1 Introduction

Static linear thermoelastic problems take place in different applications in engineering. Different works have been carried out to numerically model thermomechanical phenomena, such as forging [4], fatigue [12], welding [11] and solidification of steel [24]. Finite element method is a common method used in these works [33]. However, practical guidelines for dealing with large-scale problems involving several variables are still missing and efforts have been made to overcome computational limitations in the finite element method [9]; in spite of this, meshfree methods have also been developed to overcome these limitations, such as the simulation of complex geometries and difficulties for remeshing.

Meshless methods can be classified in terms of the partial differential equation (PDE) form, namely weak form and strong form meshless methods [27]. In terms of weak form methods, it can be said that one of the most used weak form meshless method is the elementfree Galerkin method (EFG). It has been reported that this method can be used to analyze plates under mechanical loading as well as thermal gradient [7] and solving edge crack problems with the effect of crack interactions under thermal and mechanical loads [36]. This method has also been used to study the thermal buckling on different shaped plates with different boundary conditions [21] as well as the thermomechanical flow of friction stir welding process, [46, 49]. EFG can be used also to solve classical thermoelastic problems [10, 50], such as hollow cylinder and infinite plate with a circular hole which are commonly used as benchmark problems. In [10], a combination between EFG and the improved moving least-squares approximation is proposed where the essential boundary conditions are imposed by the penalty method. Likewise EFG has been used to simulate the thermal fracture in functionally graded materials [13]. This method has also been used to analyze thermal–mechanical coupling of the orthotropic structures, [52].

The radial point interpolation method (RPIM) has been used in order to analyze thermoelastic problems with moving concentrated heat sources [23] where the Kronecker delta property is satisfied by its shape function. This method is also suitable to simulate thermomechanical crack growth, [34]. Several thermal and mechanical material properties have been identified simultaneously by using RPIM, [6]. Since this method requires background cells for numerical integration, efforts have been made to improve it such as the use of the Cartesian transformation method (CTM) where basically numerical integration is done without using background cells. In this sense, RPIM can be considered a truly meshfree method.

Another commonly used method is the local Petrov–Galerkin method that has been used to analyze thermomechanical shock fracture and heat conduction with residual stress due to welding in [31, 32], respectively. Direct meshless local Petrov–Galerkin (DMLPG) is used as a way to solve the coupled thermoelasticity problem in [19]. This method is based on the weak solution of the governing equation, and all those methods have been found to be more efficient than FEM as we expected. Finally, the particle finite element method (PFEM) has been introduced in [39] to solve thermomechanical problems involving large strains, multiple contacts, rotations and large boundary surface changes and it was later used in the modeling of chip formation and metal cutting problems [40, 41]. Besides good precision and stability, weak methods naturally satisfy Neumann-type boundary conditions. Moreover, these methods are computationally expensive due to the use of mandatory background meshes; therefore, strong form/truly meshfree methods as simpler alternatives are attractive.

With regard to the strong form methods, smoothed particle hydrodynamics (SPH) proposed by Lucy in [28] and Monaghan in [14] was the starting point of meshless methods. Nowadays, this method can be used to analyze a brake including thermomechanical coupling in [2], modeling of metal forging in [5] and large deformation problems in [1]. In [8], a numerical comparison between SPH and FEM in thermomechanical coupled problems has been made. Although the same numerical results were obtained with both methods, SPH overcomes the limitations of FEM as it was expected.

In the scientific literature, there are some works that use local radial basis function collocation method (LRBFCM) as a way to solve thermomechanical problems such as hot rolling simulation [15,16,17,18, 43], linear thermoelasticity in two dimensions [29], bending analyses of quasicrystal plates [3], analysis of transient coupled thermoelasticity [30] and simulation of thermal field in mass concrete structures with cooling pipes [20].

Meshless local strong form method (MLSM) has been used in order to solve linear elastic problems under fretting contact conditions [25], a comparison between commercial software, weak and strong form methods has been made. This comparison shows that MLSM provides an accurate result with a significantly lower number of nodes. This method can also be used to solve the Cauchy–Navier equation [44]. This study reveals that MSLM can handle computations in complex domains which is important in solving more complex real engineering problems. Other strong form methods that have been applied to elasticity problems include the finite point method [35] and the meshless finite difference method [22].

The finite pointset method is a meshfree method proposed by Kuhnert in [26]. This is a Lagrangian-type strong form method that has been used for modeling flows [48], for PDEs on evolving surfaces [45], heat transfer with moving heat sources in welding [37], fluid flow coupled with heat transfer considering phase changes [38] and recently, for solving static linear elasticity problems [42], among others. In fact, this is an efficient and versatile truly meshless method that is extended in this work and presented for the first time in a fully coupled form to numerically solve static linear thermoelastic problems.

The structure of the paper is as follows: Sect. 2 introduces the governing partial differential equations of static linear thermoelasticity, Sect. 3 shortly describes the main ideas behind the taken meshfree approach and a new fully coupled FPM discretization in detail for the involved PDEs is worked out. The corresponding numerical results and some comparisons with respect to analytical solutions are discussed in Sect. 4 followed by some conclusions and possible future work in last section.

