1 Introduction

Hybrid methods were developed to augment experimental techniques in obtaining comprehensive solutions for engineering problems. Earlier hybrid experimental—numerical methods involved supplementing numerical methods with experimental data. Later, combinations of experimental, numerical and analytical methods were explored to get the complete idea by circumventing the inherent limitations in the individual methods.

Hybrid experimental–numerical stress analysis techniques were used to supplement numerical methods. Kobayashi [1] summarizes the evolution of hybrid experimental–numerical stress analysis techniques supplementing the numerical methods with the boundary data obtained from experimental techniques. Illustrative case studies including 2D and 3D structural components, biomechanics and fracture mechanics were presented. Atluri and Nishioka [2] discussed hybrid methods for stress analysis including hybrid–analytical–numerical (H–A–N) method, hybrid–experimental–numerical (H–E–N) method and hybrid–numerical (H–N) method. The authors [3] developed “Finite-element-alternating method, a H–A–N method based on Schwarz–Neuman method, and demonstrated its application in fracture mechanics. A problem of a traction-free crack in a finite body subjected to a loading was decomposed into two parts—a finite body without crack and an infinite body with the crack. In the first part, they employed FE simulations to solve an identical uncracked finite body with the same loading. Subsequently, an infinite body with a crack subjected to arbitrary loading on the crack surface was solved analytically. The traction-free boundary conditions on the crack faces were incorporated by superposing equal and opposite residual stresses obtained from FE simulations to the analytical method. The resulting set of linear algebraic equations was solved for unknown coefficients, and these coefficients were employed for evaluation of stress intensity factor (SIF). Nishioka [4] reviewed the hybrid numerical methods, combining two or more different methods, and categorized the hybrid numerical methods employed in static and dynamic problems of fracture mechanics based on the state of hybridization of the numerical method with experimental or analytical methods. These are, namely, H–E–N, H–N–E, H–A–N, H–N–A and H–N–N and are illustrated with pertinent examples.

Lin et al. [5] reviewed a hybrid experimental–analytical technique coupling thermoelastic stress analysis (TSA) and a 2D elastostatic solution method. The method involves evaluating isopachic stresses over the domain from the data recorded by the TSA system and employing the stresses in equations furnished by Michell stress solution to obtain the unknown constants in ASF. The resulting set of linear algebraic equations was solved using least-squares technique. The method was presented for different cases including, among others, irregularly-perforated plates, plates with cracks and notches. Employing the ASF constants, they evaluated the stresses over the domain and validated the stresses with FEA.

Wen and Aliabadi [6] proposed a hybrid technique coupling finite difference method (FDM) and moving least squares method (MLS) for 2D elastostatic and elastodynamic cases. Employing finite difference grid comprising of domain points, intermediary points near irregular boundaries and boundary points, they solved the governing equations using classical central FDM at the domain points and point collocation method (PCM) incorporating MLS for the intermediary and boundary points. The method was illustrated for static cases including rectangular plate with a circular hole and parallelogram dam under constant lateral pressure and a dynamic case involving a square sheet under dynamic load. The results were compared with boundary element method (BEM) and an exact analytical solution for static and dynamic cases, respectively; however, a small variance was observed between the hybrid method and BEM (the benchmark solution). The method demonstrated a computationally efficient strategy to calculate shape functions of higher order derivatives. Khaji and Khodakarami [7] have presented a semi-analytical approach for a 2D potential problem. The method uses a numerical approach on the boundary discretized by using higher order non-isoparametric element while analytical approach is employed inside the domain. The non-isoparametric element incorporates higher-order Chebyshev mapping functions and shape functions constructed to furnish Kronecker Delta property for the function and its derivatives. Khaji and Khodakarami [8] have extended the same strategy for solving 3D elastostatics .

Maheri et al. [9] presented an aero-structure simulation combining analytical and FEA for wind turbines featuring bend-twist adaptive blades. The two-part method included establishing an analytical model between the induced twist and the flap bending at the hub in the first part. The induced twist was determined by FEA at a particular run-condition to incorporate the effects of intrinsic material and structural properties. These FE-data at the particular run-condition were coupled with the analytical model to predict the performance at other run-conditions. The results demonstrated a practical strategy for aerodynamic optimization complemented by a significant reduction in the computational time.

In dynamic problems, Ma et al. [10] proposed a method combining an analytical method and FE simulations for vibration analysis of plate structures with discontinuities for mid-frequency range. The method includes discretizing the plate structure uniformly into rectangle segment and non-rectangle segment. The rectangular segments are analyzed by using an analytical wave propagation method while non-rectangular segments are analyzed using FE. Dynamic coupling incorporated by enforcing compatibility and equilibrium conditions at the interface of the segments furnishes the hybrid solution for the complete structure. Chen et al. [11] developed a hybrid analytical–numerical approach for finite cylindrical shell with interior structures subjected to vibration. The structures modeled as cylindrical shell with interior structural attachments were analyzed by employing Wave based method (WBM) and Finite element method (FEM), respectively. Artificial spring technique was adopted for coupling the internal structure and the shell for the analysis. Vibration response obtained from the method and FE simulations were compared for different coupling and boundary conditions.

In the recent past, numerical–theoretical approaches have also been explored in elastostatic problems. Louhghalam et al. [12] approached a problem of stress concentration factor (SCF) evaluation in a plate with openings subjected to bending by coupling extended conformal mapping method and coarse-mesh FE stress. The extended conformal mapping method employs Laurent series. The method involves evaluating the unknown constants in the series by conformally mapping the series to a unit disk and satisfying the boundary conditions by using regression analysis. To evaluate SCF in a finite plate with openings, the authors couple the coarse-mesh FE data retrieved along a contour in an intermediate region with Laurent series obtained in the previous step to estimate the far-field boundary conditions. The estimated far-field conditions were employed in evaluating stress concentration factor (SCF) at the openings. Results obtained from the method were validated with fine mesh-FE results, showing a good correspondence over the domain.

Grm and Batista [13] proposed a coupled analytical–FEM method (Mixed method) for the stress analysis of a finite plate with a polygonal shaped hole. In the method, a finite domain with prescribed boundary conditions is analyzed by superposition of the analytical results for an infinite domain with FE results of the finite domain [14]. These FE results of the finite domain are obtained by imposing the difference in tractions on the finite boundary—obtained by superposing the analytically rendered tractions on the finite boundary with the prescribed tractions. They formulated the problem by employing modified Muskhelishvili complex-variables method utilizing Schwarz–Christoffel mapping functions. The resulting linear algebraic equations were solved for the unknown mapping coefficients by satisfying the boundary conditions on hole. Subsequently, these coefficients were utilized to obtain the solution for infinite domain and employed to project the tractions of the finite boundary. The ensuing tractions on the finite body were superposed with the prescribed tractions to get the differential tractions. These corrections in BCs were supplemented to FEM for evaluation of correction in the solution determined analytically. A parametric study involving a finite plate with square holes was undertaken to illustrate the method. The maximum shear stresses, computed and plotted over the domain, were compared with FEA employing refined meshes. These results demonstrated the computational time efficiency and accuracy of the method.

