{\partial x\partial y} = - fr_{x} $$
(1)
$$ \mu \frac{{\partial^{2} v}}{{\partial x^{2} }} + (\lambda + 2\mu )\frac{{\partial^{2} v}}{{\partial y^{2} }} + (\lambda + \mu )\frac{{\partial^{2} u}}{\partial x\partial y} = - fr_{y} $$
(2)

where u and v give displacements and frx and fry show the external forces in x and y directions, respectively.

λ and μ are the Lamé constants given by:

$$ \mu = \frac{E}{2(1 + \nu )} > 0,\quad \lambda = \frac{E\nu }{{(1 - 2\nu )(1 + \nu )}} > 0 $$
(3)

where ν and E represent the Poisson’s ratio and the Young modulus, respectively.

3 Function Approximation

Function approximation is the most prominent task in numerical schemes. Numerous approaches have been proposed to improve the accuracy of numerical simulation. This section extensively presents two popular schemes used in meshless approximation schemes.

3.1 Moving Least Squares (MLS) Method

In this method, the approximated value of the unknown function f can be obtained by

$$ f({\mathbf{x}}_{{\mathbf{a}}} ) = \, \sum\limits_{i = 1}^{m} {p_{i} ({\mathbf{x}}) \, a_{i} ({\mathbf{x}}) \equiv {\mathbf{p}}^{{\text{T}}} ({\mathbf{x}}){\mathbf{a}}({\mathbf{x}})} $$
(4)

where m represents the number of terms in the polynomial basis functions, pT(x) shows a monomial basis, and a(x) denotes a vector of undetermined quantities. The polynomial basis for a two-dimensional problem can be given as follows (Liu 2002)

$$ \begin{array}{*{20}l} {{\mathbf{P}} = 1} \hfill & {{\text{for}}\,m = 1} \hfill \\ {{\mathbf{P}} = [1 \, x \, y]^{{\text{T}}} } \hfill & {{\text{for}}\,m = 3} \hfill \\ {{\mathbf{P}} = [1 \, x \, y \, x^{2} \, xy \, y^{2} ]^{{\text{T}}} } \hfill & {{\text{for}}\,m = 6} \hfill \\ \end{array} $$
(5)

a(x) can be determined by minimizing the sum of the weighted squared residuals (J) presented in the following matrix form

$$ J = ({\mathbf{P}} \cdot {{\varvec{\upalpha}}}({\mathbf{x}}) - {\mathbf{f}})^{{\text{T}}} \cdot {\mathbf{W}} \cdot ({\mathbf{P}} \cdot {{\varvec{\upalpha}}}({\mathbf{x}}) - {\mathbf{f}}) $$
(6)
$$ {{\varvec{\upalpha}}}({\mathbf{x}}) = {\mathbf{A}}^{ - 1} ({\mathbf{x}}) \cdot {\mathbf{B}}({\mathbf{x}}) \cdot ({\mathbf{u}}) $$
(7)

where \({\mathbf{f}}^{{\text{T}}} = (f_{1} ,f_{2} , \ldots ,f_{n} ),\,{\mathbf{A}}({\mathbf{x}}) = {\mathbf{P}}^{{\mathbf{T}}} \cdot {\mathbf{W}} \cdot {\mathbf{P}},{\mathbf{B}}({\mathbf{x}}) = {\mathbf{P}}^{{\mathbf{T}}} \cdot {\mathbf{W}}\).

The unknown function f(x) can be approximated by

$$ f({\mathbf{x}}) = \sum\limits_{i = 1}^{n} {N_{i} ({\mathbf{x}}) \cdot f_{i} } $$
(8)

where Ni(x) shows the shape function of the reference node i as follows:

$$ N_{i} ({\mathbf{x}}) = {\mathbf{p}}^{{\text{T}}} ({\mathbf{x}}){\mathbf{A}}^{ - 1} ({\mathbf{x}}) \cdot {\mathbf{B}}({\mathbf{x}}) $$
(9)