2 Governing equations

The thermoelastic behavior in an isotropic and homogeneous conducting solid can be described once the temperature, stress and displacement fields are computed. For the study considered in this work, the displacement field obeys the Navier–Cauchy equations coupled with the stationary heat equation which can be written as:

$$\begin{aligned}&(\lambda + \mu )\nabla (\nabla \cdot {\mathbf {u}}) + \mu \nabla ^2 {\mathbf {u}}-\nabla (\beta (T-T_{{\mathrm{ref}}}))=0 \end{aligned}$$
(1)
$$\begin{aligned}&\nabla (k \nabla T)=0 \end{aligned}$$
(2)

For a two-dimensional problem, these equations could be expressed as follows:

$$\begin{aligned}&(\lambda + \mu )\frac{\partial }{\partial x}\left( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right) + \mu \left( \frac{\partial ^2 u}{\partial x^2} + \frac{\partial ^2 u}{\partial y^2 } \right) \nonumber \\&\quad -\frac{\partial (\beta (T-T_{{\mathrm{ref}}}))}{\partial x}=0 \end{aligned}$$
(3)
$$\begin{aligned}&(\lambda + \mu )\frac{\partial }{\partial y}\left( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right) + \mu \left( \frac{\partial ^2 v}{\partial x^2} + \frac{\partial ^2 v}{\partial y^2 } \right) \nonumber \\&\quad -\frac{\partial (\beta (T-T_{{\mathrm{ref}}}))}{\partial y}=0 \end{aligned}$$
(4)
$$\begin{aligned}&\frac{\partial }{\partial x}\left( k\frac{\partial T}{\partial x}\right) + \frac{\partial }{\partial y}\left( k\frac{\partial T}{\partial y}\right) =0 \end{aligned}$$
(5)

where \(\lambda \) and \(\mu \) denote the Lamé coefficients and k is the thermal conductivity. The coefficient \(\beta \) is given by \(\beta =(3\lambda + 2\mu )\gamma \), where \(\gamma \) denotes the linear coefficient of thermal expansion.

The types of boundary conditions considered in this work are mainly essential or Dirichlet conditions which represent some prescribed displacement, denoted as \({\mathbf {u}}={\mathbf {u}}_0\) and some prescribed temperature \(T=T_0\) on some portions of the defined boundary. In a similar manner, traction or natural boundary conditions, denoted as \(\sigma {\mathbf {n}}={\mathbf {t}}_0\), which represent some prescribed surface traction in a specified boundary region as well as some temperature gradient or heat flux denoted by \((k(x)\nabla T) \cdot {\mathbf {n}} = q_0\), prescribed on a given boundary. \(\sigma \) denotes the stress tensor which for a linear thermoelastic problem could be expressed as

$$\begin{aligned} \sigma = [\lambda \mathrm{tr}(\varepsilon )-\beta (T-T_{{\mathrm{ref}}})]I + 2\mu \varepsilon , \quad \varepsilon = \frac{\nabla {\mathbf {u}} + (\nabla {\mathbf {u}})^t}{2}\nonumber \\ \end{aligned}$$
(6)

where I and \(\varepsilon \) are the identity and strain tensors, respectively. \({\mathbf {t}}_0=(t_{01},t_{02})^t\) is the traction vector and \({\mathbf {n}}=(n_1,n_2)^t\) denotes the unit normal vector. If a thermoelastic problem with essential boundary conditions is considered, the boundary conditions can be written as follows

$$\begin{aligned} t_{01}&= \mu n_2 \frac{\partial u}{\partial y} + \lambda n_1 \frac{\partial v}{\partial y} + (2\mu + \lambda )n_1\frac{\partial u}{\partial x} \nonumber \\&\quad + \mu n_2 \frac{\partial v}{\partial x}-\beta (T-T_{{\mathrm{ref}}})n_1 \end{aligned}$$
(7)
$$\begin{aligned} t_{02}&= \mu n_1 \frac{\partial u}{\partial y} + (2\mu + \lambda ) n_2 \frac{\partial v}{\partial y} + \lambda n_2\frac{\partial u}{\partial x} \nonumber \\&\quad + \mu n_1 \frac{\partial v}{\partial x}-\beta (T-T_{{\mathrm{ref}}})n_2 \end{aligned}$$
(8)
$$\begin{aligned} q_0&= n_1 k(x)\frac{\partial T}{\partial x} + n_2 k(x)\frac{\partial T}{\partial y} \end{aligned}$$
(9)

With all these definitions, it is possible to model the static linear thermoelastic problem correctly.

3 FPM discretization for general elliptic partial differential equations

In this section, a short description on FPM discretization for elliptic partial differential equations in a general form is described which has been already presented in [42]. Consider the numerical solution of an elliptic PDE with the form:

$$\begin{aligned} A_1 \psi + {\mathbf {B}} \cdot \nabla \psi + C_1 \Delta \psi - F = 0 \end{aligned}$$
(10)

with prescribed boundary conditions, where \(A_i, {\mathbf {B}}\), \(C_i\), F and \(\varphi \) are known. Thus, the corresponding FPM discretization under this setting and considering the general case with Robin boundary conditions is shortly sketched.

