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Evaluation of genetic programming-based models for simulating bead dimensions in wire and arc additive manufacturing


Wire and arc additive manufacturing (WAAM) is a novel rapid prototyping process that employs gas tungsten arc welding, controlled by a robot, to build complex 3D parts by successive layer deposition technique. Experimental studies on WAAM are useful for understanding the physics of the process however the quantification and optimization of process parameters is difficult due to complex mechanisms involved in WAAM process. In this present work, the measurement of two bead dimensions (bead height and bead width) based on the three inputs (peak current, wire feed speed, and travel speed) is done using the gas tungsten arc welding machine. Experimental study is followed by proposition of two variants of advanced evolutionary algorithms (gene expression programming and multi-gene genetic programming) in formulation of the functional expressions for the two bead dimensions based on the three inputs. The performance analysis of the two proposed models is conducted based on the four statistical error metrics, hypothesis tests and cross-validation. The relationships extracted between the bead dimensions and the three inputs reveals that the peak current influences both the bead height and bead width simultaneously. The findings reported will have a positive implication on the industry in predictive monitoring of the bead dimensions during the WAAM process.

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This study was supported by Shantou University Scientific Research Funded Project (Grant No. NTF 16002)

Author information

Correspondence to Akhil Garg.



$$\begin{aligned}&\text {Bead height}_{\mathrm{MGGP}}({ mm})= -0.11195+(-8.8995e-006)^{*}\nonumber \\&\qquad ((\tan (((\text {x}2)-((\tan (\text {x}1))^{*}(\text {exp}(\text {x}3))))^{*}(\text {x}1)))\nonumber \\&\quad -\, (\cos (\sin (\text {x}1))))+(0.015439)^{*} (\tan ((\text {x}2)-((\tan (\text {x}1))^{*}\nonumber \\&\qquad (\text {exp}(\text {x}3)))))+(0.054744)^{*}(\tan ((\text {exp}(\tan (\text {x}1)))^{*}\nonumber \\&\qquad ((\tan (\text {tanh}(\text {x}1)))^{*}((\text {exp}(\text {x}3))^{*} (\text {tanh}(\tan (\text {x}1)))))))\nonumber \\&\quad +\,(0.70311)^{*}((\text {x}2)-((\text {x}2)^{*}(\text {x}3)))+(0.0036055)^{*}(((\text {x}1)\nonumber \\&\quad -\,((\text {x}1)^{*}((1.158172)))) -((\tan ((\text {x}2)\nonumber \\&\quad -((\text {x}2)^{*}(\text {x}3))))^{*}(\tan (\text {x}1))))+\,(-0.0052258)\nonumber \\&\quad {}^{*}((\tan ((\text {x}1)^{*}(\text {x}3)))^{*}(\cos ((\tan (((\text {x}2)\nonumber \\&\quad -(\text {x}2))^{*}(\text {x}1)))-((\text {x}1)^{*}(\tan ((\text {x}1)^{*}(\text {x}3)))))))\nonumber \\&\quad +(0.0018798)^{*}((\text {x}1)-((\text {x}1)^{*}(\text {x}2)))\nonumber \\&\quad +\,(-0.022407)^{*}(\tan ((\text {exp}(\text {tanh}(\text {x}2)))^{*}((\tan (\text {x}3))^{*}(\text {x}2)))) \end{aligned}$$