The first derivatives of the MLS shape function can be calculated by

$$ \frac{{{\text{d}}N}}{{{\text{d}}x}} = \frac{{{\text{d}}{\mathbf{p}}^{{\text{T}}} }}{{{\text{d}}x}} \, {\mathbf{A}}^{ - 1} {\mathbf{B}} + {\mathbf{p}}^{{\text{T}}} \frac{{{\text{d}}{\mathbf{A}}^{ - 1} }}{{{\text{d}}x}} \, {\mathbf{B}} + {\mathbf{p}}^{{\text{T}}} {\mathbf{A}}^{ - 1} \frac{{{\text{d}}{\mathbf{B}}}}{{{\text{d}}x}} $$
(10)
$$ \frac{{{\text{d}}N}}{{{\text{d}}y}} = \frac{{{\text{d}}{\mathbf{p}}^{{\text{T}}} }}{{{\text{d}}y}} \, {\mathbf{A}}^{ - 1} {\mathbf{B}} + {\mathbf{p}}^{{\text{T}}} \frac{{{\text{d}}{\mathbf{A}}^{ - 1} }}{{{\text{d}}y}} \, {\mathbf{B}} + {\mathbf{p}}^{{\text{T}}} {\mathbf{A}}^{ - 1} \frac{{{\text{d}}{\mathbf{B}}}}{{{\text{d}}y}} $$
(11)

3.2 SPH Method

In the SPH method, the value of a scalar field function f(r) can be estimated using the kernel or weight function (Shao and Lo 2003)

$$ f({\mathbf{r}}_{a} ) \cong \sum\limits_{b} {f({\mathbf{r}}_{b} )W(|{\mathbf{r}}_{b} - {\mathbf{r}}_{a} |, \, R_{{\text{s}}} )\Delta V_{b} } $$
(12)

where a and b are the reference node and its neighbors, ΔVb shows the volume of neighboring nodes, \(W(|{\mathbf{r}}_{b} - {\mathbf{r}}_{a} |, \, R_{{\text{s}}} )\) represents kernel or weight function, and Rs is the size of the support domain representing the interaction area of node a.

The following kernel function is employed in this study for the SPH calculations with reasonable efficiency and accuracy (Liu 2002).

$$ W(r,R_{{\text{s}}} ) = \alpha_{{\text{d}}} \left( {\frac{3}{4}\left( {\frac{r}{{R_{{\text{s}}} }}} \right)^{2} - \frac{3}{2}\left( {\frac{r}{{R_{{\text{s}}} }}} \right) + \frac{3}{4}} \right) $$
(13)

where in one-, two-, and three-dimensional space, \(\alpha_{{\text{d}}} = 2/R_{{\text{s}}} ,\alpha_{{\text{d}}} = 8/\pi R_{{\text{s}}}^{2}\), and \(\alpha_{{\text{d}}} = 10/\pi R_{{\text{s}}}^{3}\), respectively.

The nodal volume of the reference node may be obtained using Eq. (12)

$$ \frac{1}{{\Delta V_{a} }} = \sum\limits_{b} {\frac{1}{{\Delta V_{b} }}} \, W\Delta V_{b} = \sum\limits_{b} W \to \Delta V_{a} = \frac{1}{{\sum\limits_{b} W }} $$
(14)

The gradient of the function is approximated based on the differentiable form of the kernel function as follows:

$$ \nabla f({\mathbf{r}}_{a} ) = \sum\limits_{b} {\Delta V_{b} f({\mathbf{r}}_{b} )} \nabla_{a} W_{ab} $$
(15)

where \(\nabla_{a} W\) is the gradient of the kernel function with respect to the reference node a.

The formulation of the Laplacian operator can be determined in a straightforward approach based on the introduced SPH definitions. The irregularity of calculation node distributions, due to the second-order derivative of the kernel function, can lead to larger numerical errors. The following sections present two Laplacian models formulated by using a hybrid approach (including SPH and finite difference approximation) for the first derivative. In addition, the mixed SPH model is also formulated.

3.2.1 Standard Model (Model 1)

The standard and conventional Laplacian model was derived by a finite difference approximation of the first-order derivative in the context of SPH definition (Shao and Lo 2003).

$$ \nabla^{2} f_{a} = \sum\limits_{b} {\frac{{2\Delta V_{b} {\mathbf{r}}_{ab} \nabla W_{ab} }}{{|{\mathbf{r}}_{ab} |^{2} }}(} f_{a} - f_{b} ) $$
(16)

The following equation represents the SPH approximation of the second-order derivate operator \(\partial^{2} f/\partial x\partial y\) using the TSE definition for 2-D applications and the above equation. This approximation is required for the discretization of the governing equations of the elasticity problems.

