A complementary approach to experimental modeling and analysis of welding processes: dimensional analysis

Abstract

Welding modeling is beneficial for evaluating welding processes and monitoring welding quality. The majority of welding modeling is accomplished through theoretical analysis and experimental techniques. A high level of logical reasoning and an in-depth understanding of welding mechanisms are prerequisites for theoretical analysis. Experimental approaches are appropriate for single-objective systems with few variables and inadequate for complex systems with many variables. Dimensional analysis is a suitable and complementary approach to welding modeling that may address the shortcomings of the experimental method and theoretical analysis. To determine the relationships between welding variables, dimensional analysis starts with the dimensions of the welding variables and then combines related variables into various dimensionless numbers. In this study, dimensional analysis is summarized for welding modeling and process analysis. The steps of establishing relationships between variables and the application of qualitative and quantitative analysis are discussed in detail. Dimensional analysis drastically decreases the number of variables, which makes it easier to design experiments. A proposal of this study is to convert dimensional operations into matrix operations, providing a potential solution for their programmatic implementation. Dimensional analysis has been widely applied in heat transfer, arc, molten pool, and welding processes. This study may provide a feasible solution for welding modeling researchers.

Appendix. Derivation of Eq. (7)

The dimensional equations of welding variables ($X_{1}$, $X_{2}$, $\cdots$, $X_{m}$) in Table 1 are expressed as

$$\begin{aligned} \left\{ \begin{aligned} \left[ X_{1} \right]&=\left[ D_{1}\right] ^{a_{11}} \left[ D_{2}\right] ^{a_{21}}\cdots \left[ D_{m}\right] ^{a_{m1}} \\ \left[ X_{2} \right]&=\left[ D_{1}\right] ^{a_{12}} \left[ D_{2}\right] ^{a_{22}}\cdots \left[ D_{m}\right] ^{a_{m2}} \\ \vdots \\ \left[ X_{m} \right]&=\left[ D_{1}\right] ^{a_{1m}} \left[ D_{2}\right] ^{a_{2m}}\cdots \left[ D_{m}\right] ^{a_{mm}} \end{aligned} \right. \end{aligned}$$

(14)

Taking the logarithm of both ends of Eq. (14) and expressing it in matrix form at the same time,

$$\begin{aligned} \left[ \begin{array}{c} \ln {\left[ X_{1} \right] } \\ \ln {\left[ X_{2} \right] } \\ \vdots \\ \ln {\left[ X_{m} \right] } \end{array} \right] = \left[ \begin{array}{cccc} a_{11} &{} a_{21} &{} \cdots &{} a_{m1} \\ a_{12} &{} a_{22} &{} \cdots &{} a_{m2} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ a_{1m} &{} a_{2m} &{} \cdots &{} a_{mm} \end{array} \right] \left[ \begin{array}{c} \ln {\left[ D_{1}\right] } \\ \ln {\left[ D_{2}\right] } \\ \vdots \\ \ln {\left[ D_{m}\right] } \end{array} \right] \end{aligned}$$

(15)

Doing the same for Eq. (2) and the dimensional equations of welding variables ($X_{m+1}$,$X_{m+2}$,$\cdots$,$X_{n}$),

$$\begin{aligned} \left[ \begin{array}{c} \ln {\left[ X_{m+1} \right] } \\ \ln {\left[ X_{m+2} \right] } \\ \vdots \\ \ln {\left[ X_{n} \right] } \end{array} \right] = \left[ \begin{array}{cccc} \kappa _{11} &{} \kappa _{12} &{} \cdots &{} \kappa _{1m} \\ \kappa _{21} &{} \kappa _{22} &{} \cdots &{} \kappa _{2m} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \kappa _{k1} &{} \kappa _{k2} &{} \cdots &{} \kappa _{km} \end{array} \right] \left[ \begin{array}{c} \ln {\left[ X_{1}\right] } \\ \ln {\left[ X_{2}\right] } \\ \vdots \\ \ln {\left[ X_{m}\right] } \end{array} \right] \end{aligned}$$

