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Process Modelling and Optimization of Hardness in Laser Cladding of Inconel® 625 Powder on AISI 304 Stainless Steel
Published in Samson Jerold Samuel Chelladurai, Suresh Mayilswamy, Arun Seeralan Balakrishnan, S. Gnanasekaran, Green Materials and Advanced Manufacturing Technology, 2020
S. Sivamani, M. Vijayanand, A. Umesh Bala, R. Varahamoorthi
The first derivative test is used to find local extreme points, either maximum or minimum. A local maximum is where the function starts to decrease and a local minimum is where the function starts to increase. Critical points are where a function can have a local maximum or minimum, and are the only places where the first derivative can change sign (Stewart 2012). Let c be a critical point for a continuous function f(x): If f′(x) changes from positive to negative at c, then f(c) is a local maximum.If f′(x) changes from negative to positive at c, then f(c) is a local minimum.If f′(x) does not change sign at c, then f(c) is neither a local maximum or minimum.
Case Studies
Published in Sunan Huang, Kok Kiong Tan, Poi Voon Er, Tong Heng Lee, Intelligent Fault Diagnosis and Accommodation Control, 2020
Sunan Huang, Kok Kiong Tan, Poi Voon Er, Tong Heng Lee
Figure 9.48 shows a machining center which will serve as the system in the background for the elaboration of the framework. Vs, on the lead screw of the machine, represents a vibration source and there may be more than one present. The unwanted vibration source may arise from a loose gear, a worn lead screw or a cracked bearing of the machine, as examples. The vibration source generates a vibration spectrum which propagates to the other parts of the machine. There is a critical point (Lr) on the machine at which the condition monitoring is based. There may be more than one critical point in general. In this illustration example, the critical point is shown at the tip of the tool. Monitoring of the vibration spectrum at the tool tip is important as it directly affects the quality of machined parts and the manufacturing efficiency, and it is not possible to directly derive vibration measurements at this point since it engages the workpiece during the machining process.
Design for Optimisation
Published in Keith L. Richards, The Engineering Design Primer, 2020
Equation (12.4) gives the value of the critical points (optimum points). If the second derivative evaluated at the critical point is less than zero, then this point will be a maximum. The point will also be a minimum if the second derivative is greater than zero. d2Udx2<0implies a local maximum.d2Udx2>0Implies a local minimum.
A review of response surface methodology for biogas process optimization
Published in Cogent Engineering, 2022
Solal Stephanie Djimtoingar, Nana Sarfo Agyemang Derkyi, Francis Atta Kuranchie, Joseph Kusi Yankyera
To achieve the optimal solution, the original design should take a specific direction. Linear models generate surfaces that are used to indicate that direction (Bezerra et al., 2008). The dimensions of the critical point can be computed by the first value of the derivative of the function (Bezerra et al., 2008): The first step involves calculating the first derivative and identifying all the zeros within the experimental region, then the possibility of the existence of a saddle point is explored through the calculation of the second derivative (Witek-Krowiak et al., 2014). If the two derivatives are equal to null, a graphical representation of the model will be used to determine the local extreme (Witek-Krowiak et al., 2014). The previous procedure is valid only for single response optimisation. However, the optimal region can be found by visual inspection of the surfaces while the visualisation of the predicted model equation can be obtained by plotting the response surface (contour and 3-D plot; Bezerra et al., 2008). The critical points of the second-order models can be defined as maximum, minimum, or saddle (Figure 7).
A study of calculus students’ difficulties, approaches and ability to solve multivariable optimization problems
Published in International Journal of Mathematical Education in Science and Technology, 2022
Finding and interpreting critical points or extrema [a priori code]. A few researchers have found that finding and interpreting critical points or extrema when solving UOPs is challenging for students (cf., Brijlall & Ndlovu, 2013; Dominguez, 2010; Mkhatshwa, 2019; Swanagan, 2012). After setting up the objective function in each task, students could use different approaches, namely algebraic, graphical, or numerical to determine the critical points or extrema of the objective function. For example, using an algebraic approach to determine extrema consist of evaluating the objective function (volume function) in each task at the critical points of the function. We coded the different approaches students used to determine critical points or extrema as well as how they interpreted these quantities in the context of each task.
A hybrid solution to parallel calculation of augmented join trees of scalar fields in any dimension
Published in Computer-Aided Design and Applications, 2018
Paul Rosen, Junyi Tu, Les A. Piegl
Fig. 1 shows functions which have a local minimum, local maximum, and a saddle point, respectively. A simple observation helps us understand how to detect these 3 cases. For the minimum and maximum, notice that all regions surrounding the critical point are higher or lower, respectively. So, if the value of an element is smaller than all its neighbors, it is a local minimum. If the value of an element is larger than all its neighbors, it is a local maximum. The saddle point is a little more complicated to understand. Notice that around the saddle point, the function value goes up in two directions and down in two other directions. Therefore, if the neighbors of an element are larger in two or more disjoint directions and smaller in two or more disjoint direction, then the point may be a saddle. This criterion does not guarantee a saddle point because of interpolation error. However, it can be used to exclude non-saddle points.