$$ \left( {\frac{{\partial^{2} f}}{\partial x\partial y}} \right)_{a} = \sum\limits_{b} {\frac{{2\Delta V_{b} \left( {{\mathbf{x}}_{ab} .\frac{\partial W}{{\partial y}}|_{ab} + {\mathbf{y}}_{ab} .\frac{\partial W}{{\partial x}}|_{ab} } \right)}}{{|{\mathbf{r}}_{ab} |^{2} }}(} f_{a} - f_{b} ) $$
(17)

3.2.2 The Hybrid Model (Model 2)

Recently, a robust SPH Laplacian model has been proposed and applied for simple 2-D elliptical PDEs which exhibited higher accuracy in comparison with higher-order models (Shobeyri, 2019; Heydari et al. 2019). The model is formulated by combining the standard model [Eq. (16)] with TSE and MLS shape functions:

$$ \nabla^{2} f_{a} = \sum\limits_{b} {\frac{{2(f_{a} - f_{b} )N_{ab} }}{{|{\mathbf{r}}_{ab} |^{2} }}(} \, x_{ab} (M_{1} (2,1:3) \times [1 \, \Delta x_{b} \, \Delta y_{b} ]^{{\text{T}}} ) + \, y_{ab} (M_{1} (3,1:3) \times [1 \, \Delta x_{b} \, \Delta y_{b} ]^{{\text{T}}} )) $$
(18)

where \(x_{ab} = x_{a} - x_{b} ,y_{ab} = y_{a} - y_{b}\) and Nab represents MLS shape functions. a and b denote the reference and neighbor nodes, respectively. The matrix M1 is also defined by

$$ M_{1} = \left[ {\begin{array}{*{20}c} {\sum\limits_{b} {N_{ab} } } & {\sum\limits_{b} {\Delta x_{b} N_{ab} } } & {\sum\limits_{b} {\Delta y_{b} N_{ab} } } \\ {\sum\limits_{b} {\Delta x_{b} N_{ab} } } & {\sum\limits_{b} {\Delta x_{b}^{2} N_{ab} } } & {\sum\limits_{b} {\Delta x_{b} \Delta y_{b} N_{ab} } } \\ {\sum\limits_{b} {\Delta y_{b} N_{ab} } } & {\sum\limits_{b} {\Delta x_{b} \Delta y_{b} N_{ab} } } & {\sum\limits_{b} {\Delta y_{b}^{2} N_{ab} } } \\ \end{array} } \right]^{ - 1} = \left[ {\begin{array}{*{20}c} {m_{11} } & {m_{12} } & {m_{13} } \\ {m_{21} } & {m_{22} } & {m_{23} } \\ {m_{31} } & {m_{32} } & {m_{33} } \\ \end{array} } \right] $$
(19)

where \(\Delta x_{b} = x_{b} - x_{a} ,\Delta y_{b} = y_{b} - y_{a} ,M_{1} (2,1:3) = [m_{21} \, m_{22} \, m_{23} ]\) and \(M_{1} (3,1:3) = [m_{31} \, m_{32} \, m_{33} ].\) The details on the formulations of this model were presented by Shobeyri (2019).

The derivative operator \(\partial^{2} f/\partial x\partial y\) using this approach can be given by

$$ \left( {\frac{{\partial^{2} f}}{\partial x\partial y}} \right)_{a} = \sum\limits_{b} {\frac{{2(f_{a} - f_{b} )N_{ab} }}{{|{\mathbf{r}}_{ab} |^{2} }}(} \, y_{ab} (M(2,1:3) \times [1 \, \Delta x_{b} \, \Delta y_{b} ]^{{\text{T}}} ) + \, x_{ab} (M(3,1:3) \times [1 \, \Delta x_{b} \, \Delta y_{b} ]^{{\text{T}}} )) $$
(20)

This model has been recently applied for solution of elasticity problems (Shobeyri 2022) and shows high accuracy, and therefore, it is necessary to examine the numerical performance of the proposed mixed models in comparison with this approach.

3.2.3 The Mixed SPH Model (Model 3)

To reduce the order of the quadratic PDEs of elasticity problems [Eqs. (1) and (2)], the following equations can be applied in the mixed formulation to determine the relationship between the stresses and displacements components.

$$ \begin{gathered} \sigma_{x} = (\lambda + 2\mu )\frac{\partial u}{{\partial x}} + \lambda \frac{\partial v}{{\partial y}} \hfill \\ \sigma_{y} = \lambda \frac{\partial u}{{\partial x}} + (\lambda + 2\mu )\frac{\partial v}{{\partial y}} \hfill \\ \tau_{xy} = \mu \left( {\frac{\partial u}{{\partial y}} + \frac{\partial v}{{\partial x}}} \right) \hfill \\ \end{gathered} $$
(21)

where σx, σy are the normal stresses while τxy shows the tangential stress.