Consider the Taylor expansion of \(\psi ({\mathbf {x}}_i)\) around a point \({\mathbf {x}}\in \Gamma _r\)

$$\begin{aligned} \psi ({\mathbf {x}}_i)= & {} \psi ({\mathbf {x}}) + \sum _{k=1}^3\frac{\partial \psi }{\partial x_k}(x_{k,i}-x_k)\\&+\frac{1}{2}\sum _{j,k=1}^3\frac{\partial ^2 \psi }{\partial x_k \partial x_j}(x_{k,i}-x_k)(x_{j,i}-x_j)+e_{1,i} \end{aligned}$$

for \(i=1,\ldots ,m\), where \(e_{1,i}\) denotes the truncation error of the Taylor series expansion and m is the number of points inside the neighborhood of \({\mathbf {x}}\) which is defined through a Gaussian weight function w whose form is given by

$$\begin{aligned} w({\mathbf {x}}-{\mathbf {x}}_i) = \left\{ \begin{array}{cc} e^{-\alpha \Vert {\mathbf {x}}-{\mathbf {x}}_i\Vert ^2/h^2},&{} \text {if } \, \frac{\Vert {\mathbf {x}}-{\mathbf {x}}_i\Vert }{h}\le 1\\ 0&{} {\text {else}} \end{array} \right. \end{aligned}$$
(11)

where h defines the interaction length between nodes and it is called the smoothing length.

A linear system of \(m+2\) equations is obtained when the m Taylor series expansions for \(\psi ({\mathbf {x}}_i)\) are taken together with the elliptic partial differential equation and the corresponding boundary conditions. This linear system can be written in matrix form as

$$\begin{aligned} {\mathbf {e}} = M{\mathbf {a}}_\psi -{\mathbf {b}} \end{aligned}$$
(12)

where

$$\begin{aligned} M = \left( \begin{array}{cccccccccc} 1 &{} h_{1,1} &{} h_{2,1} &{} h_{3,1} &{} \frac{1}{2}h_{1,1}^2 &{} h_{1,1}h_{2,1} &{} h_{1,1}h_{3,1} &{} \frac{1}{2}h_{2,1}^2 &{} h_{2,1}h_{3,1} &{} \frac{1}{2}h_{3,1}^2\\ 1 &{} h_{1,2} &{} h_{2,2} &{} h_{3,2} &{} \frac{1}{2}h_{1,2}^2 &{} h_{1,2}h_{2,2} &{} h_{1,2}h_{3,2} &{} \frac{1}{2}h_{2,2}^2 &{} h_{2,2}h_{3,2} &{} \frac{1}{2}h_{3,2}^2\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 1 &{} h_{1,m} &{} h_{2,m} &{} h_{3,m} &{} \frac{1}{2}h_{1,m}^2 &{} h_{1,m}h_{2,m} &{} h_{1,m}h_{3,m} &{} \frac{1}{2}h_{2,m}^2 &{} h_{2,m}h_{3,m} &{} \frac{1}{2}h_{3,m}^2\\ A_1 &{} B_1&{} B_2 &{} B_3&{} C_1 &{} 0 &{} 0 &{} C_1 &{} 0 &{} C_1\\ A_2 &{} C_2 n_1&{} C_2 n_2&{} C_2 n_3&{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ \end{array} \right) \end{aligned}$$
(13)

with \(h_{1,j}=(x_{1,j}-x_1)\), \(h_{2,j}=(x_{2,j}-x_2)\) and \(h_{3,j}=(x_{3,j}-x_3)\),

$$\begin{aligned} {\mathbf {a}}_\psi= & {} \left( \psi ,\frac{\partial \psi }{\partial x_1},\frac{\partial \psi }{\partial x_2},\frac{\partial \psi }{\partial x_3},\frac{\partial ^2 \psi }{\partial {x_1}^2},\frac{\partial ^2 \psi }{\partial x_1 \partial x_2},\frac{\partial ^2 \psi }{\partial x_1 \partial x_3}, \frac{\partial ^2 \psi }{\partial {x_2}^2},\frac{\partial ^2 \psi }{\partial x_2 \partial x_3},\frac{\partial ^2 \psi }{\partial {x_3}^2}\right) ^t \end{aligned}$$
(14)
$$\begin{aligned} {\mathbf {b}}= & {} (\psi _1,\psi _2,\ldots ,\psi _m,F,\varphi )^t \end{aligned}$$
(15)
$$\begin{aligned} {\mathbf {e}}= & {} (e_1,e_2,\ldots ,e_m,e_{m+1},e_{m+2})^t \end{aligned}$$
(16)

and \({\mathbf {n}} = (n_1,n_2,n_3)^t\) denotes the unitary normal vector to the boundary. The unknown vector \({\mathbf {a}}_\psi \) is obtained trough the weighted least-squares method as

$$\begin{aligned} {\mathbf {a}}_\psi = (M^tW M)^{-1}(M^t W){\mathbf {b}}, \end{aligned}$$
(17)

where W is

$$\begin{aligned} W=\left[ \begin{array}{cccccc} w({\mathbf {x}}-{\mathbf {x}}_1) &{} 0 &{} \cdots &{} 0 &{} 0 &{} 0\\ 0 &{} w({\mathbf {x}}-{\mathbf {x}}_2) &{} \cdots &{} 0 &{} 0 &{} 0\\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} w({\mathbf {x}}-{\mathbf {x}}_n) &{} 0 &{} 0\\ 0 &{} 0 &{} \cdots &{} 0 &{} 1 &{} 0\\ 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} 1 \end{array} \right] \end{aligned}$$
(18)

Depending on where the point lies, that is, if \({\mathbf {x}}_i\in \Gamma _d\), \({\mathbf {x}}_i\in \Gamma _n\) or \({\mathbf {x}}_i\in \Omega \), some rows in M must be carefully replaced or deleted. Details can be found in [42].

