Neural network modeling and analysis of the material removal process during laser machining

Abstract

To manufacture parts with nano- or micro-scale geometry using laser machining, it is essential to have a thorough understanding of the material removal process in order to control the system behaviour. At present, the operator must use trial-and-error methods to set the process control parameters related to the laser beam, motion system, and work piece material. In addition, dynamic characteristics of the process that cannot be controlled by the operator such as power density fluctuations, intensity distribution within the laser beam, and thermal effects can significantly influence the machining process and the quality of part geometry. This paper describes how a multi-layered neural network can be used to model the nonlinear laser micro-machining process in an effort to predict the level of pulse energy needed to create a dent or crater with the desired depth and diameter. Laser pulses of different energy levels are impinged on the surface of several test materials in order to investigate the effect of pulse energy on the resulting crater geometry and the volume of material removed. The experimentally acquired data is used to train and test the neural network's performance. The key system inputs for the process model are mean depth and mean diameter of the crater, and the system outputs are pulse energy, variance of depth and variance of diameter. This study demonstrates that the proposed neural network approach can predict the behaviour of the material removal process during laser machining to a high degree of accuracy.

Neural network training algorithm

The feedforward and back-propagation stages are shown in the following algorithm.

Step 0. Set the learning rate parameter α to 0.1 and the momentum constant µ to small values from 0.1 to 0.5. Determine the number of hidden layers as well as the number of neurons per layer. Determine the maximum number of iterations. Set the minimum system error E _av .

Step 1. Initialize the weights w ^l _ji and biases w ^l _jo for all layers to small random values between -0.1 and +0.1.

Step 2. While the stopping conditions are false, do Steps 3–10.

Step 3. For each training pair (X(n), d(n)) do Steps 4–10, where X(n) is the input signal vector at iteration n and d(n) is the desired response vector at iteration n.

Step 4. Determine the response vector Y¹(n) for all the neurons in the first layer using

$$u_j^{\rm{1}} (n) = {\bf{W}}^1 (n)^{\rm{T}} {\bf{X}}(n) $$

(6)

$$ y_j^{\rm 1} (n){\rm = }f\left( {u_j^{\rm 1} (n)} \right) $$

(7)

where W ¹(n)is the interconnection weight vector for the first hidden layer neurons and f is the activation function as given by Eq. 2 which is used by the neurons of the first layer.

Step 5. Determine the response of the neurons in each of the following hidden layers as well as the output layer using

$$u_j^l (n) = {\bf{W}}^l (n)^{\rm{T}} {\bf{Y}}^{l - 1} (n) $$

(8)

$$y_j^l (n) = f(u_j^l (n)) $$

(9)

where f is the activation function in layer l and Y ^l−1(n) are the responses of neurons of the preceding layer l.

Step 6. Determine the mean squared error (or system error) E(n) associated with each pattern n using

$$E(n) = {1 \over 2}\sum\limits_j {(d_j (n) - y_j^o (n))} ^2 $$

(10)

where summation is done for all the neurons in the output layer l.

Step 7. Determine the average (normalized) system error E _av using

$$E_{av} = {1 \over N}\sum\limits_{n = 1}^N {E(n)} $$

(11)

where N is the total number of training patterns.

Step 8. Compute the error information terms δ ^o _j (n) and calculate the weight correction terms, Δw ^o _ji (n), for all neurons included in the output layer o using

$$\delta _j^o (n) = (d_j (n) - y_j^o (n))f'u_j^o (n) $$

(12)

$$\Delta w_{ji}^o (n) = \mu \Delta w_{ji}^o (n - 1) + \alpha \delta _j^o (n)y_i^l (n) $$

(13)

$$\Delta w_{j0}^o (n) = \mu \Delta w_{jo}^o (n - 1) + \alpha \delta _j^o (n) $$

(14)

where the layer l is the layer preceding the output layer o.

Step 9. Compute the error information terms and the weight correction terms for all neurons in the previous (hidden) layers starting with layer l that precedes the output layer and propagate backwards up to the first hidden layer with

$$ \delta _j^l (n) = (\sum\limits_{k = 1}^m {\delta _k ^{l + 1} } (n)w_{kj}^{l + 1} (n))f'(u_j^l (n)) $$

(15)

$$\Delta w_{ji}^l (n) = \mu \Delta w_{ji}^l (n - 1) + \alpha \delta _j^l (n)y_i^{l - 1} (n) $$

(16)

$$\Delta w_{jo}^l (n) = \mu \Delta w_{jo}^l (n - 1) + \alpha \delta _j^l (n) $$

(17)

where m is the number of neurons in layer l+1, index k refers to neurons in the layer following layer l and index i refers to neurons in the layer preceding layer l.

Step 10. Update weights for all layers using:

$$w_{ji}^l (n)_{({\rm{new}})} = w_{ji}^l (n)_{{\rm{(old)}}} + \Delta w_{ji}^l (n) $$

(18)

Step 11. Test for stopping condition

If the chosen maximum number of iterations is reached or if the normalized system error calculated in Step 7 is smaller than the pre-set value in Step 0, then STOP, otherwise continue.