Combination of Eqs. (1), (2), and (21) led to the first-order differential governing equations:

$$ \begin{aligned} \frac{{\partial \sigma_{x} }}{\partial x} + \frac{{\partial \tau_{xy} }}{\partial y} & = - fr_{x} \\ \frac{{\partial \tau_{xy} }}{\partial x} + \frac{{\partial \sigma_{y} }}{\partial x} & = - fr_{y} \\ \end{aligned} $$
(22)

Equations (21) and (22) can be rewritten in a compact form:

$$ L(\varphi ) + F = 0 $$
(23)

where L(·)gives a first-order differential operator given by

$$ L( \cdot ) = L_{1} ( \cdot )_{x} + L_{2} ( \cdot )_{y} + L_{3} ( \cdot ) $$
(24)

The vector of unknowns represented by φ has the following expression

$$ \varphi = [u,v,\sigma_{x} ,\sigma_{y} ,\tau_{xy} ]^{{\text{T}}} $$
(25)

F is the vector of source term which is defined by

$$ F = [0,0,0, - fr_{x} , - fr_{y} ]^{{\text{T}}} $$
(26)

and the following matrices define the operators \(L_{1} ( \cdot )_{x} ,L_{2} ( \cdot )_{y} ,L_{3} ( \cdot )\).

$$ L_{1} = \left[ {\begin{array}{*{20}c} {\lambda + 2\mu } & 0 & 0 & 0 & 0 \\ \lambda & 0 & 0 & 0 & 0 \\ 0 & \mu & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} } \right],\quad L_{2} = \left[ {\begin{array}{*{20}c} 0 & \lambda & 0 & 0 & 0 \\ 0 & {\lambda + 2\mu } & 0 & 0 & 0 \\ \mu & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ \end{array} } \right],\quad L_{3} = \left[ {\begin{array}{*{20}c} 0 & 0 & { - 1} & 0 & 0 \\ 0 & 0 & 0 & { - 1} & 0 \\ 0 & 0 & 0 & 0 & { - 1} \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right] $$
(27)

Employing the standard SPH gradient model [Eq. (15)] to solve Eq. (23) (the final system of equations) leads to a zero diagonal coefficients matrix with very large errors. This could be explained by the fact that reference nodes cannot be employed in the gradient model. A higher-order first derivative model was employed in this study to enhance the accuracy of the simulation of the elasticity problems. To this end, the Taylor series expansion around a reference node multiplied by a kernel function over its support domain is employed:

$$ \sum\limits_{b} {f({\mathbf{r}}_{b} )W} \approx \sum\limits_{b} {f({\mathbf{r}}_{a} )W} + \sum\limits_{b} {f_{x} ({\mathbf{r}}_{a} )\Delta x_{b} W} + \, \sum\limits_{b} {f_{y} ({\mathbf{r}}_{a} )\Delta y_{b} W} $$
(28)

Multiplying both sides of the equation by \(\Delta x_{b} = (x_{b} - x_{a} )\) and \(\Delta y_{b} = (y_{b} - y_{a} )\) leads to the following equation.

$$ \sum\limits_{b} {f({\mathbf{r}}_{b} )\Delta x_{b} W} \approx \sum\limits_{b} {f({\mathbf{r}}_{a} )\Delta x_{b} W} + \sum\limits_{b} {f_{x} ({\mathbf{r}}_{a} )\Delta x_{b}^{2} W} + \, \sum\limits_{b} {f_{y} ({\mathbf{r}}_{a} )\Delta x_{b} \Delta y_{b} W} $$
(29)
$$ \sum\limits_{b} {f({\mathbf{r}}_{b} )\Delta y_{b} W} \approx \sum\limits_{b} {f({\mathbf{r}}_{a} )\Delta y_{b} W} + \sum\limits_{b} {f_{x} ({\mathbf{r}}_{a} )\Delta x_{b} \Delta y_{b} W} + \, \sum\limits_{b} {f_{y} ({\mathbf{r}}_{a} )\Delta y_{b}^{2} W} $$
(30)

The following matrix is derived from Eqs. (28)–(30) and offers the higher-order approximation of first derivatives (Zheng et al. 2010).