On a similar line, a novel FE–analytical method incorporating a mesh-reduction technique is proposed and explored for 2D elastostatic problems. The main idea of the paper is built on a hybrid FE–analytical approach where the coarse FE-approach is coupled with ASF approach. The constants in ASF are obtained from Fourier series coefficients—obtained by fitting coarse-mesh FE displacement data retrieved on the inner circular boundary with trigonometric functions as the basis functions—and tractions on the inner circular boundary. The evaluated constants are used to compute von Mises stresses over the domain and are compared with fine-mesh FE results.

The following section delineates an outline of the method and is followed by Sect. 3 describing the detailed features of the method. Illustrative cases including square and hexagonal perforated plates under symmetrical, anti-symmetrical and asymmetrical loadings are presented in Sect. 4. While Sect. 5 presents the results for these cases, Sect. 6presents the discussions of the HM.

2 FEA–analytical hybrid method: an outline

The hybrid method (HM) is based on a combined approach of numerical and analytical method. In this method, FE approach is coupled with an analytical approach involving infinite series for stress analysis of a 2D linear elastic structure. The method utilizes generalized ASF in polar coordinate given by Michell [15, 16] and [17].

The following subsections delineate the idea of the hybrid method including the field-equations and its coupling with coarse-mesh FE results.

2.1 Airy constants from fourier series

The polygonal plate with a hole is modeled as a plane stress problem and is formulated in terms of stresses using polar coordinates. Eq. (1) describes the governing equation in the domain in terms of ASF.

$$\begin{aligned} {\nabla ^4}\phi = {\nabla ^2}\left( {{\nabla ^2}\phi } \right) = \left( {\frac{{{\partial ^2}}}{{\partial {r^2}}} + \frac{1}{r}\frac{\partial }{{\partial r}} + \frac{1}{{{r^2}}}\frac{{{\partial ^2}}}{{\partial {\theta ^2}}}} \right) \,\left( {\frac{{{\partial ^2}}}{{\partial {r^2}}} + \frac{1}{r}\frac{\partial }{{\partial r}} + \frac{1}{{{r^2}}}\frac{{{\partial ^2}}}{{\partial {\theta ^2}}}} \right) \,\phi = 0 \end{aligned}$$
(1)

A generalized solution for the above equation is given in Eq. (2), as seen in [15, 16] and [17].

$$\begin{aligned} \phi \left( {r,\theta } \right)=\, & {} {a_0} + {b_0}\log r + {c_0}{r^2} + {d_0}{r^2}\log r + ({A_0} + {B_0}\log r + {C_0}{r^2} + {D_0}{r^2}\log r)\,\theta \nonumber \\&+ \,\left( {{a_1}\,r+ {b_1}\,r\,\log \,r + \frac{{{c_1}}}{r} + \,{d_1}\,{r^3}} \right) \,\sin \theta \,\nonumber \\&+ \,\left( {{a^\prime _1}\,r+ {b^\prime _1}\,r\,\log \,r + \frac{{{c^\prime _1}}}{r} + \,{d^\prime }_1\,{r^3}} \right) \,\cos \theta \nonumber \\&+ \,\left( {{A_1}\,r+ {B_1}\,r\,\log \,r} \right) \,\theta \,\sin \theta \, + \left( {{A^\prime _1}\,r+ {B^\prime _1}\,r\,\log \,r} \right) \,\theta \,\cos \theta \nonumber \\&+ \,\sum \limits _{n = 2,3,4\ldots }^\infty {\left[ {{a_n}\,{r^n} + {b_n}\,{r^{2 + n}} + {c_n}\,{r^{ - n}} + {d_n}\,{r^{2 - n}}} \right] } \,\sin \,(n\,\theta ) + \sum \limits _{n = 2,3,4\ldots }^\infty {\left[ {{a^\prime _n}{r^n} + {b^\prime _n}\,{r^{2 + n}} + {c^\prime _n}{r^{ - n}} + {d^\prime _n}\,{r^{2 - n}}} \right] } \,\cos \,(n\,\theta ) \end{aligned}$$
(2)

In Eq. (2), the constants \({a_0},\,{b_0},{c_0},\,{d_0},{A_0},{B_0},{C_0},{D_0},{a_1},{b_1}, {c_1},\,{d_1},{a^\prime _1},{b^\prime _1},{c^\prime _1},{d^\prime _1}, {A_1},{B_1},{A^\prime _1},{B^\prime _1}, {a_n},{b_n},{c_n},{d_n},{a^\prime _n},{b^\prime _n},{c^\prime _n},{d^\prime _n}\) are evaluated by satisfying boundary conditions.

The field variables are evaluated from the ASF. For a polygonal plate with a hole radial, tangential and shear stress components are given collectively in Eq. (3).

$$\begin{aligned} &{\sigma _{rr}}\left( {r,\theta } \right)= \frac{1}{r}\frac{{\partial \phi }}{{\partial r}} + \frac{1}{{{r^2}}}\frac{{{\partial ^2}\phi }}{{\partial {\theta ^2}}} \nonumber \\&{\sigma _{\theta \theta }}\left( {r,\theta } \right) = \frac{{{\partial ^2}\phi }}{{\partial {r^2}}} \nonumber \\&{\sigma _{r\theta }}\left( {r,\theta } \right) = - \frac{\partial }{{\partial r}}\left( {\frac{1}{r}\frac{{\partial \phi }}{{\partial \theta }}} \right) \end{aligned}$$
(3)

The expressions for radial, tangential and shear strains are given collectively in Eq. (4)

$$\begin{aligned}&{\varepsilon _{rr}}\left( {r,\theta } \right) = \,\,\frac{1}{E}\left( {{\sigma _{rr}} - \nu \,{\sigma _{\theta \theta }}} \right) \,\, = \,\,\frac{{\partial {u_{rr}}}}{{\partial r}} \nonumber \\&{\varepsilon _{\theta \theta }}\left( {r,\theta } \right) = \,\,\frac{1}{E}\left( {{\sigma _{\theta \theta }} - \nu \,{\sigma _{rr}}} \right) \,\, = \,\,\frac{{{u_{rr}}}}{r} + \frac{1}{r}\frac{{\partial {u_{\theta \theta }}}}{{\partial \theta }} \nonumber \\&{\gamma _{r\theta }}\left( {r,\theta } \right) = \,\,\frac{{2\left( {1 + \nu } \right) }}{E}{\tau _{r\theta }}\,\, = \,\,\frac{1}{r}\frac{{\partial {u_{rr}}}}{{\partial \theta }} + \frac{{\partial {u_{\theta \theta }}}}{{\partial r}} - \frac{{{u_{\theta \theta }}}}{r} \end{aligned}$$
(4)

Applying Hooke’s law, radial and circumferential displacements are evaluated. The corresponding expressions are given collectively in Eq. (5).

