On semi-definiteness and minimal H-eigenvalue of a symmetric space tensor using nonnegative polynomial optimization techniques

2019 ◽  
Vol 73 ◽  
pp. 3-11
Author(s):  
Y. Xu ◽  
L. Qi ◽  
W. Sun
Keyword(s):  
Symmetric Space ◽  
Polynomial Optimization ◽  
Optimization Techniques ◽  
Space Tensor ◽  
Nonnegative Polynomial

scholarly journals Nonnegative Polynomial Optimization over Unit Spheres and Convex Programming Relaxations

2012 ◽  
Vol 22 (3) ◽  
pp. 987-1008 ◽  
Author(s):  
Guanglu Zhou ◽  
Louis Caccetta ◽  
Kok Lay Teo ◽  
Soon-Yi Wu
Keyword(s):  
Convex Programming ◽  
Polynomial Optimization ◽  
Nonnegative Polynomial

SDP RELAXATIONS FOR QUADRATIC OPTIMIZATION PROBLEMS DERIVED FROM POLYNOMIAL OPTIMIZATION PROBLEMS

2010 ◽  
Vol 27 (01) ◽  
pp. 15-38 ◽  
Author(s):  
MARTIN MEVISSEN ◽  
MASAKAZU KOJIMA
Keyword(s):  
Large Scale ◽  
Optimization Problems ◽  
Polynomial Optimization ◽  
Convergent Sequence ◽  
Quadratic Optimization ◽  
Optimization Techniques ◽  
Multivariate Polynomial ◽  
Sdp Relaxation ◽  
Polynomial Optimization Problem

Based on the convergent sequence of SDP relaxations for a multivariate polynomial optimization problem (POP) by Lasserre (2006), Waki et al. (2006) constructed a sequence of sparse SDP relaxations to solve sparse POPs efficiently. Nevertheless, the size of the sparse SDP relaxation is the major obstacle in order to solve POPs of higher degree. This paper proposes an approach to transform general POPs to quadratic optimization problems (QOPs), which allows to reduce the size of the SDP relaxation substantially. We introduce different heuristics resulting in equivalent QOPs and show how sparsity of a POP is maintained under the transformation procedure. As the most important issue, we discuss how to increase the quality of the SDP relaxation for a QOP. Moreover, we increase the accuracy of the solution of the SDP relaxation by applying additional local optimization techniques. Finally, we demonstrate the high potential of this approach through numerical results for large scale POPs of higher degree.

Review on the aerodynamics of intermediate compressor duct

2020 ◽  
Vol 14 (4) ◽  
pp. 7446-7468
Author(s):  
Manish Sharma ◽  
Beena D. Baloni
Keyword(s):  
High Pressure ◽  
Pressure Loss ◽  
Flow Analysis ◽  
Outer Wall ◽  
Optimization Techniques ◽  
Optimum Pressure ◽  
Research Areas ◽  
Static Pressure Distribution ◽  
High Pressure Compressor ◽  
Pressure Compressor

In a turbofan engine, the air is brought from the low to the high-pressure compressor through an intermediate compressor duct. Weight and design space limitations impel to its design as an S-shaped. Despite it, the intermediate duct has to guide the flow carefully to the high-pressure compressor without disturbances and flow separations hence, flow analysis within the duct has been attractive to the researchers ever since its inception. Consequently, a number of researchers and experimentalists from the aerospace industry could not keep themselves away from this research. Further demand for increasing by-pass ratio will change the shape and weight of the duct that uplift encourages them to continue research in this field. Innumerable studies related to S-shaped duct have proven that its performance depends on many factors like curvature, upstream compressor’s vortices, swirl, insertion of struts, geometrical aspects, Mach number and many more. The application of flow control devices, wall shape optimization techniques, and integrated concepts lead a better system performance and shorten the duct length.  This review paper is an endeavor to encapsulate all the above aspects and finally, it can be concluded that the intermediate duct is a key component to keep the overall weight and specific fuel consumption low. The shape and curvature of the duct significantly affect the pressure distortion. The wall static pressure distribution along the inner wall significantly higher than that of the outer wall. Duct pressure loss enhances with the aggressive design of duct, incursion of struts, thick inlet boundary layer and higher swirl at the inlet. Thus, one should focus on research areas for better aerodynamic effects of the above parameters which give duct design with optimum pressure loss and non-uniformity within the duct.

Application of Computational Mechanics to Tire Design—Yesterday, Today, and Tomorrow

2011 ◽  
Vol 39 (4) ◽  
pp. 223-244 ◽  
Author(s):  
Y. Nakajima
Keyword(s):  
Finite Element ◽  
New Technologies ◽  
Computational Mechanics ◽  
Design Procedure ◽  
Side Wall ◽  
Rolling Resistance ◽  
Optimization Techniques ◽  
Stress Strain ◽  
Element Analysis ◽  
Crown Shape

Abstract The tire technology related with the computational mechanics is reviewed from the standpoint of yesterday, today, and tomorrow. Yesterday: A finite element method was developed in the 1950s as a tool of computational mechanics. In the tire manufacturers, finite element analysis (FEA) was started applying to a tire analysis in the beginning of 1970s and this was much earlier than the vehicle industry, electric industry, and others. The main reason was that construction and configurations of a tire were so complicated that analytical approach could not solve many problems related with tire mechanics. Since commercial software was not so popular in 1970s, in-house axisymmetric codes were developed for three kinds of application such as stress/strain, heat conduction, and modal analysis. Since FEA could make the stress/strain visible in a tire, the application area was mainly tire durability. Today: combining FEA with optimization techniques, the tire design procedure is drastically changed in side wall shape, tire crown shape, pitch variation, tire pattern, etc. So the computational mechanics becomes an indispensable tool for tire industry. Furthermore, an insight to improve tire performance is obtained from the optimized solution and the new technologies were created from the insight. Then, FEA is applied to various areas such as hydroplaning and snow traction based on the formulation of fluid–tire interaction. Since the computational mechanics enables us to see what we could not see, new tire patterns were developed by seeing the streamline in tire contact area and shear stress in snow in traction.Tomorrow: The computational mechanics will be applied in multidisciplinary areas and nano-scale areas to create new technologies. The environmental subjects will be more important such as rolling resistance, noise and wear.

Hypoelliptic Laplacian and Orbital Integrals (AM-177)

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Fixed Point ◽  
Symmetric Space ◽  
Heat Kernel ◽  
Geodesic Flow ◽  
Casimir Operator ◽  
Index Theory ◽  
Weighted Sum ◽  
Orbital Integrals ◽  
Hypoelliptic Heat Kernel ◽  
Wave Kernel

This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed. Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. The Malliavin calculus is used at critical stages of the proof.

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