$$ \left[ {\begin{array}{*{20}c} {f({\mathbf{r}}_{a} )} \\ {f_{x} ({\mathbf{r}}_{a} )} \\ {f_{y} ({\mathbf{r}}_{a} )} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\sum\limits_{b} W } & {\sum\limits_{b} {\Delta x_{b} W} } & {\sum\limits_{b} {\Delta y_{b} W} } \\ {\sum\limits_{b} {\Delta x_{b} W} } & {\sum\limits_{b} {\Delta x_{b}^{2} W} } & {\sum\limits_{b} {\Delta x_{b} \Delta y_{b} W} } \\ {\sum\limits_{b} {\Delta y_{b} W} } & {\sum\limits_{b} {\Delta x_{b} \Delta y_{b} W} } & {\sum\limits_{b} {\Delta y_{b}^{2} W} } \\ \end{array} } \right]^{ - 1} \left[ {\begin{array}{*{20}c} {\sum\limits_{b} {f({\mathbf{r}}_{b} )W_{ab} } } \\ {\sum\limits_{b} {f({\mathbf{r}}_{b} )\Delta x_{b} W_{ab} } } \\ {\sum\limits_{b} {f({\mathbf{r}}_{b} )\Delta y_{b} W_{ab} } } \\ \end{array} } \right] $$
(31)

Here fx and fy are derivatives of the function f with respect to x and y coordinates. To enhance the numerical performance of the model, MLS shape functions using m = 1 [see Eq. (5)] are employed instead of the kernel function W. The final coefficients matrix representing the solution of the problem are square and linear matrix with size 5Nt × 5Nt in which Nt is the total number of calculation nodes, and therefore, efficient solvers can be applied to solve the systems of equations. In this study, a least squares approach using backslash operator in MATLAB software was used to solve the final system of equations. The boundary conditions can be easily imposed in the computations using the known analytical values of the displacements for the boundary nodes. In other words, the stresses of all nodes and the displacements of the inner nodes are considered unknown values of the corresponding PDE.

After examination of several SPH gradient models such as those explained by Liu (2002) and Zheng et al. (2017), it has been found that the higher-order gradient model [Eq. (31)] coupled with the abovementioned mixed formulation can get better accuracy. To avoid nonzero diagonal coefficients matrix, reference nodes are considered in the corresponding equations [Eqs. (21)–(31)] in the proposed model. It seems that contribution of reference nodes is included in the computations by a more appropriate approach using the present model in comparison with the other higher-order gradient models.

Different mathematical formulations can be applied to derive a mixed formulation in which second-order PDEs are replaced by the unknown function and its gradients. A quadratic PDE can be generally given as:

$$ a_{x} \frac{{\partial^{2} f}}{{\partial x^{2} }} + b_{y} \frac{{\partial^{2} f}}{{\partial y^{2} }} + c_{xy} \frac{{\partial^{2} f}}{\partial x\partial y} = g(x,y) $$
(32)

where ax, by and cxy represent the coefficients of the PDE and g(x, y) stands for the source term. f is also the unknown function.

The corresponding mixed formulation is as follows (Eini et al. 2020):

$$ A^{x} \frac{\partial F}{{\partial x}} + A^{y} \frac{\partial F}{{\partial y}} + A^{0} F = S $$
(33)

where

$$ F = [f,f_{x} , \, f_{y} ] $$
(34)
$$ S^{{\text{T}}} = [0,0, \, g(x,y)] $$
(35)
$$ A^{x} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & {a_{x} } & {c_{xy} } \\ \end{array} \, } \right]\quad A^{y} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & {b_{y} } \\ \end{array} \, } \right]\quad A^{0} = \left[ {\begin{array}{*{20}c} 0 & { - 1} & 0 \\ 0 & 0 & { - 1} \\ 0 & 0 & 0 \\ \end{array} \, } \right] $$
(36)

fx and fy are the derivatives of function f in x and y directions, respectively.

Recently, this approach is employed to derive a mixed model in the context of SPH to simulate potential flows which are described by Laplacian operator [cxy = 0 in Eq. (32)] (Shobeyri 2022). It was found that using the mixed model for solving the combined elasticity PDEs [Eqs. (1) and (2)] cannot get promising accuracy. This could be probably explained by the fact that the contribution of reference nodes is not included properly in the corresponding numerical model and larger errors are inevitable while the proposed mixed model in the present study gives high accuracy as shown in Sect. 4. In this mixed formulation, the unknown functions are displacements and their gradients (u, ux, uy, v, vx and vy) while the unknown parameters of the mixed model of this study are the displacements and stress [see Eq. (25)] leading to a natural approach for satisfaction of Neumann boundary conditions. In other words, the model given in Eq. (33) cannot directly impose the boundary conditions corresponding to structural stresses. In addition, the final system of equations of this model has larger size (6Nt × 6Nt) comparing to the proposed mixed method due to the six unknowns (displacements and their gradients).

