Strong bi-homogeneous B\'{e}zout theorem and its use in effective real algebraic geometry

Let f1, ..., fs be a polynomial family in Q[X1,..., Xn] (with s less than n) of degree bounded by D. Suppose that f1, ..., fs generates a radical ideal, and defines a smooth algebraic variety V. Consider a projection P. We prove that the degree of the critical locus of P restricted to V is bounded by D^s(D-1)^(n-s) times binomial of n and n-s. This result is obtained in two steps. First the critical points of P restricted to V are characterized as projections of the solutions of Lagrange's system for which a bi-homogeneous structure is exhibited. Secondly we prove a bi-homogeneous B\'ezout Theorem, which bounds the sum of the degrees of the equidimensional components of the radical of an ideal generated by a bi-homogeneous polynomial family. This result is improved when f1,..., fs is a regular sequence. Moreover, we use Lagrange's system to design an algorithm computing at least one point in each connected component of a smooth real algebraic set. This algorithm generalizes, to the non equidimensional case, the one of Safey El Din and Schost. The evaluation of the output size of this algorithm gives new upper bounds on the first Betti number of a smooth real algebraic set. Finally, we estimate its arithmetic complexity and prove that in the worst cases it is polynomial in n, s, D^s(D-1)^(n-s) and the binomial of n and n-s, and the complexity of evaluation of f1,..., fs

Comparison of analytical and wind-tunnel results for flutter and gust response of a transport wing with active controls

Two flutter suppression control laws wre designed and tested on a low speed aeroelastic model of a DC-10 derivative wing. Both control laws demontrated increases in flutter speed in excess of 25 percent above the passive wing flutter speed. In addition, one of the control laws was effective in reducing loads due to turbulence generated in the wind tunnel. The effect of variations in gain and phase on the closed-loop performance was measured and is compared with predictions. In general, both flutter and gust response predictions agree reasonably well with experimental data

Transient elastohydrodynamic analysis of piston skirt lubricated contact under combined axial, lateral and tilting motion

Most modern engines utilise pistons with an offset gudgeon pin. In internal combustion engines, the offset is to the major thrust side of the piston. The piston thrust side is the part of the piston perpendicular to the gudgeon pin that carries the majority of side loading during the power stroke. Primary reason for having the gudgeon pin positioned eccentrically is to prevent the piston from slamming into the cylinder bore after the connecting rod journal passes the top dead centre. This phenomenon is referred to as piston slap, and is more pronounced in compression ignition and high performance engines due to higher combustion pressure than that of commercial spark ignition engines. The coming together of the piston and the bore results in scuffing, at best, or, catastrophic failure at worst. Clearance space between bore and piston is filled by a lubricant film. The main role of the lubricant is to separate the piston and bore by reacting to the applied load. Investigating the above problem requires a holistic approach, whereby a dynamic three degree-of-freedom piston model is coupled with a lubrication model to represent the actual system. The dynamic model determines the motion of the piston in combined axial, lateral and rotation about the gudgeon pin. The reactive forces due to lubricant films on the major and minor thrust sides of the piston play significant roles in piston dynamics and are evaluated by either quasi-static or transient solution of the lubricant contact conjunctions. The novel quasi-static analysis is carried out in the sense of its detailed approach, including many key practical features. not incorporated in other analyses, hitherto reported in literature. These features include first and foremost the development of a specific contact mechanics model for evaluation of conforming contacts for piston skirt against liner or bore. The quasi-static analysis includes many practical feature not encountered in other literature on the subject, such as detailed surface irregularities and modification features, and with thermal distortion. The analysis has been extended to thermohydrodynamics, as well as micro-hydrodynamics, all with high computational mesh densities, and robust methods of solution in space and time domains, including effective influence Newton-Raphson method and linear acceleration integration scheme. The transient tribo-elasto-multi-body dynamics problem includes physics of motion study from film thickness prediction and secondary motion evaluation of the order of micrometers and minutes of arc to large rigid body dynamics, including simultaneous solution of the contact problem at both major and minor thrust sides. Such a comprehensive solution has not hitherto been reported in literature. The thesis discusses many aspects of piston dynamics problem, through the broad spectrum of vehicle manufacture, with many pertinent practical engineering issues. In particular, it provides solutions for high performance Formula 1 racing engines. This is the first ever comprehensive analysis of piston tribodynamics for this range of engines at very high combustion pressures. This study has shown the paramount influence of profile of piston in promoting lubrication between the contiguous bodies, as evident from the pattern of lubricant flow through the contact. Deformation of the bodies increases the volume of lubricant in the contact. During the reversal in direction of piston motion, when the entraining velocity momentarily cases and reversal takes place, the load is held by an elastic squeez