(16)

$$\begin{aligned} \begin{aligned}&\left[ \begin{array}{c} \ln {\left[ X_{m+1} \right] } \\ \ln {\left[ X_{m+2} \right] } \\ \vdots \\ \ln {\left[ X_{n} \right] } \end{array} \right] = \\&\left[ \begin{array}{cccc} a_{1(m+1)} &{} a_{2(m+1)} &{} \cdots &{} a_{m(m+1)} \\ a_{1(m+2)} &{} a_{2(m+2)} &{} \cdots &{} a_{m(m+2)} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ a_{1n} &{} a_{2n} &{} \cdots &{} a_{mn} \end{array} \right] \left[ \begin{array}{c} \ln {\left[ D_{1}\right] } \\ \ln {\left[ D_{2}\right] } \\ \vdots \\ \ln {\left[ D_{m}\right] } \end{array} \right] \end{aligned} \end{aligned}$$

(17)

Combining Eq. (15), Eq. (16), and Eq. (17),

$$\begin{aligned} \begin{aligned}&\left[ \begin{array}{cccc} \kappa _{11} &{} \kappa _{12} &{} \cdots &{} \kappa _{1m} \\ \kappa _{21} &{} \kappa _{22} &{} \cdots &{} \kappa _{2m} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \kappa _{k1} &{} \kappa _{k2} &{} \cdots &{} \kappa _{km} \end{array} \right] \left[ \begin{array}{cccc} a_{11} &{} a_{21} &{} \cdots &{} a_{m1} \\ a_{12} &{} a_{22} &{} \cdots &{} a_{m2} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ a_{1m} &{} a_{2m} &{} \cdots &{} a_{mm} \end{array} \right] \\&= \left[ \begin{array}{cccc} a_{1(m+1)} &{} a_{2(m+1)} &{} \cdots &{} a_{m(m+1)} \\ a_{1(m+2)} &{} a_{2(m+2)} &{} \cdots &{} a_{m(m+2)} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ a_{1n} &{} a_{2n} &{} \cdots &{} a_{mn} \end{array} \right] \end{aligned} \end{aligned}$$

(18)

Assuming that,

$$\begin{aligned} \begin{array}{l} {\textbf {K}} = \left[ \begin{array}{cccc} \kappa _{11} &{}\kappa _{12} &{}\cdots &{}\kappa _{1m} \\ \kappa _{21} &{}\kappa _{22} &{}\cdots &{}\kappa _{2m} \\ \vdots &{}\vdots &{}\ddots &{} \vdots \\ \kappa _{k1} &{}\kappa _{k2} &{}\cdots &{}\kappa _{km} \end{array} \right] \\ {\textbf {A}} = \left[ \begin{array}{cccc} a_{11} &{} a_{12} &{}\cdots &{} a_{1m} \\ a_{21} &{} a_{22} &{}\cdots &{} a_{2m} \\ \vdots &{}\vdots &{}\ddots &{} \vdots \\ a_{m1} &{} a_{m2} &{}\cdots &{} a_{mm} \end{array} \right] \\ {\textbf {B}} = \left[ \begin{array}{cccc} a_{1(m+1)} &{} a_{1(m+2)} &{}\cdots &{} a_{1n} \\ a_{2(m+1} &{} a_{2(m+2)} &{}\cdots &{} a_{2n} \\ \vdots &{}\vdots &{}\ddots &{} \vdots \\ a_{m(m+1} &{} a_{m(m+2)} &{}\cdots &{} a_{mn} \end{array} \right] \end{array} \end{aligned}$$

Then, there are,

$$\begin{aligned} {\textbf {K}}{} {\textbf {A}}^T={\textbf {B}}^T \end{aligned}$$

(19)

where T is the transpose of the matrix. Performing a matrix transpose on Eq. (19),

$$\begin{aligned} {\textbf {A}}{} {\textbf {K}}^T={\textbf {B}} \end{aligned}$$

(20)

Since the matrix ${\textbf {A}}$ is full rank and the inverse matrix ${\textbf {A}}^{-1}$ exists. Then both ends of Eq. (20) are simultaneously left multiplied by ${\textbf {A}}^{-1}$, which gives

$$\begin{aligned} {\textbf {K}}^T={\textbf {A}}^{-1}{} {\textbf {B}} \end{aligned}$$

(21)

$$\begin{aligned} {\textbf {K}}=\left( {\textbf {A}}^{-1}{} {\textbf {B}} \right) ^T \end{aligned}$$

(22)

So that’s the end of the derivation.