$$\begin{aligned}&{u_{rr}}\left( {r,\theta } \right) = \frac{1}{E}\int {\left( {{\sigma _{rr}} - \nu \,{\sigma _{\theta \theta }}} \right) } \,dr + {S_1}\,\sin \theta + {S_2}\,\cos \theta \nonumber \\&{u_{\theta \theta }}\left( {r,\theta } \right) = \frac{r}{E}\int {\left( {{\sigma _{\theta \theta }} - \nu \,{\sigma _{rr}}} \right) } \,d\theta - \frac{1}{E}\iint {\left( {{\sigma _{rr}} - \nu \,{\sigma _{\theta \theta }}} \right) }\, dr\,d\theta + {S_1}\,\cos \theta - {S_2}\,\sin \theta + {S_3}\,r \end{aligned}$$
(5)

The field variables are given by the following Eqs. (6)–(10),

$$\begin{aligned} {\sigma _{rr}}= & {} \frac{{{b_0}}}{{{r^2}}} + 2{c_0} + {d_0}\left( {2\ln r + 1} \right) + {B_0}\frac{\theta }{{{r^2}}} + 2{C_0}\theta + {D_0}\left( {2\ln r + 1} \right) \theta \nonumber \\&+\, \left( {\frac{{{b_1}}}{r} - \frac{{2{c_1}}}{{{r^3}}} + 2{d_1}r} \right) \sin \theta + \left( {\frac{{{b^\prime _1}}}{r} - \frac{{{2c^\prime _1}}}{{{r^3}}} + {2d^\prime _1} r} \right) \cos \theta \nonumber \\&+\, \left( {\frac{{2{A_1}}}{r}} \right) \cos \theta - \left( {\frac{{{2A^\prime _1}}}{r}} \right) \sin \theta + \left( {\frac{{{B_1}\theta }}{r}} \right) \sin \theta \nonumber \\&+\, \left( {\frac{{{B^\prime _1}\theta }}{r}} \right) \cos \theta + \left( {\frac{{2{B_1}}}{r}\ln r} \right) \cos \theta - \left( {\frac{{{2B^\prime _1}}}{r}\ln r} \right) \sin \theta \nonumber \\&-\, \sum \limits _{n = 2}^\infty {\left( \begin{matrix} {a_n}n(n - 1)\,{r^{n - 2}} + {b_n}(n + 1)(n - 2){r^n} \\ +\, {c_n}n(n + 1){r^{ - (n + 2)}} + {d_n}(n - 1)(n + 2){r^{ - n}} \end{matrix}\right) } \sin n\theta \nonumber \\&-\, \sum \limits _{n = 2}^\infty {\left( \begin{matrix} {a^\prime _n} n(n - 1)\,{r^{n - 2}} + {b^\prime _n}(n + 1)(n - 2){r^n} \\ +\, {c^\prime _n} n(n + 1){r^{ - (n + 2)}} + {d^\prime _n}(n - 1)(n + 2){r^{ - n}} \\ \end{matrix} \right) } \cos n\theta \end{aligned}$$
(6)
$$\begin{aligned} {\sigma _{\theta \theta }}= & {} - \frac{{{b_0}}}{{{r^2}}} + 2{c_0} + {d_0}\left( {2\ln r + 3} \right) - \frac{{{B_0}\,\theta }}{{{r^2}}} + 2{C_0}\theta + {D_0}\left( {2\ln r + 3} \right) \theta \nonumber \\&+\, \left( {\frac{{{b_1}}}{r} + \frac{{2{c_1}}}{{{r^3}}} + 6{d_1}r} \right) \sin \theta + \left( {\frac{{{b^\prime _1}}}{r} + \frac{{{2c^\prime _1}}}{{{r^3}}} + {6d^\prime _1} r} \right) \cos \theta \nonumber \\&+\, \left( {\frac{{{B_1}\theta }}{r}} \right) \sin \theta + \left( {\frac{{{B^\prime _1}\theta }}{r}} \right) \cos \theta \nonumber \\&+\, \sum \limits _{n = 2}^\infty {\left( \begin{matrix} {a_n}n(n - 1)\,{r^{n - 2}} + {b_n}(n + 1)(n + 2){r^n}\\ +\, {c_n}n(n + 1){r^{ - (n + 2)}} + {d_n}(n - 1)(n - 2){r^{ - n}} \end{matrix} \right) } \sin n\theta \nonumber \\&+\, \sum \limits _{n = 2}^\infty {\left( \begin{matrix} {a^\prime _n} n(n - 1)\,{r^{n - 2}} + {b^\prime _n}(n + 1)(n + 2){r^n} \\ +\, {c^\prime _n} n(n + 1){r^{ - (n + 2)}} + {d^\prime _n}(n - 1)(n - 2){r^{ - n}} \end{matrix} \right) } \cos n\theta \end{aligned}$$
(7)
$$\begin{aligned} {\sigma _{r\theta }}= & {} \frac{{{A_0}}}{{{r^2}}} + {B_0}\left( {\frac{{\ln r - 1}}{{{r^2}}}} \right) - {C_0} - {D_0}\left( {\ln r + 1} \right) \nonumber \\&+\, \left( { - \frac{{{b_1}}}{r} + \frac{{2{c_1}}}{{{r^3}}} - 2{d_1}r} \right) \cos \theta - \left( { - \frac{{{b^\prime _1}}}{r} + \frac{{{2c^\prime _1}}}{{{r^3}}} + {2d^\prime _1} r} \right) \sin \theta \nonumber \\&-\, \left( {\frac{{{B_1}}}{r}} \right) \sin \theta - \left( {\frac{{{B^\prime _1}}}{r}} \right) \cos \theta - \left( {\frac{{{B_1}}}{r}\theta } \right) \cos \theta + \left( {\frac{{{B^\prime _1}}}{r}\theta } \right) \sin \theta \nonumber \\&-\, \sum \limits _{n = 2}^\infty {\left( \begin{matrix} {a_n}n(n - 1)\,{r^{n - 2}} + {b_n}n(n + 1){r^n} \\ -\, {c_n}n(n + 1){r^{ - (n + 2)}} - {d_n}n(n - 1){r^{ - n}} \end{matrix} \right) } \cos n\theta \nonumber \\&+\, \sum \limits _{n = 2}^\infty {\left( \begin{matrix} {a^\prime _n} n(n - 1)\,{r^{n - 2}} + {b^\prime _n} n(n + 1){r^n}\\ -\, {c^\prime _n} n(n + 1){r^{ - (n + 2)}} - {d^\prime _n} n(n - 1){r^{ - n}} \end{matrix} \right) } \sin n\theta \end{aligned}$$
(8)
$$\begin{aligned} {u_{rr}}= & {} \frac{1}{E}\left\{ { - \frac{{{b_0}}}{r}} \right. (1 + \nu ) + 2{c_0}(1 - \nu )r + {d_0}(1 - \nu )(2r\,\ln r - r) - 2{d_0}\nu r \nonumber \\&+ \,\left[ { - \frac{{{B_0}}}{r}(1 + \nu ) + 2C{}_0(1 - \nu )r + {D_0}(1 - \nu )(2r\,\ln r - r) - 2{D_0}\nu r} \right] \theta \nonumber \\&+\, \left[ {{b_1}(1 - \nu )\,\ln r + \frac{{{c_1}}}{{{r^2}}}(1 + \nu ) + {d_1}{r^2}(1 - \nu ) - 2{d_1}\,\nu \,{r^2}} \right] \sin \theta \nonumber \\&+\, \left[ {{b^\prime _1}(1 - \nu )\,\ln r + \frac{{{c^\prime _1}}}{{{r^2}}}(1 + \nu ) + {d^\prime _1}{r^2}(1 - \nu ) - {2d^\prime _1}\,\nu \,{r^2}} \right] \cos \theta \nonumber \\&+\, \left( {2{A_1}\,\ln r} \right) \cos \theta - \left( {{2A^\prime _1}\,\ln r} \right) \sin \theta \nonumber \\&+\, \left[ {{B_1}\ln r(1 - \nu )} \right] \theta \sin \theta + \left[ {{B^\prime _1}\ln r(1 - \nu )} \right] \theta \cos \theta \nonumber \\&+\, \left[ {{B_1}{{\ln }^2}r} \right] \cos \theta - \left[ {{B^\prime _1}{{\ln }^2}r} \right] \sin \theta \nonumber \\&+\, \sum \limits _{n = 2}^\infty {\left[ \begin{matrix} {a_n}n(1 + \nu ){r^{n - 1}} + {b_n}\left[ {(n - 2) + \nu (n + 2)} \right] {r^{n + 1}} \\ -\, {c_n}n(1 + \nu ){r^{ - n + 1}} - {d_n}\left[ {(n + 2) + \nu (n - 2)} \right] {r^{ - (n - 1)}} \end{matrix} \right] } \sin n\theta \nonumber \\&+\, \sum \limits _{n = 2}^\infty {\left[ \begin{matrix} {a^\prime _n} n(1 + \nu ){r^{n - 1}} + {b^\prime _n}\left[ {(n - 2) + \nu (n + 2)} \right] {r^{n + 1}} \\ -\, {c^\prime _n} n(1 + \nu ){r^{ - n + 1}} - {d^\prime _n} \left[ {(n + 2) + \nu (n - 2)} \right] {r^{ - (n - 1)}} \end{matrix} \right] } \cos n\theta \nonumber \\&+\, \left. {g\left( \theta \right) } \right\} \end{aligned}$$
(9)
$$\begin{aligned} {u_{\theta \theta } }= & {} \frac{1}{E} \Bigg \{4{d_0}\theta r + 2{D_0}{\theta ^2}r - \left[ {{b_1}(1 - \nu )(1 - \,\ln r) + \frac{{{c_1}(1 + \nu )}}{{{r^2}}} + {d_1}{r^2}(5 + \nu )} \right] \cos \theta \nonumber \\&+\, \left[ {{b^\prime _1}(1 - \nu )(1 - \,\ln r) + \frac{{{c^\prime _1}\,(1 + \nu )}}{{{r^2}}} + {d^\prime _1}{r^2}(5 + \nu )} \right] \sin \theta \nonumber \\&-\, \left( {2{A_1}\,\left( {\ln r + \nu } \right) } \right) \sin \theta - \left( {{2A^\prime _1}\,\left( {\ln r + \nu } \right) } \right) \cos \theta \nonumber \\&+\, {B_1}\left[ {(1 - \nu ) - (1 + \nu )\ln r - {{\ln }^2}r} \right] \sin \theta \nonumber \\&+\, {{{B^\prime _1}}}\left[ {(1 - \nu ) - (1 + \nu )\ln r - {{\ln }^2}r} \right] \cos \theta \nonumber \\&-\, {B_1}\left[ {(1 - \nu ) - \ln r\,(1 - \nu )} \right] \theta \cos \theta + {B^\prime _1}\left[ {(1 - \nu ) - \ln r\,(1 - \nu )} \right] \theta \sin \theta \nonumber \\&-\, \sum \limits _{n = 2}^\infty {\left[ \begin{matrix} {a_n}n(1 + \nu ){r^{n - 1}} + {b_n}\left[ {n(1 + \nu ) + 4} \right] {r^{n + 1}} \\ +\, {c_n}n(1 + \nu ){r^{ - (n + 1)}} + {d_n}\left[ {n(1 + \nu ) - 4} \right] {r^{ - (n - 1)}} \end{matrix} \right] } \cos n\theta \nonumber \\&+\, \sum \limits _{n = 2}^\infty {\left[ \begin{matrix} {a^\prime _n} n(1 + \nu ){r^{n - 1}} + {b^\prime _n}\left[ {n(1 + \nu ) + 4} \right] {r^{n + 1}}\\ +\, {c^\prime _n} n(1 + \nu ){r^{ - (n + 1)}} + {d^\prime _n} \left[ {n(1 + \nu ) - 4} \right] {r^{ - (n - 1)}} \end{matrix} \right] } \sin n\theta \nonumber \\&-\, \int \limits _\theta {g\left( \theta \right) \,\,d\theta } + f\left( r \right) \Bigg \} \end{aligned}$$
(10)