$$\begin{aligned}&\text {Bead height}_{\mathrm{GEP}}(mm)= -0.27529+(0.0029583)^{*}\nonumber \\&\qquad (\tan ((\tan ((\text {x}2)^{*}((14.502281))))^{*}(\text {x}1)))+(-0.029465)^{*}\nonumber \\&\qquad (\cos (((\text {tanh}(\text {x}2))+((\text {x}2)^{*}(\tan ((\text {x}3)^{*}(\text {x}1)))))^{*}(\text {x}1)))\nonumber \\&\quad +\,(-0.20681)^{*} (((\tan ((\text {x}3)^{*}(\text {x}1)))^{*}(\text {x}3))^{*}(\text {x}3))\nonumber \\&\quad +\,(-0.0015171)^{*}((\text {x}2)^{*}(\text {x}1)) +(-0.6226)^{*}((\text {tanh}(\cos ((\text {x}3)\nonumber \\&\quad +\,((3.678354)))))+((\text {x}2)^{*}((\text {tanh}(\text {x}3))\nonumber \\&\quad -\,(\text {tanh}((14.502281))))))+(-0.019202)^{*}\nonumber \\&\qquad ((\sin (\sin ((plog(\text {x}1))^{*}((\text {x}2)^{*}(\text {exp}(\text {x}2))))))\nonumber \\&\quad +\,(\sin (\sin ((plog(\text {x}1))^{*}((\text {x}1)^{*}(\text {exp}(\text {x}2)))))))\nonumber \\&\quad +\,(0.041126)^{*} ((\tan ((\text {x}2)^{*}(\text {x}1)))^{*}(\text {x}3))+(0.0012584)^{*}\nonumber \\&\qquad ((\tan ((\text {tanh}(\text {x}3))^{*}(\text {x}1)))^{*} ((\text {x}2)^{*}((\text {tanh}(\text {x}3))\nonumber \\&\quad -\,(\text {tanh}((14.502281)))))) \end{aligned}$$
$$\begin{aligned}&\text {Bead width}_{\mathrm{MGGP}}(mm)= 4.5428+(0.08313)^{*}((\hbox {x}1)-(\hbox {x}2))\nonumber \\&\quad +\,(-0.0093962)^{*}((\text {exp}(\tan (\hbox {x}1)))+((\hbox {x}1)-(\hbox {x}2)))\nonumber \\&\quad +\,(-0.19108)^{*}((\sin (((-15.924689))-(\hbox {x}3)))^{*}(((\hbox {x}3)^{*}(\hbox {x}1))\nonumber \\&\quad -\,(\hbox {x}2)))+(34.0543)^{*}(\text {plog}(\sin (((-15.924689))-(\hbox {x}3))))\nonumber \\&\quad +\,(-14.6929)^{*}(plog(\hbox {x}3))+(0.032877)^{*}(\sin ((((\sin (\hbox {x}3))^{*}\nonumber \\&\qquad (\hbox {x}1))-(\hbox {x}2))-(\hbox {x}2)))+(0.093567)^{*}(\text {exp}(\cos (\hbox {x}2)))\nonumber \\&\quad +\,(0.74232)^{*}(\cos (\sin (\cos ((\text {exp}(\hbox {x}1))+(\hbox {x}2))))) \end{aligned}$$
$$\begin{aligned}&\text {Bead width}_{\mathrm{GEP}}(mm)= -4.2408+(-0.00047266)^{*}\nonumber \\&\quad ((\text {x}1)^{*}(((\text {x}1)-\,(\text {x}2))^{*}(\text {x}3)))\nonumber \\&\quad +(-0.18222)^{*}(\cos (\tan (\text {exp}(\text {x}1))))\nonumber \\&\quad +\,(-0.06944)^{*}(\cos (\tan (((\text {x}1)-(\text {x}2))^{*}(\text {x}3))))\nonumber \\&\quad +\,(0.089221)^{*}(\text {x}1)+(-0.12007)^{*}(\cos (\cos (\tan (((\text {x}1)\nonumber \\&\quad -\,(\text {x}2))^{*}(\text {x}3)))))+(-0.054305)^{*}\nonumber \\&\quad (\sin (\sin (\sin (\tan (\text {plog}(\tan (\text {x}3)))))))\nonumber \\&\quad +\,(-0.030142)^{*}(\sin (((((\text {x}1)-(\text {x}2))^{*}(\text {x}3))\nonumber \\&\quad +\,(\text {plog}(\tan (((-4.416740))^{*}(\text {x}2)))))\nonumber \\&\quad +(\sin (\tan (\text {plog}(\sin (\text {x}2)))))))\nonumber \\&\quad +\,(0.075278)^{*}(\sin ((\text {plog}(\tan (\text {x}3)))+(\tan (\tan (\tan (\text {x}2)))))) \end{aligned}$$
$$\begin{aligned}&Coefficient\; of\; \det \; er\;\min \; ation\, (R^{2})\nonumber \\&\quad = \left( {\frac{\sum _{i=1}^n {(A_i -\overline{A_i } )(M_i -\overline{M_i } )} }{\sqrt{\sum _{i=1}^n {(A_i -\overline{A_i } )^{2}} \sum _{i=1}^n {(M_i -\overline{M_i } )^{2}} }}} \right) ^{2} \end{aligned}$$
$$\begin{aligned}&Root\; mean\; square\; error \;(RMSE)= \sqrt{\frac{\sum _{i=1}^N {\left| {M_i -A_i } \right| ^{2}} }{N}} \end{aligned}$$
$$\begin{aligned}&Relative\; error\; (\% )= \frac{\left| {M_i -A_i } \right| }{A_i }\times 100 \end{aligned}$$

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Panda, B., Shankhwar, K., Garg, A. et al. Evaluation of genetic programming-based models for simulating bead dimensions in wire and arc additive manufacturing. J Intell Manuf 30, 809–820 (2019).

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  • Wire and arc additive manufacturing
  • Bead geometry
  • Multi-gene genetic programming
  • Modelling