Calculating the Phase Equilibria of Al Rich Al-Cu-Mg Alloys

2011 ◽  
Vol 287-290 ◽  
pp. 2411-2414
Author(s):  
Zhi He ◽  
Lan Yun Li ◽  
Yong Qin Liu
Keyword(s):  
Phase Equilibria ◽  
Linear Equations ◽  
Least Square ◽  
Mg Alloys ◽  
Experimental Results ◽  
Ternary Alloys ◽  
Calculated Phase ◽  
Marquardt Method ◽  
Levenberg Marquardt ◽  
Non Linear Equations

This paper investigates a new method, the Levenberg-Marquardt method, to calculate the phase equilibria of the Al-Cu-Mg ternary alloys. The Levenberg-Marquardt method is the best algorithm to obtain the least-square solution of non-linear equations. Its application to ternary Al-Cu-Mg system is executed in detail in this paper. The calculated phase equilibria agrees well with the experimental results. Furthermore, the Levenberg-Marquardt method is not sensitive to the initial values.

scholarly journals VIGOROUS 3D ANGULAR RESECTION MODEL USING LEVENBERG – MARQUARDT METHOD

2020 ◽  
Vol 1 (1) ◽  
pp. 08-13
Author(s):  
Yaseen Mustafa
Keyword(s):  
Global Minimum ◽  
Least Squares Method ◽  
Reference Points ◽  
Least Square ◽  
Starting Values ◽  
Marquardt Method ◽  
3D Space ◽  
Levenberg Marquardt ◽  
Minimum Solution ◽  
Spherical Triangles

The resection in 3D space is a common problem in surveying engineering and photogrammetry based on observed distances, angles, and coordinates. This resection problem is nonlinear and comprises redundant observations which is normally solved using the least-squares method in an iterative approach. In this paper, we introduce a vigorous angular based resection method that converges to the global minimum even with very challenging starting values of the unknowns. The method is based on deriving oblique angles from the measured horizontal and vertical angles by solving spherical triangles. The derived oblique angles tightly connected the rays enclosed between the resection point and the reference points. Both techniques of the nonlinear least square adjustment either using the Gauss-Newton or Levenberg – Marquardt are applied in two 3D resection experiments. In both numerical methods, the results converged steadily to the global minimum using the proposed angular resection even with improper starting values. However, applying the Levenberg – Marquardt method proved to reach the global minimum solution in all the challenging situations and outperformed the Gauss-Newton method.

The Solidification Path due to the Solute Redistribution of Al-Si-Mg Alloys

2011 ◽  
Vol 361-363 ◽  
pp. 1354-1356
Author(s):  
Zhi He ◽  
Hao Bin Zhou ◽  
Zhong Yao Zhang ◽  
Lan Yun Li
Keyword(s):  
Partition Coefficients ◽  
Solidification Path ◽  
Mg Alloys ◽  
Experimental Results ◽  
Ternary Alloys ◽  
Solute Redistribution ◽  
Important Phenomenon ◽  
Binary Partition ◽  
Solidification Processes

The solution redistribution was an important phenomenon during the solidification of multi-component alloys. The different paths of solidification of different component Al-Si-Mg alloys were calculated in this paper. The calculations were coupled with CALPHAD technology. The interaction of solutes would change the solute redistribution coefficients during the solidification especially in the ends of solidification. The solidification paths were calculated by employing the CALPHAD technology and the binary partition coefficients separately. The results show that errors exist under assuming the partition coefficients of solutes as a constant due to the interaction between solutes in ternary alloys. The predicted solidification processes of Al-Si-Mg alloys agree well with the experimental results in this paper.