Parameter identification and model based control of direct drive robots

Imperial Users onl

Fundamental Theorem of Algebra: A Survey of History and Proofs

Higher Educatio

Research in structural and solid mechanics, 1982

Advances in structural and solid mechanics, including solution procedures and the physical investigation of structural responses are discussed

Limit cycle vibrations in turbomachinery

The focus is on an examination of rotordynamic systems which are simultaneously susceptible to limit cycle instability and subharmonic response. Characteristics of each phenomenon are determined as well as their interrelationship. A normalized, single mass rotor model is examined as well as a complex model of the high pressure fuel turbopump and the Space Shuttle Main Engine. Entrainment of limit cycle instability by subharmonic response is demonstrated for both models. The nonuniqueness of the solution is also demonstrated

Rigorous numerical approaches in electronic structure theory

Electronic structure theory concerns the description of molecular properties according to the postulates of quantum mechanics. For practical purposes, this is realized entirely through numerical computation, the scope of which is constrained by computational costs that increases rapidly with the size of the system. The significant progress made in this field over the past decades have been facilitated in part by the willingness of chemists to forego some mathematical rigour in exchange for greater efficiency. While such compromises allow large systems to be computed feasibly, there are lingering concerns over the impact that these compromises have on the quality of the results that are produced. This research is motivated by two key issues that contribute to this loss of quality, namely i) the numerical errors accumulated due to the use of finite precision arithmetic and the application of numerical approximations, and ii) the reliance on iterative methods that are not guaranteed to converge to the correct solution. Taking the above issues in consideration, the aim of this thesis is to explore ways to perform electronic structure calculations with greater mathematical rigour, through the application of rigorous numerical methods. Of which, we focus in particular on methods based on interval analysis and deterministic global optimization. The Hartree-Fock electronic structure method will be used as the subject of this study due to its ubiquity within this domain. We outline an approach for placing rigorous bounds on numerical error in Hartree-Fock computations. This is achieved through the application of interval analysis techniques, which are able to rigorously bound and propagate quantities affected by numerical errors. Using this approach, we implement a program called Interval Hartree-Fock. Given a closed-shell system and the current electronic state, this program is able to compute rigorous error bounds on quantities including i) the total energy, ii) molecular orbital energies, iii) molecular orbital coefficients, and iv) derived electronic properties. Interval Hartree-Fock is adapted as an error analysis tool for studying the impact of numerical error in Hartree-Fock computations. It is used to investigate the effect of input related factors such as system size and basis set types on the numerical accuracy of the Hartree-Fock total energy. Consideration is also given to the impact of various algorithm design decisions. Examples include the application of different integral screening thresholds, the variation between single and double precision arithmetic in two-electron integral evaluation, and the adjustment of interpolation table granularity. These factors are relevant to both the usage of conventional Hartree-Fock code, and the development of Hartree-Fock code optimized for novel computing devices such as graphics processing units. We then present an approach for solving the Hartree-Fock equations to within a guaranteed margin of error. This is achieved by treating the Hartree-Fock equations as a non-convex global optimization problem, which is then solved using deterministic global optimization. The main contribution of this work is the development of algorithms for handling quantum chemistry specific expressions such as the one and two-electron integrals within the deterministic global optimization framework. This approach was implemented as an extension to an existing open source solver. Proof of concept calculations are performed for a variety of problems within Hartree-Fock theory, including those in i) point energy calculation, ii) geometry optimization, iii) basis set optimization, and iv) excited state calculation. Performance analyses of these calculations are also presented and discussed