These field variables expressions are the linear functions of the constants at specific values of polar coordinate. At a particular value of radial coordinate (r), the field variables can be expressed as Fourier series with ASF constants as its coefficients. Consequently, the tractions \(({\sigma _{rr}}\,\mathrm{{and}}\,{\sigma _{r\theta }})\) and displacements \(({u_{rr}}\,\mathrm{{and}}\,{u_{\theta \theta }})\) at the inner radius (\(r=a\)) can be expressed as the sum of harmonics with the polar angle as the independent variable with coefficients containing unknown ASF constants.

2.2 Coupling with coarse-mesh FE results

At a particular coordinate, the Fourier series coefficients encapsulate ASF constants. These Fourier series coefficients are obtained by harmonic regression analysis of displacement data furnished on the inner circular boundary \( (r = a) \) by a coarse-mesh FE simulation. The displacements retrieved from the FE simulation include radial and tangential displacement data. These equations form an overdetermined system of equations \((A\,x = b )\) and are solved using a least squares method employing harmonics as bases functions. The solution for an overdetermined system of equations is obtained by the following expression.

$$\begin{aligned} x = {\left( {{A^T}A} \right) ^{ - 1}}{A^T}b \end{aligned}$$

The Harmonic regression analysis is implemented using an in-built function in symbolic software MATHEMATICA. The regression analyses for these two sets of data—radial and tangential displacements—furnish two sets of linear algebraic equations. These two sets of equations along with another two sets of equations—obtained by satisfying the traction free conditions on the circular boundary strongly—provide equations for determination of ASF constants. The constants in ASF are determined by solving the above set of equations Eqs. (6), (8), (9) (10). Incorporating stress function constants, stresses over domain are evaluated and the same are plotted as contour and colour map over the domain.

3 The hybrid method: nuances

The preceding section delineated the idea of the hybrid method. The present section presents the features of HM including the weak dependence of HM on the mesh density, selection of the number of data-retrieving points, choice of the basis functions for the regression analysis and a criterion for number of terms in the series. To elucidate these features, an illustrative case of a square plate with a central hole subjected to an equi-biaxial displacement is considered for all the aspects.