3.3 The Mixed MLS Model (Model 4)

The MLS method is one of the most efficient meshless function approximations with extensive applications in numerical simulation due to its completeness, robustness, and continuity. To derive the mixed MLS model, the derivatives of this method [Eqs. (10) and (11)] can be applied in Eq. (23). Model 4 is also included in the error analysis to further show the effectiveness of the mixed SPH method. In Table 1, the applied models for solution of the test problems with the related equations are given.

Table 1 The models with name and the related equations
Full size table

4 Numerical Examples

In this section, two benchmark elasticity problems (e.g., a cantilever beam under end load and an infinite plate with a circular hole under uniaxial load) were solved by the presented models and their numerical performance was compared with the available analytical solutions.

The following simple error indicator was applied to assess the accuracy of the models.

$$ E_{{\text{r}}} = \sum\limits_{i = 1}^{n} {\frac{{f_{i} - f_{{{\text{ana}}}} }}{{\overline{f}_{{{\text{ana}}}} }} \times \frac{100}{n}} $$
(37)

where Er is the mean error, fi shows the numerical result at calculation node i, fana represents the corresponding analytical solution, \(\mathop f\limits^{ - }_{{{\text{ana}}}}\) denotes the mean of analytical solution, and n stands for the number of calculation nodes in the commotional domain. To have a better accuracy analysis,\(\mathop f\limits^{ - }_{{{\text{ana}}}}\) was employed instead of fana as the numerical solutions representing the displacements are too small in some parts of the computational domain. The horizontal or the vertical displacements can be chosen as function f in the above equation to examine the accuracy of the models based on the physical properties of the test problems.

4.1 A Cantilever Beam Under a Transverse Load

The first benchmark problem is a 2-D cantilever beam under end point load with a rectangular cross section as shown in Fig. 1. The analytical displacement solution can be expressed by Afshar et al. (2012):

$$ u = \frac{{ - P_{L} y}}{6EI} \, \left[ {3x(2L - x) + (2 + \nu )(y^{2} - c^{2} )} \right] $$
(38)
$$ v = \frac{{P_{L} }}{6EI} \, \left[ {x^{2} (3L - x) + 3\nu (L - x)y^{2} + (4 + 5\nu )c^{2} x} \right] $$
(39)

where \(I = 2c^{3} /3\) gives the moment of inertia. The computational constants are \(P_{L} = 1, \, E = 10000, \, c = 1, \, L = 30, \, \nu = 0.3\) (in SI units).

Fig. 1
figure 1

A cantilever beam under end load

Full size image

At first, accuracy analysis of the introduced models was investigated under three uniform node configurations represented by node size of L0 = 1/2.5, 1/3, 1/4 m. The numerical errors of calculated vertical displacement defined in Eq. (38) are illustrated in Fig. 2 showing the better accuracy of the proposed mixed model over the other models. It should be noted that all models exhibited approximately second-order convergence rate. The exact solutions of displacements can be determined at the boundaries using the above equations, and the Dirichlet boundary conditions are imposed based on the calculated displacements. The same approach was also applied to satisfy the boundary conditions of the second test problem presented later in this paper.

Fig. 2
figure 2

Error analysis of the models for the cantilever beam test problem using uniform node distributions

