Search results for: Lagrangian submanifolds

Authors: De-Shin Liu, Po-Chun Wen, Zhen-Wei Zhuang, Hsueh-Chih Liu, Pei-Chen Huang

Abstract:

Wafer probing tests play an important role in semiconductor manufacturing procedures in accordance with the yield and reliability requirement of the wafer after the backend-of-the-line process. Accordingly, the stable physical and electrical contact between the probe and the tested wafer during wafer probing is regarded as an essential issue in identifying the known good die. The probe card can be integrated with multiple probe needles, which are classified as vertical, cantilever and micro-electro-mechanical systems type probe selections. Among all potential probe types, the vertical probe has several advantages as compared with other probe types, including maintainability, high probe density and feasibility for high-speed wafer testing. In the present study, the mechanical response of the wafer probing test with the vertical cobra probe on 720 μm thick silicon (Si) substrate with a 1.4 μm thick aluminum (Al) pad is investigated by the experiment and transient dynamic simulation approach. Because the deformation mechanism of the vertical cobra probe is determined by both bending and buckling mechanisms, the stable correlation between contact forces and overdrive (OD) length must be carefully verified. Moreover, the decent OD length with corresponding contact force contributed to piercing the native oxide layer of the Al pad and preventing the probing test-induced damage on the interconnect system. Accordingly, the scratch depth of the Al pad under various OD lengths is estimated by the atomic force microscope (AFM) and simulation work. In the wafer probing test configuration, the contact phenomenon between the probe needle and the tested object introduced large deformation and twisting of mesh gridding, causing the subsequent numerical divergence issue. For this reason, the arbitrary Lagrangian-Eulerian method is utilized in the present simulation work to conquer the aforementioned issue. The analytic results revealed a slight difference when the OD is considered as 40 μm, and the simulated is almost identical to the measured scratch depths of the Al pad under higher OD lengths up to 70 μm. This phenomenon can be attributed to the unstable contact of the probe at low OD length with the scratch depth below 30% of Al pad thickness, and the contact status will be being stable when the scratch depth over 30% of pad thickness. The splash of the Al pad is observed by the AFM, and the splashed Al debris accumulates on a specific side; this phenomenon is successfully simulated in the transient dynamic simulation. Thus, the preferred testing OD lengths are found as 45 μm to 70 μm, and the corresponding scratch depths on the Al pad are represented as 31.4% and 47.1% of Al pad thickness, respectively. The investigation approach demonstrated in this study contributed to analyzing the mechanical response of wafer probing test configuration under large strain conditions and assessed the geometric designs and material selections of probe needles to meet the requirement of high resolution and high-speed wafer-level probing test for thinned wafer application.

Keywords: wafer probing test, vertical probe, probe mark, mechanical response, FEA simulation

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Authors: Kamyar Tolouei, Ehsan Moosavi

Abstract:

In open-pit mines, the long-term production scheduling optimization problem (LTPSOP) is a complicated problem that contains constraints, large datasets, and uncertainties. Uncertainty in the output is caused by several geological, economic, or technical factors. Due to its dimensions and NP-hard nature, it is usually difficult to find an ideal solution to the LTPSOP. The optimal schedule generally restricts the ore, metal, and waste tonnages, average grades, and cash flows of each period. Past decades have witnessed important measurements of long-term production scheduling and optimal algorithms since researchers have become highly cognizant of the issue. In fact, it is not possible to consider LTPSOP as a well-solved problem. Traditional production scheduling methods in open-pit mines apply an estimated orebody model to produce optimal schedules. The smoothing result of some geostatistical estimation procedures causes most of the mine schedules and production predictions to be unrealistic and imperfect. With the expansion of simulation procedures, the risks from grade uncertainty in ore reserves can be evaluated and organized through a set of equally probable orebody realizations. In this paper, to synthesize grade uncertainty into the strategic mine schedule, a stochastic integer programming framework is presented to LTPSOP. The objective function of the model is to maximize the net present value and minimize the risk of deviation from the production targets considering grade uncertainty simultaneously while satisfying all technical constraints and operational requirements. Instead of applying one estimated orebody model as input to optimize the production schedule, a set of equally probable orebody realizations are applied to synthesize grade uncertainty in the strategic mine schedule and to produce a more profitable and risk-based production schedule. A mixture of metaheuristic procedures and mathematical methods paves the way to achieve an appropriate solution. This paper introduced a hybrid model between the augmented Lagrangian relaxation (ALR) method and the metaheuristic algorithm, the Harris Hawks optimization (HHO), to solve the LTPSOP under grade uncertainty conditions. In this study, the HHO is experienced to update Lagrange coefficients. Besides, a machine learning method called Random Forest is applied to estimate gold grade in a mineral deposit. The Monte Carlo method is used as the simulation method with 20 realizations. The results specify that the progressive versions have been considerably developed in comparison with the traditional methods. The outcomes were also compared with the ALR-genetic algorithm and ALR-sub-gradient. To indicate the applicability of the model, a case study on an open-pit gold mining operation is implemented. The framework displays the capability to minimize risk and improvement in the expected net present value and financial profitability for LTPSOP. The framework could control geological risk more effectively than the traditional procedure considering grade uncertainty in the hybrid model framework.

Keywords: grade uncertainty, metaheuristic algorithms, open-pit mine, production scheduling optimization

Procedia PDF Downloads 82

Authors: Dario Milani, Guido Morgenthal

Abstract:

Fluid dynamic computation of wind caused forces on bluff bodies e.g light flexible civil structures or high incidence of ground approaching airplane wings, is one of the major criteria governing their design. For such structures a significant dynamic response may result, requiring the usage of small scale devices as guide-vanes in bridge design to control these effects. The focus of this paper is on the numerical simulation of the bluff body problem involving multiscale phenomena induced by small scale devices. One of the solution methods for the CFD simulation that is relatively successful in this class of applications is the Vortex Particle Method (VPM). The method is based on a grid free Lagrangian formulation of the Navier-Stokes equations, where the velocity field is modeled by particles representing local vorticity. These vortices are being convected due to the free stream velocity as well as diffused. This representation yields the main advantages of low numerical diffusion, compact discretization as the vorticity is strongly localized, implicitly accounting for the free-space boundary conditions typical for this class of FSI problems, and a natural representation of the vortex creation process inherent in bluff body flows. When the particle resolution reaches the Kolmogorov dissipation length, the method becomes a Direct Numerical Simulation (DNS). However, it is crucial to note that any solution method aims at balancing the computational cost against the accuracy achievable. In the classical VPM method, if the fluid domain is discretized by Np particles, the computational cost is O(Np2). For the coupled FSI problem of interest, for example large structures such as long-span bridges, the aerodynamic behavior may be influenced or even dominated by small structural details such as barriers, handrails or fairings. For such geometrically complex and dimensionally large structures, resolving the complete domain with the conventional VPM particle discretization might become prohibitively expensive to compute even for moderate numbers of particles. It is possible to reduce this cost either by reducing the number of particles or by controlling its local distribution. It is also possible to increase the accuracy of the solution without increasing substantially the global computational cost by computing a correction of the particle-particle interaction in some regions of interest. In this paper different strategies are presented in order to extend the conventional VPM method to reduce the computational cost whilst resolving the required details of the flow. The methods include temporal sub stepping to increase the accuracy of the particles convection in certain regions as well as dynamically re-discretizing the particle map to locally control the global and the local amount of particles. Finally, these methods will be applied on a test case and the improvements in the efficiency as well as the accuracy of the proposed extension to the method are presented. The important benefits in terms of accuracy and computational cost of the combination of these methods will be thus presented as long as their relevant applications.

Keywords: adaptation, fluid dynamic, remeshing, substepping, vortex particle method

Procedia PDF Downloads 241