3.1 Influence of mesh parameters: coarse versus fine and structured versus unstructured

The mesh-quality employed in the numerical method affects the solution. To demonstrate the efficacy of the HM with a coarse mesh and an unstructured mesh, a comparative study involving coarse-mesh based HMs and a fine-mesh based HM with very-fine mesh FEA and a comparative study involving a structured-mesh based HM and unstructured-mesh based HM with fine-mesh FEA is undertaken.

For the illustrative cases, the discretization is carried out in ANSYS using an auto-mesh option incorporated with a feature to accommodate user-defined mesh-refinements. Figure 1 demonstrates the effect of mesh densities—coarse and fine meshes on the HM. von Mises stress results obtained from coarse-mesh based HMs, having 1731, 2377, 3950 and 15,525 nodes, and fine-mesh based HM, having 146,017 nodes, are compared with fine-mesh based FEA, having 146,017 nodes. The corresponding mesh with details including the number of nodes, elements and data-retrieving points on the boundary are shown in Fig. 1. In Fig. 1a, b, the black regions near the corner indicate the stresses which are not in the stress-range indicated by FEA. Further, Fig. 1c–e—showing good correspondence with fine-mesh based FEA—demonstrate the weak dependence of the HM on mesh-grid size. On the other hand, it is essential to ensure enough nodes to capture the domain information. This is illustrated by comparing coarse mesh based HMs , Fig. 1a–d, with fine-mesh based FEA, Fig. 1f. Evidently, the coarse-mesh based HM reduces discretization time.

In Fig. 2, the effect of structured and unstructured mesh is illustrated for a case with a/c ratio equal to 0.2. The HM results are compared with a corresponding fine-mesh FEA showing convergence. For a comparative study, the numbers of nodes in the mapped and unmapped meshes are nearly same; they are 3873 (mapped) and 3950 (unmapped) for the case of a/c = 0.2. The figure shows no conspicuous variance between the results from the two meshes, indicating HM results are weakly dependent on the mesh-structure. Based on this observation, unstructured mesh (free or non-uniform mesh) is consistently adopted for illustrating all cases.

Fig. 1
figure 1

Effect of mesh density on HM: comparision of coarse-mesh based HMs and fine-mesh based HM with fine-mesh FEA a coarse-mesh HM (nodes-1731, elements-546, n-4) b coarse-mesh HM (nodes-2377, elements-756, n-4) c coarse-mesh HM (nodes-3950, elements-1271, n-4) d coarse-mesh HM (nodes-15,525, elements-5084, n-4) e fine-mesh HM (nodes-146,017, elements-48,384, n-4) f fine-mesh FEA (nodes-146,017, elements-48,384)

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Fig. 2
figure 2

Effect of mesh structure: comparison of mapped coarse-mesh based HM and free coarse-mesh based HM with free FEA (fine-mesh) a mapped coarse-mesh HM b free coarse-mesh HM c free fine-mesh FEA

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3.2 Selection of points on the circular boundary

HM involves incorporating data interrogated on the circular boundary at discrete points. To discern the influence of the number of data retrieving points, the coarse-mesh based HM employed in Sect. 3.1 is realized with 4 different sets of discrete points—17, 30, 301 and 601 points—on the circular boundary. The displacements are retrieved using a Path Operation, in-built command in ANSYS. A Path Operation captures the field variables information along the path at equally spaced discrete points. As in Sect. 3.1, von Mises results obtained from HM are compared with standard refined mesh FEA results. The results are shown in Fig. 3. Figure 3a–d show HM results for the cases with 601, 301, 30 and 17 interrogation points, respectively. In Fig. 3c, d, the black regions near the upper-left corner and the lower-right corner indicate stresses which are not in the stress-range indicated by FEM. Further, the figures show that the HM with 601 and 301 points have good correspondence with FEA in comparison to HM employing 30 and 17 points on the boundary. Thus, it is necessary to supplement data from a sufficient number of discrete points in the HM. In general, the number of points depends on the complexity of the problem.

3.2.1 Effect of mesh density on the number of retrieving points

The number of retrieving points employed in the HM depends on the FE mesh. To illustrate this, the following example is presented. Figure 4 shows the influence of retrieving points on the mesh-density. Here, a coarse-mesh HM (319 nodes) with 601 interrogation points shows deviation from a fine-mesh FEA while a fine-mesh HM (146,017 nodes) with 17 interrogation points shows a good correspondence.

3.2.2 Effect of \( \frac{a}{c} \) ratio on the number of retrieving points

The HM incorporates the data from the circular boundary. Figure 5 presents the effect of a/c ratios on number of retrieving points. The correspondence between HM analyses and FEA increases with increasing number of interrogation points for particular a/c ratio (Fig. 5, row-wise). This is due to the influence of more data sample points in the regression analysis, which helps in a better smoothening of the data.

On the other hand, the correspondence between the two methods decreases as the a/c ratio decreases for a particular number of retrieving points (Fig. 5, column-wise). This is attributed to distantness of the retrieving points from the outer boundary.

Fig. 3
figure 3

Effect of data points on the circular boundary for interrogation a coarse-mesh with 601 points HM b coarse-mesh with 301 points HM c coarse-mesh with 30 points HM d coarse-mesh with 17 points HM e fine-mesh FEA

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Fig. 4
figure 4

Effect of FE mesh on data points on the circular boundary for interrogation a coarse-mesh with 601 points based HM b fine-mesh with 17 points based HM c fine-mesh FEA (standard)

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Fig. 5
figure 5

Effect of a/c ratio on data retrieving points on circular boundary

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3.3 Harmonic regression analysis: curve fitting of the data from the boundary

In Sect. 2.1, the field variables are Fourier series with \(\theta \) as an independent variable at a particular value of radius r. This suggests employing harmonics as the basis functions for regression analysis of coarse-mesh FE data and is accomplished in MATHEMATICA.

3.4 Selection of terms in regression analysis (n): a criterion

The number of basis functions in regression analysis plays a significant role in evaluating the Fourier coefficients. A criterion based on comparison of relative error percentage defined with respect to coarse-mesh FE data with a predefined percentage error tolerance limit is employed. To illustrate the criterion, the illustrative case is considered.

The relative error percentage is given in Eq. (11), where \({\mathrm{{U}}_{\mathrm{{FEM}}}}\,\mathrm{{and }}\,{\mathrm{{U}}_{\mathrm{{HM}}}}\) are resultant displacements obtained from FEM and HM, respectively. The percentage error tolerance limit is specified as 0.01% The number of terms in the infinite series is decided iteratively by comparing the displacements obtained from the coarse-mesh HM with the coarse-mesh FE displacements. The number n giving minimum relative error percentage or maximum percentage match is chosen. Table 1 enumerates the percentage match for the case considered in Sect. 3.1 for various n values. The table shows that choice of n as 4 gives a maximum percentage match.

$$\begin{aligned} \mathrm {Error\,(\%)=\bigg |\frac{U_{FEM}-U_{HM}}{U_{FEM}}\bigg |}\times 100 \end{aligned}$$
(11)
Table 1 Iteration for selection of n
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4 Numerical examples: illustrations

To demonstrate the HM, stress evaluations of two centrally perforated polygonal plates including square and regular hexagon are considered. In addition, two ratios of hole-radius to side (a/c = 0.2 and 0.8) are considered, and these plates are subjected to symmetrical, anti-symmetrical and asymmetrical loading conditions.