Considering the following solution

$$\begin{aligned} {\mathbf {a}}_\psi = Q(M^t W){\mathbf {b}}, \end{aligned}$$
(19)

and taking \({\mathbf {q}}=(q_{1,1},q_{1,2},\ldots ,q_{1,10})\) as the first row of Q, the following linear system of equations arises

$$\begin{aligned}&\psi _j - \sum _{i=1}^{m_j}w_{j_i}\left( q_{1,1} + q_{1,2}h_{1,i} + q_{1,3}h_{2,i} + q_{1,4}h_{3,i} \right. \nonumber \\&\qquad \left. + q_{1,5}\frac{h^2_{1,i}}{2} + q_{1,6}h_{1,i}h_{2,i} + q_{1,7}h_{1,i}h_{3,i} \right. \nonumber \\&\qquad \left. + q_{1,8}\frac{h^2_{2,i}}{2} + q_{1,9}h_{2,i}h_{3,i}+ q_{1,10}\frac{h^2_{3,i}}{2}\right) \psi _{j_i} \nonumber \\&\quad = (q_{1,1}A_1 + q_{1,2}B_1 + q_{1,3}B_2 + q_{1,4}B_3 + q_{1,5}C_1 \nonumber \\&\qquad + q_{1,8}C_1 + q_{1,10}C_1)F \nonumber \\&\qquad + (q_{1,1}A_2 q_{1,2}C_2n_x + q_{1,3}C_2n_y + q_{1,4}C_2n_z )\varphi \end{aligned}$$
(20)

where \(\psi _j\) denotes the unknown function values at point j and \(m_j\) the number of jth-neighboring nodes inside its neighborhood defined by the weight function. It explicitly couples each point with its neighbor nodes which leads the coupling between inner and boundary points. Since Eq. (20) is valid for \(j=1,2,\ldots ,N\), this can be arranged in a global sparse system of linear equations \(K{\tilde{\Psi }} = {\mathbf {f}}\) which can be solved by iterative methods.

3.1 FPM discretization for static linear thermoelasticity equations

The procedure to obtain the global system of equations described in the last section can be extended for the two-dimensional static linear thermoelasticity equations as it is shown along this section. Consider the Taylor expansions for the two approximating solution functions of the coupled system of PDE equations,

$$\begin{aligned} u_i&= u + \sum _{k=1}^3\frac{\partial u}{\partial x_k}(x_{k,i}-x_k)\\&\quad +\frac{1}{2}\sum _{j,k=1}^3\frac{\partial ^2 u}{\partial x_k \partial x_j}(x_{k,i}-x_k)(x_{j,i}-x_j)+e_{1,i}\\ v_i&= v + \sum _{k=1}^3\frac{\partial v}{\partial x_k}(x_{k,i}-x_k)\\&\quad +\frac{1}{2}\sum _{j,k=1}^3\frac{\partial ^2 v}{\partial x_k \partial x_j}(x_{k,i}-x_k)(x_{j,i}-x_j)+e_{2,i}\\ T_i&= T + \sum _{k=1}^3\frac{\partial T}{\partial x_k}(x_{k,i}-x_k)\\&\quad +\frac{1}{2}\sum _{j,k=1}^3\frac{\partial ^2 T}{\partial x_k \partial x_j}(x_{k,i}-x_k)(x_{j,i}-x_j)+e_{3,i} \end{aligned}$$

where \(e_{1,i}\), \(e_{2,i}\) and \(e_{3,i}\) denote the errors of the Taylor series expansion. As in the previous section, together with these equations, (3), (4) and (5) with the corresponding boundary conditions should be also considered. \({\mathbf {u}} = \mathbf {u_0}=(u_0,v_0)\) and \(T=T_0\) in case of Dirichlet boundary conditions for displacement and temperature, or Eqs. (7), (8) and (9) in case of Neumann boundary conditions which is the most involved case explained next. If \({\mathbf {x}}\in \Gamma _n\), a linear system of \(m+6\) equations is obtained which in terms of the truncation error can be written as,

$$\begin{aligned} {\mathbf {e}} = {\tilde{M}}{\mathbf {a}}-{\mathbf {b}} \end{aligned}$$

where

$$\begin{aligned} {\tilde{M}}= \left( \begin{array}{ccc} M &{} {\mathbf {0}} &{} {\mathbf {0}}\\ {\mathbf {0}} &{} M &{} {\mathbf {0}}\\ {\mathbf {0}}&{} {\mathbf {0}} &{} M\\ J_1 &{} J_2 &{} J_3 \end{array} \right) \end{aligned}$$