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To examine numerical performance of the models for more complex conditions, the calculation process began by solving the test case on four irregular node distributions as depicted in Fig. 3. The number of calculation nodes corresponded to the uniform configurations using node spacing of L0 = 1/2.5, 1/3, 1/4, 1/5 m, respectively. Figure 4 shows the numerical errors representing first-order convergence rate for the models. As seen, all of the improved models showed smaller errors in comparison with the standard method. Both the mixed models, especially the SPH, achieved higher accuracy compared with the hybrid model 2. The results indicated that the SPH higher-order model of first derivative [Eq. (31)] has a more promising performance in the mixed formulation than the corresponding MLS model. According to Fig. 5, the vertical displacement of the nodes near y = 0 m was calculated by model 3. L0 = 1/5 m showed a proper agreement with the analytical solution, reflecting the effectiveness of the proposed mixed model in the context of the SPH method. Variations of the normal stress σx using model 3 and the fine node distribution over the computational domain are shown and compared with the analytical solution in Fig. 6 which shows promising similarity. Note that the exact solution of the normal and tangential stresses can be easily obtained through Eqs. (21), (38), and (39). Figure 7 depicts the accuracy of the models in calculating the normal stress σx along the upper surface of the beam (y = 1 m). As expected, both the mixed models offered higher accuracy while the performance of the higher-order model 2 and the standard model was almost the same due to their almost similar accuracy in calculation of displacements (Fig. 4) for the node size of 1/5 m. The elimination of the second-order derivatives and direct calculation of the stresses in the mixed formulations can explain the better performance of the mixed models. Noteworthy, the presented stresses of models 1 and 2 are obtained based on the calculated displacements and the higher-order first derivative model in Eq. (31). The CPU times of the models with the corresponding errors are presented in Fig. 8. The larger coefficient matrix of the mixed model 3 led to heavier computation compared to the standard model. The efficiency of the mixed SPH model over the other models especially the standard model can be easily deduced from the interpolation process using the illustrated errors and CPU times (the system processor is Intel-Core i5, CPU@2.00 GHz-2.6 GHz, and RAM 4.00 GB). Despite higher accuracy, models 1 and 2 provide approximately the same efficiency. Regarding the higher accuracy of the high-order model 2 in solving of elliptic PDEs over some improved SPH models (Heydari et al. 2019), the mixed formulation can achieve higher efficiency even over the improved models. It should also be noted that the minimum errors of models 1 and 2 were obtained from the dimensionless radius of support domain of \(R_{{\text{s}}} /L_{0} \approx 4\) while the mixed models require smaller values for this parameter (\(R_{{\text{s}}} /L_{0} \approx 1.7\)).

Fig. 3
figure 3

Sample node distributions using node size L0 equal to a 1/2.5 m, b 1/3 m, c 1/4 m, and d 1/5 m for the cantilever beam

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Fig. 4
figure 4

Error analysis of the models for the cantilever beam test problem

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Fig. 5
figure 5

The vertical displacement of the cantilever beam for the nodes near y = 0 using model 3 and L0 = 1/5 m

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Fig. 6
figure 6

Comparison of contours of σx (in SI units) of the cantilever beam using a model 3 and L0 = 1/5 m with b the analytical solution

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Fig. 7
figure 7

Error analysis of the models in calculating σx (in SI units) along y = 1 m for the cantilever beam using L0 = 1/5 m

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Fig. 8
figure 8

Computational costs of the models for the cantilever beam test problem [the colored text represents the numerical error defined in Eq. (32)]

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4.2 An Infinite Plate with a Circular Hole

The second test problem considered in this study is an infinite plate with a central circular hole with unit radius (ap = 1 m) subjected to a uniaxial load (PL) as depicted in Fig. 9. The exact solution for components of displacement will be (Amani et al. 2012)

$$ u_{{\text{r}}} = \frac{{P_{L} }}{4G}\left\{ {r[\frac{\kappa - 1}{2} + \cos (2\theta )] + \frac{{a_{p}^{2} }}{r}[1 + (1 + \kappa )\cos (2\theta )] - \frac{{a_{p}^{4} }}{{r^{3} }}\cos (2\theta )} \right\} $$
(40)
$$ u_{\theta } = \frac{{P_{L} }}{4G}\left\{ {(1 - \kappa )\frac{{a_{p}^{2} }}{r} - r - \frac{{a_{p}^{4} }}{{r^{3} }}} \right\}\sin (2\theta ) $$
(41)