The following two subsections describe the details for square and regular hexagon plates, respectively. In each subsection, 3 different loading cases are shown.

4.1 Square plate with a central hole

A schematic of square plate with a centrally placed hole is shown in Fig. 6 with a coordinate system located at the centre of the hole. The side of the square is 2c while diameter of the hole is 2a. The problem is formulated as plane stress case with \(2c=2\) m, \(E = 210\) GPa and \(\nu = 0.3\).

Fig. 6
figure 6

Schematic of a square plate with a centrally placed hole

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Fig. 7
figure 7

A square plate with a centrally placed hole subjected to a symmetrical loading case b anti-symmetrical loading case c asymmetrical loading case

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4.1.1 Symmetrical loading cases

A schematic of the plate with the loading condition is shown in Fig. 7a. Two-fold geometrical symmetry reduces the problem to the analysis of a quarter-plate. The quarter plate is subjected to a uniform displacement of 0.001 m on straight edges in the direction normal to it and is traction-free on the hole. Accounting for single-valuedness of displacements, finiteness of stresses and symmetries, the generalized ASF in Eq. (2) is reduced to ASF given Eq. (12).

$$\begin{aligned} \phi = {b_0}\log r + {c_0}{r^2} + \sum \limits _{n = 4,8,12\ldots }^\infty {\left[ {{a^\prime _n} {r^n} + {b^\prime _n}\,{r^{2 + n}} + {c^\prime _n}{r^{ - n}} + {d^\prime _n}\,{r^{2 - n}}} \right] } \,\cos \,(n\,\theta ) \end{aligned}$$
(12)

4.1.2 Anti-symmetrical loading cases

A schematic of the plate with the loading condition is shown in Fig. 7b with uniform displacement of 0.001 m parallel to the sides. Similar to Sect. 4.1.1, the generalized ASF reduces to the form in Eq. (13).

$$\begin{aligned} \phi = \,\,\sum \limits _{n = 2,6,10\ldots }^\infty {\left[ {{a_n}\,{r^n} + {b_n}\,{r^{2 + n}} + {c_n}\,{r^{ - n}} + {d_n}\,{r^{2 - n}}} \right] } \,\mathrm{{sin}}\,(n\,\theta ) \end{aligned}$$
(13)

4.1.3 Asymmetrical loading cases

A schematic of the plate with the loading condition is shown in Fig. 7c. The figure shows a general loading case including uniform displacement (0.001 m) parallel to vertical side and uniform pressure \({10^6}\,\mathrm {N/{m^2}}\) on the horizontal top-side. Loading asymmetry stipulates entire plate for the analysis and the generalized ASF given in Eq. (2) is employed.

4.2 Hexagonal plate with a central hole

A schematic of regular hexagonal plate with a centrally placed hole is shown in Fig. 8 where coordinate system is located at the centre of hole. The side of the regular hexagon is c and diameter is 2a. As in the previous subsection, c is maintained at 1 m while inner diameter of hole is varied to achieve a/c ratios 0.8 and 0.2. Further, problem is formulated as plane stress case with \(E = 210 \) GPa and \(\nu = 0.3\).

Fig. 8
figure 8

Schematic of a hexagonal plate with a centrally placed hole

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Fig. 9
figure 9

A hexagonal plate with a centrally placed hole subjected to a symmetrical loading case b anti-symmetrical loading case c asymmetrical loading case

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4.2.1 Symmetrical loading cases

A schematic of the plate with the loading condition is shown in Fig. 9a. Two-fold geometrical symmetry reduces the problem to analysis of quarter plate. A quarter plate having dimension \( {{\sqrt{3} } \mathord 2}\,\,\mathrm {m}\,\, \times \,\,1\,\mathrm {m}\) is subjected to uniform displacement of 0.001 m on outer edges in the direction normal to it. The loading and geometric symmetries, single-valuedness of displacements and finiteness of stresses yields the ASF in Eq. (14).

$$\begin{aligned} \phi = {b_0}\log r + {c_0}{r^2} + \sum \limits _{n = 6,12,18\ldots }^\infty {\left[ {{a^\prime _n}{r^n} + {b^\prime _n}\,{r^{2 + n}} + {c^\prime _n}{r^{ - n}} + {d^\prime _n}\,{r^{2 - n}}} \right] } \,\cos \,(n\,\theta ) \end{aligned}$$
(14)

4.2.2 Anti-symmetrical loading cases

A schematic of the plate with the loading condition is shown in Fig. 9b. Similar to Sect. 4.2.1, the generalized ASF reduces to the form in Eq. (15).

$$\begin{aligned} \phi = \sum \limits _{n = 2,4,6\ldots }^\infty {\left[ {{a_n}\,{r^n} + {b_n}\,{r^{2 + n}} + {c_n}\,{r^{ - n}} + {d_n}\,{r^{2 - n}}} \right] } \, \sin \,(n\,\theta ) \end{aligned}$$
(15)

4.2.3 Asymmetrical loading cases

A schematic of the plate with a general loading condition is shown in Fig. 9c. Loading asymmetry stipulates entire plate for the analysis and the generalized ASF given in Eq. (2) is employed.

5 Results and observations

The preceding section discussed the cases considered for illustration of the HM. von Mises stress (or equivalent stress), polar stresses and polar displacements obtained from coarse-mesh based HM are plotted as contour plot over the domain. The plot obtained from the HM is compared with corresponding fine-mesh FE simulations. The FE simulations are conducted in ANSYS using 8-noded quadrilateral shaped Plane 183 continuum elements. The following subsections outline the results and observations, respectively.

5.1 Results

The loading cases for \(a/c = 0.8\) and \(a/c = 0.2\) are designated as Case 1 and Case 2, respectively. In each case, von Mises stress results obtained from coarse-mesh HM along with fine-mesh FEA for different loading configurations are plotted collectively. In addition to von Mises stress results, field variables results are plotted for three distinct cases. Further, mesh details in the HM including numbers of nodes, elements, number of retrieving points and number of ASF terms used in the analysis are furnished with the figures.

5.1.1 Square plate

5.1.1.1 von Mises stress

Figure 10 summarizes the von Mises stress results for all three loading configurations for Case 1. The figure also indicates number of nodes, number of elements, number of retrieving points and number of terms used in harmonic analysis and colour maps. Similarly, Fig. 11 summarizes the von Mises stress results for all three loading configurations for Case 2.

Fig. 10
figure 10

Square plate (Case 1): HM and FEA von Mises results for different loading configurations

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Fig. 11
figure 11

Square plate (Case 2): HM and FEA von Mises results for different loading configurations

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5.1.1.2 Field variables

Field variables results—polar stresses and polar displacements—are plotted for a symmetrical and an asymmetrical loading configuration for Case 1 and Case 2. Figure 12 presents the field variables results for symmetrical loading configuration for Case 1. Similarly, Fig. 13 presents the field variables results for asymmetrical loading configuration for Case 2.