and

$$\begin{aligned} {\mathbf {e}}&= (e_{1,1},\dots ,e_{1,m},e_{2,1},\dots ,e_{2,m},e_{3,1},\dots ,\\&\qquad e_{3,m},0,0,0,0,0,0)^t\\ {\mathbf {b}}&=(u_1,\dots ,u_m,v_1,\dots ,v_m,T_1,\dots ,T_m,0,0,0,t_{0,1}\\&\quad -\beta n_1 T_{{\mathrm{ref}}},t_{0,2}-\beta n_2 T_{{\mathrm{ref}}},q_0)^t\\ {\mathbf {a}}&= ({\mathbf {a}}_u,{\mathbf {a}}_v,{\mathbf {a}}_T)^t\\ {\mathbf {a}}_u&= \left( u,\frac{\partial u}{\partial x_1},\frac{\partial u}{\partial x_2},\frac{\partial ^2 u}{\partial x_1^2},\frac{\partial ^2 u}{\partial x_1 \partial x_2}, \frac{\partial ^2 u}{\partial x_2^2} \right) ^t\\ {\mathbf {a}}_v&= \left( v,\frac{\partial v}{\partial x_1},\frac{\partial v}{\partial x_2},\frac{\partial ^2 v}{\partial x_1^2},\frac{\partial ^2 v}{\partial x_1 \partial x_2}, \frac{\partial ^2 v}{\partial x_2^2} \right) ^t\\ {\mathbf {a}}_T&= \left( T,\frac{\partial T}{\partial x_1},\frac{\partial T}{\partial x_2},\frac{\partial ^2 T}{\partial x_1^2},\frac{\partial ^2 T}{\partial x_1 \partial x_2}, \frac{\partial ^2 T}{\partial x_2^2} \right) ^t \end{aligned}$$

M is of the same form as in the original FPM discretization for elliptic PDEs. Moreover, matrices \(J_1\), \(J_2\) and \(J_3\) are defined as follows,

$$\begin{aligned} J_1= & {} \left( \begin{array}{cccccc} 0 &{} 0 &{} 0 &{} 2\mu + \lambda &{} 0 &{} \mu \\ 0 &{} 0 &{} 0 &{} 0 &{} \lambda + \mu &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} (2\mu + \lambda )n_1 &{} \mu n_2 &{} 0 &{} 0 &{} \mu \\ 0 &{} \lambda n_2 &{} \mu n_1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \end{array} \right) ,\\ J_2= & {} \left( \begin{array}{cccccc} 0 &{} 0 &{} 0 &{} 0 &{} \lambda + \mu &{} 0 \\ 0 &{} 0 &{} 0 &{} \mu &{} 0 &{} 2\mu + \lambda \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} \mu n_2 &{} \lambda n_1 &{} 0 &{} 0 &{} 0 \\ 0 &{} \mu n_1 &{} (2\mu + \lambda )n_2 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \end{array} \right) \end{aligned}$$

and

$$\begin{aligned} J_3 = \left( \begin{array}{cccccc} 0 &{} -\beta &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} -\beta &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} k &{} 0 &{} k \\ -\beta n_1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ -\beta n_2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} n_1 &{} n_2 &{} 0 &{} 0 &{} 0 \end{array} \right) \end{aligned}$$

Following the procedure in the previous section, the corresponding solution for the linear system can be written as

$$\begin{aligned} {\mathbf {a}} = ({\tilde{M}}^tW {\tilde{M}})^{-1}({\tilde{M}}^t W){\mathbf {b}} \end{aligned}$$
(21)

Therefore, for the computation of the displacements and temperature fields, only the first, seventh and thirteenth components in \({\mathbf {a}}\) are really needed. Consequently if

$$\begin{aligned} {\mathbf {q}}_i = (q_{i,1},q_{i,2},\ldots ,q_{i,18})^t \end{aligned}$$

denotes the ith row in \(({\tilde{M}}^tW {\tilde{M}})^{-1}\), Eq. (21) can be slightly worked out and the following linear system arises for the elements of the displacement vector \({\mathbf {u}}\) and T,

$$\begin{aligned}&u_j-\sum _{i=1}^{m_j}w\left[ \left( q_{k,1} + h_{1,i} q_{k,2} + h_{2,i} q_{k,3}\right. \right. \nonumber \\&\qquad \left. + \frac{1}{2} h_{1,i}^2 q_{k,4}+ h_{1,i}h_{2,i} q_{k,5}+ \frac{1}{2}h_{2,i}^2 q_{k,6}\right) u_{i}\nonumber \\&\qquad + \left( q_{k,7} + h_{1,i} q_{k,8} + h_{2,i} q_{k,9}\right. \nonumber \\&\qquad \left. + \frac{1}{2} h_{1,i}^2 q_{k,10}+ h_{1,i}h_{2,i}q_{k,11}+ \frac{1}{2}h_{2,i}^2 q_{k,12}\right) v_{i}\nonumber \\&\qquad + \left( q_{k,13} + h_{1,i} q_{k,14} + h_{2,i} q_{k,15}\right. \nonumber \\&\qquad \left. \left. +\frac{1}{2} h_{1,i}^2 q_{k,16}+ h_{1,i}h_{2,i}q_{k,17}+ \frac{1}{2}h_{2,i}^2 q_{k,18}\right) T_{i} \right] \nonumber \\&\quad = [(2\mu +\lambda )n_1 q_{k,2} + \mu n_2 (q_{k,3} + q_{k,8}) \nonumber \\&\qquad +\lambda n_1 q_{k,9}-\beta n_1 q_{k,13}](t_{0,1}-\beta n_1 T_{{\mathrm{ref}}})\nonumber \\&\qquad +[\lambda n_2 q_{k,2} + \mu n_1 (q_{k,3} + q_{k,8})\nonumber \\&\qquad +(2\mu +\lambda )n_2 q_{k,9}-\beta n_2 q_{k,13}](t_{0,2}-\beta n_1 T_{{\mathrm{ref}}}) \nonumber \\&\qquad +[n_1 q_{k,14} + n_2 q_{k,15}]q_0 \end{aligned}$$
(22)