Non-Linear Least-Square Optimization of Mechanisms Based on Levenberg-Marquardt Method

2008 ◽  
Author(s):  
Ramon Sancibrian ◽  
Ana De-Juan ◽  
Fernando Viadero
Keyword(s):  
Jacobian Matrix ◽  
Hessian Matrix ◽  
Optimization Method ◽  
Second Order ◽  
Least Square ◽  
Marquardt Method ◽  
Design Variables ◽  
Levenberg Marquardt ◽  
Pseudo Second Order ◽  
Linear Least Square

One of the main problems to improve the convergence rate in deterministic optimization of mechanisms is to obtain the Hessian matrix. The required second-order derivatives are difficult to obtain or they are not available. Levenberg-Marquardt optimization method is a pseudo-second order method which means that uses the jacobian information to estimate the Hessian matrix. In this paper, the formulation to obtain the exact form of the jacobian matrix is presented and how can be implemented in the Levenberg-Marquardt method. This formulation gives a very effective method to optimize mechanism geometry considering a large number of prescribed positions and design variables. At the same time it is possible to have control over singularities and permits to compare the desired and generated path avoiding translation and rotation effects.

The Solute Redistribution during Solidification of Al-Si-Mg Alloys

2011 ◽  
Vol 368-373 ◽  
pp. 979-982
Author(s):  
Zhi He ◽  
Hao Bin Zhou ◽  
Zhong Yao Zhang ◽  
Lan Yun Li
Keyword(s):  
Partition Coefficients ◽  
Mg Alloys ◽  
Experimental Results ◽  
Ternary Alloys ◽  
Solute Redistribution ◽  
Important Phenomenon

The solution redistribution was an important phenomenon during the solidification of multi-component alloys. The changing disciplines during solidification of different component Al-Si-Mg alloys were calculated in this paper. The calculations were coupled with CALPHAD technology. The interaction of solutes would change the solute redistribution coefficients during the solidification especially in the ends of solidification. So in the ends of the solidification, the slope of the curves turned to bigger and bigger. The results of the calculating of the eutectic fraction of the alloys show that errors exist under assuming the partition coefficients of solutes as a constant due to the interaction between solutes in ternary alloys. The predicted eutectic fractions of Al-Si-Mg alloys agree well with the experimental results for using the CALPHAD methods.

scholarly journals Refining the Orbit of Visual Double Stars

1988 ◽  
Vol 98 ◽  
pp. 133-133
Author(s):  
Edgar Soulie
Keyword(s):  
Iterative Method ◽  
Least Square ◽  
Orbital Parameters ◽  
Marquardt Method ◽  
Double Stars ◽  
Visual Double Stars ◽  
Levenberg Marquardt

AbstractAn iterative method of refining the orbital parameters of visual double stars was described. The sum of the least-square differences is minimized by the Levenberg-Marquardt method. The application to two examples was described, including one highly inclined orbit, ADS 8862 = Hussey 664 (i = 94.3 degrees).

scholarly journals A master exceptional field theory

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Guillaume Bossard ◽  
Axel Kleinschmidt ◽  
Ergin Sezgin
Keyword(s):  
Field Theory ◽  
Gauge Invariance ◽  
Sigma Model ◽  
Linear Equations ◽  
Field Theories ◽  
Gauge Invariant ◽  
Non Linear ◽  
Non Linear Equations

Abstract We construct a pseudo-Lagrangian that is invariant under rigid E11 and transforms as a density under E11 generalised diffeomorphisms. The gauge-invariance requires the use of a section condition studied in previous work on E11 exceptional field theory and the inclusion of constrained fields that transform in an indecomposable E11-representation together with the E11 coset fields. We show that, in combination with gauge-invariant and E11-invariant duality equations, this pseudo-Lagrangian reduces to the bosonic sector of non-linear eleven-dimensional supergravity for one choice of solution to the section condi- tion. For another choice, we reobtain the E8 exceptional field theory and conjecture that our pseudo-Lagrangian and duality equations produce all exceptional field theories with maximal supersymmetry in any dimension. We also describe how the theory entails non-linear equations for higher dual fields, including the dual graviton in eleven dimensions. Furthermore, we speculate on the relation to the E10 sigma model.

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