Fig. 12
figure 12

Square plate (Case 1): HM and FEA polar stresses and displacements for symmetric loading condition

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Fig. 13
figure 13

Square plate (Case 2): HM and FEA polar stresses and displacements for asymmetric loading condition

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5.1.1.3 Hoop stress along the hole boundary

Employing the HM, circumferential (hoop) stresses are plotted for two a/c ratios (a/c = 0.8 and 0.2) along the circular boundary and the same are compared with corresponding FEA (fine-mesh) results. Figure 14 shows the circumferential stress along the circular boundary for a square plate geometry.

Fig. 14
figure 14

Square plate with a hole under symmetric loading condition: HM and FEA circumferential (hoop) stress on the circular boundary for a a/c = 0.8 b a/c = 0.2

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5.1.1.4 Stress concentration factor (SCF)

To capture the stress concentration effect, a non-dimensional parameter (\( \lambda \))—a ratio of maximum hoop stress to average stress on the outer edge—is defined. This average stress is evaluated from FEM (fine-mesh).

The maximum normalized stress (\( \lambda \)) is defined in Eq. (16) ,

$$\begin{aligned} \lambda= & {} \frac{{{{\left( {{\sigma _{\theta \theta }}_{at}r = a} \right) }}}_{\max }}{{{\sigma _0}}}\nonumber \\ {\text {where}},\,\,{\sigma _0}= & {} \frac{{{{ \left( {\sum {{F_x}} } \right) }_{{\text {on the outer edge}}}}}}{{{\text {Area}}}} \end{aligned}$$
(16)

SCF for a square plate under symmetrical loading configuration is illustrated. For a comparative study, maximum normalized stress (\( \lambda \)) is evaluated at the five distinct values of the a/c ratios and the same are compared with corresponding fine-mesh FEA. The values of the a/c ratios are 0.9, 0.8, 0.6, 0.4 and 0.2, respectively. Figure 15 shows the results of maximum normalized stress values for a square geometry. Values of maximum normalized stress and mesh details including the number of nodes, elements, interrogation points and the number of terms employed in the regression analysis for different a/c ratios are given in the Tables 2 and 3, respectively.

Fig. 15
figure 15

Maximum normalized hoop stress on the circular boundary for different a/c ratios for Square plate with a hole under symmetric loading condition

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Table 2 Square plate with a hole: comparison between HM and FEA \( ({\sigma _{\theta \theta }})_{max} \)/\( {\sigma _0} \) results for different a/c ratios
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Table 3 Square plate with a hole : mesh details for HM and FEA for different a/c ratios
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5.1.2 Hexagonal plate

5.1.2.1 von Mises stress

Figure 16 summarizes the von Mises stress results for all three loading configurations for Case 1. The figure also indicates number of nodes, number of elements, number of retrieving points and number of terms used in harmonic analysis and colour maps. Similarly, Fig. 17 summarizes the von Mises stress results for all three loading configurations for Case 2.

Fig. 16
figure 16

Hexagonal plate (Case 1): HM and FEA von Mises results for different loading configurations

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Fig. 17
figure 17

Hexagonal plate (Case 2): HM and FEA von Mises results for different loading configurations

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5.1.2.2 Field variables

An approach similar to Sect. 5.1.1.2 is adopted for a hexagonal plate for a case an asymmetrical loading configuration. Figure 18 presents the results.

Fig. 18
figure 18

Hexagonal plate (Case 2): HM and FEA polar stresses and displacements for asymmetric loading condition

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5.1.2.3 Hoop stress along the hole boundary

Circumferential stress results for Case 1 and Case 2 is presented in Fig. 19 and the same are compared with corresponding FEA(fine-mesh).

Fig. 19
figure 19

Hexagonal plate with a hole under symmetric loading condition: HM and FEA circumferential (hoop) stress on the circular boundary for a \( a/c = 0.8\) b \( a/c = 0.2\)

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5.1.2.4 Stress concentration factor (SCF)

Similar to Sect. 5.1.1.4, SCF is evaluated for four distinct values of the a/c ratios for hexagonal geometry. Figure 20 presents the result of maximum normalized stress for 0.8, 0.6, 0.4 and 0.2 a/c ratios. In addition, values of the maximum normalized stress and mesh details are given in Tables 4 and 5, respectively.

Fig. 20
figure 20

Maximum normalized hoop stress on the circular boundary for different a/c ratios for hexagonal plate with a hole under symmetric loading condition

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Table 4 Hexagonal plate with a hole: comparison between HM and FEA \( ({\sigma _{\theta \theta }})_{max} \)/\( {\sigma _0} \) results for different a/c ratios
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Table 5 Hexagonal plate with a hole: mesh details for HM and FEA for different a/c ratios
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5.2 Observations and inference

The results for the all cases from HM show a good correspondence with FE simulations over a substantial part of the domain, indicating the efficacy of the HM with relatively coarse-mesh.

The results show a good correspondence with FEA for higher a/c ratios as against lower a/c ratios for the hexagonal domain. For the square domain, the correlation between the correspondence between the two methods (the HM and FEA) and a/c ratios is indistinct. Further for a particular value of a/c ratio, the stresses in a sizeable part of the domain in the vicinity of the hole show a good match with FEA in comparison with regions around the outer boundary near the corners. This observation stems from the HM incorporating FE data from the inner boundary to determine ASF constants. The proximity of inner boundary from the other boundary determines the accuracy of the FE data interrogated at the points on the inner boundary. Thus, domains far from the inner boundary show deviations.

HM results for square plate under anti-symmetric and asymmetric loading cases show deviation at the corners as compared to FEA. However, the deviation for the corresponding cases in hexagonal plate is less conspicuous. HM leverages the solution for the ASF in polar coordinates, which expresses field-variables as a Fourier series in \(\theta \). As the number of sides in a polygon increases, the polygon approaches an annular domain; thus, higher polygons are well represented by ASF on which HM is based.

HM results for hoop stresses along the circular boundary for square and hexagonal geometry under symmetrical loading configurations show good correspondence with fine-mesh FEA for both the cases (Case 1 and Case 2).

6 Discussions

The HM offers alternative scope including the extensions to other methods, different-formulation based on complex-variable approach and to develop the technique as a mesh-reduction method. The following subsections highlight the scope.

6.1 Extension of the HM to experiments

The HM relies on input displacement data at the inner circular boundary; however, other field-variables like strain, stress, isochromatics and isopachics could be used to supplement the HM. These information can be rendered by experimental methods, as given in [18, 19] and [20]. For example, Photoelasticity , strain-gauges rosettes, thermo-elastic stress analysis and Moire‘ method furnish isochromatic, strains, isopachics and displacement data, respectively. Invoking the equations in Sect. 2.1, the ASF constants can be determined on a line similar to the procedure outlined in Sect. 2. Adopting the above approach, a scheme to estimate the stresses in a remote part of the domain using accessible boundary data can be developed.