for the component u. A similar expression can be obtained for the component v and T on the jth point. This is valid for all \(u_j, v_j , T_j \in \Gamma _n\) where \(k=1,7,13\). Furthermore, an homogeneous linear system in (22) is obtained for internal domain points where in such case the last three rows in \({\tilde{M}}\) and elements in \({\mathbf {b}}\) must be deleted. Finally, using this last formulation for the displacement and temperature fields, a coupled sparse linear system, \(K{\tilde{\Psi }} = {\mathbf {f}}\), can be obtained which is numerically solved by iterative methods. Once these displacements and temperatures are computed, the stress can be obtained locally from the moving least-squares solution (21) with a updated vector \({\mathbf {b}}\) and the corresponding deformation values. Thus, the numerical solution of another sparse system for the computation of these quantities is not necessary.

4 Numerical examples

In order to test the numerical performance of the proposed FPM coupled discretization, four examples are presented in this section. These examples have been taken from [29] and [51], and they were computed on a AMD A10-7400P CPU 2.5GHz processor running Windows 10 Home operating system with 12 GB of RAM. The physical difficulty of the presented examples is gradually increasing.

In order to measure the approximating solution error regarding the obtained displacements, von Mises stresses and the corresponding temperatures, the following relative error norms are used,

$$\begin{aligned} e_{\infty }({\mathbf {u}})&= \frac{\max _{x\in X}\{\max \{|u({\mathbf {x}})-{\hat{u}}({\mathbf {x}})|,|v({\mathbf {x}})-{\hat{v}}({\mathbf {x}})|\}\}}{\max _{x\in X} \{|u({\mathbf {x}})|,|v({\mathbf {x}})|\} } \end{aligned}$$
(23)
$$\begin{aligned} e_{\infty }(\sigma _v)&= \frac{ \max _{x\in X} |\sigma _{v}({\mathbf {x}})-{\hat{\sigma }}_{v}({\mathbf {x}})| }{\max _{x\in X} |\sigma _{v}({\mathbf {x}})| } \end{aligned}$$
(24)
$$\begin{aligned} e_{\infty }(T)&= \frac{ \max _{x\in X} |T({\mathbf {x}})-{\hat{T}}({\mathbf {x}})| }{\max _{x\in X} | T({\mathbf {x}})| } \end{aligned}$$
(25)
$$\begin{aligned} e_{2}({\mathbf {u}})&= \frac{ \left\| {\mathbf {u}}({\mathbf {x}})-\hat{{\mathbf {u}}}({\mathbf {x}})\right\| }{ \left\| \hat{{\mathbf {u}}}({\mathbf {x}}) \right\| } \end{aligned}$$
(26)
$$\begin{aligned} e_{2}(\sigma _v)&= \frac{ | \sigma _v({\mathbf {x}})-{\hat{\sigma }}_v({\mathbf {x}}) | }{ | {\hat{\sigma }}_v({\mathbf {x}}) | } \end{aligned}$$
(27)
$$\begin{aligned} e_{2}(T)&= \frac{ | T({\mathbf {x}})-{\hat{T}}({\mathbf {x}}) | }{ | {\hat{T}}({\mathbf {x}}) | } \end{aligned}$$
(28)

where X denotes the set of nodes.

Fig. 1
figure 1

Problem configuration

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Fig. 2
figure 2

Left: FPM solution error with respect to the exact solution. Right: error comparison. n is the number of support nodes and \({\hat{\sigma }}\) is a Gaussian weight shape parameter

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4.1 Bending of an isothermal square

The first benchmark problem is geometrically described in Fig. 1 together with the corresponding boundary conditions. In this case a linearly increasing force is applied on the top boundary in order to deform the isothermal square. The corresponding analytical solution for the displacements reads,

$$\begin{aligned} u&= \frac{-\lambda }{8\mu (\lambda + \mu )}\left( x_1-\frac{1}{2} \right) ^2-\frac{(\lambda + 2\mu )}{8\mu (\lambda + \mu )}x_{2}^2 \end{aligned}$$
(29)
$$\begin{aligned} v&=\frac{(\lambda + 2\mu )}{4\mu (\lambda + \mu )}\left( x_1-\frac{1}{2}\right) x_2 \end{aligned}$$
(30)

The reported numerical results in this case were obtained considering \(\lambda = 0.4\), \(\mu =0.4\), \(\gamma =1\) using a discretization of 1681 points with a mean spacing of 0.025 m, \(\alpha =16.25\) and a smoothing length of 0.075 m.