where G represents the shear modulus \(\kappa = (3 - \nu )/(1 + \nu )\). v also represents the Poisson’s ratio. Due to symmetry, the upper right square quadrant of the plate was chosen to investigate the problem. The edge length of the square is also 5ap. The problem was solved using the following constants (in SI units) \(P_{L} = 1, \, E = 1000, \, \nu = 0.3\). The sample node distributions corresponding to L0 = 1/3, 1/4, 1/5 m (Fig. 10) were applied to study the numerical performance of the models. The numerical errors of the horizontal displacement calculation using the derivative models are compared in Fig. 11 which exhibited the significantly higher accuracy of the mixed SPH model, particularly over the standard method. The improved models can get the first-order convergence rate while the standard model shows slower rate. As seen in the previous example, the mixed MLS method led to better accuracy than the higher-order model 2. This indicates that the mixed models have more promising performance even in comparison with model 2 which has shown considerable efficiency over the well-known higher-order models (Heydari et al. 2019). The color maps of Fig. 12 compare the numerical results of the normal stress σx using model 3 and L0 = 1/5 m with the analytical solution (Amani et al. 2012), which suggests satisfactory similarity. Figure 13 represents error analysis of the models in calculation of normal stress σx along the left edge of the plate (x = 0). As expected, the simulated prediction of the SPH mixed model was higher even in comparison with the improved methods of 2 and 4. The calculated horizontal and vertical displacements using the mixed SPH model for the fine node distribution are compared with the corresponding analytical solutions in Fig. 14. The computational costs of the models are given in Fig. 15 with the corresponding errors to evaluate the efficiency of the methods. The improved models, especially the mixed SPH method, represented higher efficiency over the standard model. The smallest errors of the improved models can be obtained using \(R_{{\text{s}}} /L_{0} \approx 1.6 - 1.9\) while the standard model needs \(R_{{\text{s}}} /L_{0} \approx 3.5\)

Fig. 9
figure 9

An infinite plate with a circular hole subjected to a uniaxial load

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.

Fig. 10
figure 10

Nodal distributions of the plate using a (L0 = 1/3 m), b (L0 = 1/4 m) and c (L0 = 1/5 m)

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Fig. 11
figure 11

Error analysis of the models for the infinite plate problem

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Fig. 12
figure 12

a Exact and b numerical solution of model 3 using L0 = 1/5 m representing the normal stress σx in SI units for the plate

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Fig. 13
figure 13

Error analysis of the models in calculating σx (in SI units) along the left edge (x = 0) for the plate problem using L0 = 1/5 m

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Fig. 14
figure 14

Analytical and numerical solution of model 3 using L0 = 1/5 m representing a the horizontal and b vertical displacements for the plate

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Fig. 15
figure 15

Computational costs of the models for the plate test problem [the colored text represents the numerical error defined in Eq. (32)]

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5 The Mixed Formulation for Fluid–Structure Interaction

Modeling of fluid–structure interactions (FSI) is of great interest in design of various engineering systems such as marine and costal structures, fast vehicles with aerodynamic loadings, and liquid containers with sloshing. Hydrodynamic loads can be considered as external forces and cause distribution of significant tensions in solid domain of such structures. The static elastic PDEs [Eqs. (1) and (2)] can be easily solved using the proposed mixed model with significant accuracy to calculate the induced tensions when the external loads are constant in time. In case of variable hydrodynamics forces exerted on a solid structure in time, the proposed model must be applied in a different approach to solve the dynamic elastic governing equations representing linear and angular momentum conservation laws (Khayyer et al. 2018). At first, a projection-based incompressible SPH fluid model (Shao and Lo 2003) is employed to predict the pressure field and corresponding external loads on the structure [frx and fry in Eq. (22)]. The equation of conservation of linear momentum is solved explicitly by using the gradient model [Eq. (31)] to calculate the structural stresses based on the known displacements at the current time step [Eqs. (21) and (22)]. Then, the elastic deformation of structure at the next time step can be estimated by the solution of angular momentum equation. This procedure must be repeated for the entire simulation time. When the displacements are known, the structural analysis is straightforward and no additional implementation is required to simulate such problems. On the other hand, usually the displacements and corresponding stresses must be calculated implicitly by solving the governing PDEs for static elasticity problems. In fact, the proposed mixed model in which the unknown parameters [Eq. (25)] are calculated implicitly and simultaneously leading to higher accuracy of approximation of the structural stresses is only applicable and required for simulation of static elastic applications.

6 Conclusion

This paper introduced a mixed SPH method for the solution of static elasticity problems. In the mixed formulation, the quadratic PDEs of the test problems were rewritten as a system of first-order differential equations based on displacements and stresses. The elimination of the second-order derivatives can enhance the numerical performance of the proposed approach. This claim was examined in comparison with the standard SPH model, the higher-order SPH model, and the MLS-based mixed method. The results proved the higher accuracy of the mixed SPH approach over the other models, especially the standard method. The mixed model has potential to be applied for simulation of dynamic structural problems with higher accuracy but the direct calculation of stresses leading to better numerical performance cannot be achieved due to fundamental properties of the model which was discussed in Sect. 5. The recently introduced mixed model (Shobeyri 2022) can also be applied to discretize Laplacian operator for prediction of the pressure field with smaller errors in flow domain of such complex problems.