6.2 The HM: a potential mesh-reduction method

Mesh-reduction technique is a class of methods for analysis incorporating strategies requiring lesser computational requirements. BEM is an example of a mesh-reduction technique involving only boundary discretization. In Sect. 3.1, the results from HM on mesh-grid density demonstrated a potential scope for the mesh-reduction based techniques. In addition, the illustrative example considered in Sect. 3.1 was explored for discerning the influence of structured and unstructured coarse-meshes on the HM results. The study indicated no significant change in the results obtained from the structured and unstructured meshes, denoting a weak dependence of the HM on the structured mesh. The results encompassing the secondary influence of mesh-density and structured-mesh on HM place less requirements on the computational time—indicating a scope for exploration of the HM as a mesh-reduction technique.

6.3 Complex-variable method based HM formulation

The results in Sect. 5 are obtained by employing HM based on real variable approach. However, the same HM can be formulated using complex variable approach, as given in [21,22,23] and [24]. To demonstrate the formulation of HM based on complex variable approach, a square plate with a centrally placed hole having \(a/c = 0.8\) and subjected to loading conditions—symmetrical, anti-symmetrical and asymmetrical loading—is employed.

In complex variable approach, ASF is formulated in terms of Kolosov–Muskhelishvili (KM) potentials and are expressed as the Laurent series. These complex potentials are given by [24,25,26] and [27]. The general KM potentials and ASF are given by Eq. (17) and Eq. (18), respectively. The following equations are employed for asymmetrical loading case.

$$\begin{aligned} \gamma (z)= & {} \sum \limits _{n = 0,1,2\ldots }^\infty {\left( {{K_n}\,{z^{\left( {n + 1} \right) }} + {F_n}\,{z^{ - \left( {n + 1} \right) }}} \right) } - \,\,i\sum \limits _{n = 0,1,2\ldots }^\infty {\left( {{K^\prime _n}\,{z^{\left( {n + 1} \right) }} + {F^\prime _n}\,{z^{ - \left( {n + 1} \right) }}} \right) } \nonumber \\ \psi (z)= & {} \sum \limits _{n = 0,1,2\ldots }^\infty {\left( {{L_n}\,{z^{\left( {n + 1} \right) }} + {H_n}\,{z^{ - \left( {n + 1} \right) }}} \right) } - \,\,i\sum \limits _{n = 0,1,2\ldots }^\infty {\left( {{L^\prime _n}\,{z^{\left( {n + 1} \right) }} + {H^\prime _n}\,{z^{ - \left( {n + 1} \right) }}} \right) } \end{aligned}$$
(17)
$$\begin{aligned} \phi (z,\overline{z} \,)= & {} \frac{1}{2}\left( {z\,\overline{\gamma \left( z \right) } + \overline{z} \,\,\gamma \left( z \right) + \chi \left( z \right) + \overline{\chi \left( z \right) } } \right) \nonumber \\&= {\text {Re}} \left( {\overline{z} \,\,\gamma \left( z \right) + \chi \left( z \right) } \right) \end{aligned}$$
(18)

where, \(z = r\,{e^{i\,\,\theta }}\), \(\gamma \left( z \right) \,{\text {and}}\,\psi \left( z \right) \, = {\chi ^\prime }\left( z \right) \) are complex potentials.

As in Sect. 4, these complex potentials are reduced by appealing to geometrical and loading symmetries. For symmetrical loading, complex potentials are reduced to a form shown by the Eq. (19).

$$\begin{aligned} \gamma (z)= & {} \sum \limits _{n = 0,2,4\ldots }^\infty {\left( {{K_n}\,{z^{\left( {n + 1} \right) }} + {F_n}\,{z^{ - \left( {n + 1} \right) }}} \right) } \nonumber \\ \psi (z)= & {} \sum \limits _{n = 0,2,4\ldots }^\infty {\left( {{L_n}\,{z^{\left( {n + 1} \right) }} + {H_n}\,{z^{ - \left( {n + 1} \right) }}} \right) } \end{aligned}$$
(19)

For anti-symmetrical loading, potential functions are reduced to a form shown by the Eq. (20).

$$\begin{aligned} \gamma (z)= & {} \, - \,i\sum \limits _{n = 0,2,4\ldots }^\infty {\left( {{K^\prime _n}\,{z^{\left( {n + 1} \right) }} + {F^\prime _n}\,{z^{ - \left( {n + 1} \right) }}} \right) } \nonumber \\ \psi (z)= & {} - \,i\sum \limits _{n = 0,2,4\ldots }^\infty {\left( {{L^\prime _n}\,{z^{\left( {n + 1} \right) }} + {H^\prime _n}\,{z^{ - \left( {n + 1} \right) }}} \right) } \end{aligned}$$
(20)

von Mises stress results for the mesh in Sect. 5.1 are plotted and compared with the corresponding fine-mesh FEA results. HM with complex variable approach also shows a good correspondence with FEA results. However, potential functions constants are evaluated by least squares technique in contrast with real-variable approach employing Gauss elimination technique. Figure 21 encapsulates the results for the square plate under different loading configurations along with the results of FEA and mesh details.

6.3.1 An illustrative example for square plate: case 1 for all 3 loading conditions

Figure 21 summarizes the von Mises stress results, obtained from complex-variable based HM, for a square plate under all three loading configurations for Case 1 . The figure also indicates number of nodes, number of elements, number of terms used in harmonic analysis and colour maps.

Based on the above results, an HM formulation incorporating conformal-mapping technique can be explored by employing the methods in the literature [28, 29] and [30].

Case 1 : Square plate with a centrally placed hole for \(a/c = 0.8\).

Fig. 21
figure 21

Complex-variable formulation: HM and FEA von Mises results for different loading configurations

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7 Conclusions

The paper proposes and explores the nuances of an HM coupling FEA and an analytical solution for 2D elastostatic cases involving perforated polygonal plates. The idea employes Fourier series, furnished by Michell Solution for a particular value of radial coordinate, to fit the coarse FE-displacement data on the inner circular boundary with a curve having harmonic basis functions. The resulting coefficients from the regression analysis and traction conditions on the inner boundary furnished equations to determine the Airy constants.

Subsequently, the details of the HM including the effects of coarse-mesh, number of data points for interrogation on the hole-boundary and a criterion of choosing the number of terms in the harmonic regression analysis were discussed by employing an equi-biaxially loaded perforated square plate. The proposed method was illustrated for two a/c ratios (0.8 and 0.2) each for hexagonal and square plates under symmetrical, anti-symmetrical and asymmetrical loading cases. Using the determined Airy constants, von Mises stresses were plotted as contour plot over the domain and compared with the FEA employing fine-mesh. In all the cases, the results show good concurrence between the methods over a considerable part of the domain with slight deviation observed in the regions far from the inner boundary. Further, the equivalence of the von Mises stress between the two is better for hexagonal plate than the square plate, and for the larger a/c ratio as compared to the smaller a/c ratio. The potential scope of the proposed method—resulting from the use of a coarse-mesh—as a mesh-reduction method, the extension of the method to experimentally decipher distant domain-data from conveniently accessible boundary and the complex-variable formulation were discussed.