Figure 2 shows the magnitude of the obtained displacement on the deformed square and the corresponding error with respect to the analytical solution. The displacement has been scaled by a factor of 0.5 for visualization purposes. As it can be observed, the highest value in the numerical error is of the order of \(10^{-8}\) which is valid for several configurations regarding the number of points and their distribution. This shows an excellent numerical behavior of FPM in this case.

Fig. 3
figure 3

Problem configuration

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4.2 Thermal expansion of a square

The second tested problem consists of a thermal expansion of a square with a temperature profile defined by \(T(x_1,x_2) = \sin (2\pi x_1)\). The geometry and the boundary conditions are depicted in Fig. 3. The exact analytical solution corresponding to the displacements reads,

$$\begin{aligned} u&= -\frac{\gamma (3\lambda + 2\mu )[\cos (2\pi x_1)-1]}{2\pi (\lambda + 2\mu )} \end{aligned}$$
(31)
$$\begin{aligned} v&=0 \end{aligned}$$
(32)
Fig. 4
figure 4

Left: Displacement magnitude. Right: Pointwise displacement relative error

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The reported numerical results in this case were obtained considering \(\lambda = 0.4\), \(\mu =0.4\), \(\gamma =1\) using a discretization of 1681 points with a mean spacing of 0.025 m, \(\alpha =16.25\) and a smoothing length of 0.075 m.

Figure 4 shows the magnitude of the obtained displacement on the thermal expanded square and the corresponding error with respect to the analytical solution. As it can be observed, the maximum computed error is of the order of \(10^{-3}\). The value of \(\alpha \) in the shape function is used to emphasize the effect of the closest neighbors, such that neighbor particles with the smallest distance to the central node have the highest weight. Thus, it is dependent on the nature of the case being analyzed. For example, for solving the incompressible Navier–Stokes equations values around 6 have shown good results [47], for coupled fluid flow and heat transfer values around 5 were selected [38], and for solid mechanics problems, values around 12 were chosen [42]. Therefore, the influence of \(\alpha \) on the numerical accuracy for this example is described in Fig. 5 in order to select a suitable value for thermoelasticity problems. This figure suggests that a suitable value for \(\alpha \) should be higher than 15 since from this value, the error no longer significantly decreases with the increase of \(\alpha \), and therefore, for the numerical examples presented in this work a value of 16.25 was chosen. In a similar manner, the number of neighboring points has been chosen according to the reported error in Fig. 6, where the displacement relative error with respect to the number of neighboring points increases as the number of points in the shape function support also increases. Stability dispersion is shown in Fig. 7 with respect to several degrees of increment on the perturbation parameter \(\delta \) which defines the distortion of the uniform distribution, see [44] for more details. It shows that this meshfree formulation is independent of the uniformity of the nodes discretizing the domain since, for a uniformly distributed distribution and a very distorted one, this proposal is able to get results with the same level of accuracy. For completeness, Fig. 8 shows the displacement error behavior with respect to the total number of points and Fig. 9 reports the FPM spent total time when solving the problem with respect to the number of discretization nodes. They indicate a good convergence behavior and an efficient computational strategy with respect to the increasing discretization points.

Fig. 5
figure 5

Displacement relative error in \(L_{\infty }\)-norm with respect to alpha

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Fig. 6
figure 6

Displacement relative error in \(L_{\infty }\)-norm with respect to number of neighboring points

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Fig. 7
figure 7

Displacement relative error in \(L_{\infty }\)-norm with respect to points dispersion

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Fig. 8
figure 8

Displacement relative error in \(L_{\infty }\)-norm with respect to total number of points

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Fig. 9
figure 9

Total computation time with respect to total number of points

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Fig. 10
figure 10

Problem configuration

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4.3 Ring plate

The third benchmark problem consists in the simulation of the thermoelastic state of a ring. The corresponding problem configuration and geometrical description are shown in Fig. 10.

The analytical solution for the involved variables is defined in polar coordinates as follows [51]:

  • The analytical expression for temperature,

    $$\begin{aligned} T(r)=1-\frac{\ln (r)}{\ln (2)} \end{aligned}$$
    (33)

  • The displacement of the ring,

    $$\begin{aligned} u_r(r)&=-\frac{r\ln (r)}{2\ln (2)} \end{aligned}$$
    (34)
    $$\begin{aligned} u_{\theta }(r)&= 0 \end{aligned}$$
    (35)

  • The strain,

    $$\begin{aligned} \varepsilon _r(r)&=\frac{\partial u_r}{\partial r}=-\frac{\ln (r)+1}{2\ln (2)} \end{aligned}$$
    (36)
    $$\begin{aligned} \varepsilon _{\theta }(r)&= \frac{1}{r}\frac{\partial u_{\theta }}{\partial \theta } + \frac{u_r}{r}=-\frac{\ln (r)}{2\ln (2)}, \end{aligned}$$
    (37)
    $$\begin{aligned} \varepsilon _{r\theta }(r)&= \frac{1}{2}\left( \frac{1}{r}\frac{\partial u_r}{\partial \theta } + \frac{\partial u_{\theta }}{\partial r}-\frac{u_{\theta }}{r} \right) =0 \end{aligned}$$
    (38)

  • The plane stress,

    $$\begin{aligned} \sigma _r(r)&=\frac{E}{1-\nu ^2}[\varepsilon _r + \nu \varepsilon _{\theta }-(1+\nu )\gamma (T-T_{{\mathrm{ref}}})]\nonumber \\&=\varepsilon _r(r)-T(r), \end{aligned}$$
    (39)
    $$\begin{aligned} \sigma _{\theta }(r)&= \frac{E}{1-\nu ^2}[\varepsilon _{\theta } + \nu \varepsilon _r-(1+\nu )\gamma (T-T_{{\mathrm{ref}}})]\nonumber \\&=\varepsilon _r(\theta )-T(r), \end{aligned}$$
    (40)
    $$\begin{aligned} \sigma _{r\theta }(r)&= \frac{E}{1+\nu }\varepsilon _{r\theta }=0 \end{aligned}$$
    (41)

The reported numerical results in this case were obtained considering \(\lambda = 0\), \(\mu =0.5\), \(\gamma =1\) using a discretization of 2081 points with a mean spacing of 0.05 m, \(\alpha =16.25\) and a smoothing length of 0.15 m.

Fig. 11
figure 11

Left: Temperature profile. Middle: Displacement magnitude. Right: von Mises stress profile

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Fig. 12
figure 12

Left: Temperature relative error. Middle: Displacement relative error. Right: von Mises stress relative error

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Fig. 13
figure 13

Relative error in \(L_{\infty }\)-norm with respect to alpha. Left: Temperature. Middle: Displacement. Right: von Mises stress

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Fig. 14
figure 14

Relative error in \(L_{\infty }\)-norm with respect to number of neighboring points. Left: Temperature. Middle: Displacement. Right: von Mises stress

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Fig. 15
figure 15

Relative error in \(L_{\infty }\)-norm with respect to total number of points. Left: Temperature. Middle: Displacement. Right: von Mises stress

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Figure 11 shows the obtained displacement, temperature and von Mises stress profiles computed with this FPM approach, and Fig. 12 shows the relative error of the obtained displacement, temperature and von Mises stress with respect to the analytical solution. As it can be observed the maximum computed error is of the order of \(10^{-4}\) which shows an excellent match of the computed solution.

Fig. 16
figure 16

Total computation time with respect to total number of points

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Figure 13 shows the error behavior with respect to \(\alpha \) in the shape function for the three quantities. This one again suggests and confirms that a suitable value for \(\alpha \) should be higher than 15. In a similar manner, the number of neighboring points has been chosen according to the reported error in Fig. 14, where the displacement, temperature and von Mises stress relative error with respect to the number of neighboring points increase as the number of points in the shape function support also increases. Finally, Fig. 15 shows the error behavior with respect to the total number of points and Fig. 16 reports the FPM spent total time when solving the problem with respect to the number of discretization nodes. As for the previous example, they indicate a good convergence behavior and an acceptable computational efficiency.

Fig. 17
figure 17

Geometry of heat exchanging device

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Fig. 18
figure 18

Top: Temperature profile. Middle: Displacement magnitude. Bottom: von Mises stress profile

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4.4 Heat exchanging device

The last and more realistic example is depicted in Fig. 17. The aim in this example is to compute the thermoelastic state in a complex geometry. In this example, a hot medium heats the device to 60\(^\circ \) inside the boreholes. A surrounding medium establishes a temperature of 35\(^\circ \) on the outside.

The reported numerical results in this case were obtained considering \(\lambda = 6.05 \cdot 10^{10}\), \(\mu =2.59 \cdot 10^{10}\), \(\gamma = 23 \cdot 10^{-6}\), \(k=237\) and \(T_{{\mathrm{ref}}}=0\) using a discretization of 4091 points with a mean spacing of 0.3 m, \(\alpha =16.25\) and a smoothing length of 0.9 m.

The corresponding numerical results are reported in Fig. 18. The reached temperature profile distribution is shown at the top of the figure, the magnitude of the displacement and the corresponding deformation scaled by a factor of 50 (for visualization purposes) are shown at the middle of the picture and the von Mises stress is also reported at the bottom of this figure. This numerical example shows the generality of the method for dealing with distorted and complex domains. These results indicates the robustness and effectiveness of this coupled formulation for dealing with problems involving linear thermomechanical processes.

5 Conclusions

A fully coupled FPM mathematical derivation for two-dimensional static linear thermoelastic problems has been presented in detail and numerically implemented for the first time. Considering the numerical reported results, it can be said that FPM is a powerful and versatile strong form meshfree approach for these kinds of solid mechanics problems. The identification of the involved parameters for an efficient FPM implementation has been done in a robust manner using the corresponding relative error analysis. The easy-to-handle boundary conditions in FPM are one featured that shall be stood out and this can be observed in the excellent matching with respect to the corresponding analytical solutions in the numerical examples. Although a fully coupled approach has been proposed in this paper, only one way coupling has been of interest in this work, i.e., the temperature dependence of the deformation is present; however, if a two-way coupling is considered this formulation could also work in an efficient manner with the correct form of the involved deformation tensor and this can be considered as an natural extension of this work in the near future. Furthermore, this approach is a promising numerical tool which could be coupled with mass transfer computations to achieve a fully coupled multi-physics formulation. As a future research, the extension to transient linear and nonlinear thermoelastic problems is of particular interest.