Bicriterial Design of Digital Filters

A problem of technical interest is the solution of approximation problems which make a tradeoff between the L ₂ norm and the L norm error criteria. This problem is investigated in the framework of filter design with respect to two conflicting optimality goals. The particular interest in L ₂-L norm compromise filters has been raised by a paper of Adams (IEEE Trans. on Circuits and Systems vol. 39, pp. 376–388, 1991), who suggested to compute such FIR filters by solution of certain constrained L ₂ approximation problems which require a proper choice of weights. It is shown in this paper that bicriterial filter design problems can be approached by classical methods from multicriteria optimization and that especially reference point approximation with the ideal point as reference point is a suitable tool to deal with Adams' problem. Solutions from this latter approach do especially not depend on the choice of weights and yield the best possible compromise filters with respect to a prescribed measure. The resulting optimization problems can be solved with (semi-infinite) programming methods having proven convergence under standard assumptions. Examples of L ₂-L norm compromise designs of a linear-phase FIR and an IIR filter are presented.     

Trade-off analysis approach for interactive nonlinear multiobjective optimization

When solving multiobjective optimization problems, there is typically a decision maker (DM) who is responsible for determining the most preferred Pareto optimal solution based on his preferences. To gain confidence that the decisions to be made are the right ones for the DM, it is important to understand the trade-offs related to different Pareto optimal solutions. We first propose a trade-off analysis approach that can be connected to various multiobjective optimization methods utilizing a certain type of scalarization to produce Pareto optimal solutions. With this approach, the DM can conveniently learn about local trade-offs between the conflicting objectives and judge whether they are acceptable. The approach is based on an idea where the DM is able to make small changes in the components of a selected Pareto optimal objective vector. The resulting vector is treated as a reference point which is then projected to the tangent hyperplane of the Pareto optimal set located at the Pareto optimal solution selected. The obtained approximate Pareto optimal solutions can be used to study trade-off information. The approach is especially useful when trade-off analysis must be carried out without increasing computation workload. We demonstrate the usage of the approach through an academic example problem.     

Practical solutions for multi-objective optimization: An application to system reliability design problems

For multiple-objective optimization problems, a common solution methodology is to determine a Pareto optimal set. Unfortunately, these sets are often large and can become difficult to comprehend and consider. Two methods are presented as practical approaches to reduce the size of the Pareto optimal set for multiple-objective system reliability design problems. The first method is a pseudo-ranking scheme that helps the decision maker select solutions that reflect his/her objective function priorities. In the second approach, we used data mining clustering techniques to group the data by using the k-means algorithm to find clusters of similar solutions. This provides the decision maker with just k general solutions to choose from. With this second method, from the clustered Pareto optimal set, we attempted to find solutions which are likely to be more relevant to the decision maker. These are solutions where a small improvement in one objective would lead to a large deterioration in at least one other objective. To demonstrate how these methods work, the well-known redundancy allocation problem was solved as a multiple objective problem by using the NSGA genetic algorithm to initially find the Pareto optimal solutions, and then, the two proposed methods are applied to prune the Pareto set.     

Level Diagrams analysis of Pareto Front for multiobjective system redundancy allocation

Reliability-based and risk-informed design, operation, maintenance and regulation lead to multiobjective (multicriteria) optimization problems. In this context, the Pareto Front and Set found in a multiobjective optimality search provide a family of solutions among which the decision maker has to look for the best choice according to his or her preferences. Efficient visualization techniques for Pareto Front and Set analyses are needed for helping decision makers in the selection task.In this paper, we consider the multiobjective optimization of system redundancy allocation and use the recently introduced Level Diagrams technique for graphically representing the resulting Pareto Front and Set. Each objective and decision variable is represented on separate diagrams where the points of the Pareto Front and Set are positioned according to their proximity to ideally optimal points, as measured by a metric of normalized objective values. All diagrams are synchronized across all objectives and decision variables. On the basis of the analysis of the Level Diagrams, we introduce a procedure for reducing the number of solutions in the Pareto Front; from the reduced set of solutions, the decision maker can more easily identify his or her preferred solution.     

Pareto navigator for interactive nonlinear multiobjective optimization

We describe a new interactive learning-oriented method called Pareto navigator for nonlinear multiobjective optimization. In the method, first a polyhedral approximation of the Pareto optimal set is formed in the objective function space using a relatively small set of Pareto optimal solutions representing the Pareto optimal set. Then the decision maker can navigate around the polyhedral approximation and direct the search for promising regions where the most preferred solution could be located. In this way, the decision maker can learn about the interdependencies between the conflicting objectives and possibly adjust one’s preferences. Once an interesting region has been identified, the polyhedral approximation can be made more accurate in that region or the decision maker can ask for the closest counterpart in the actual Pareto optimal set. If desired, (s)he can continue with another interactive method from the solution obtained. Pareto navigator can be seen as a nonlinear extension of the linear Pareto race method. After the representative set of Pareto optimal solutions has been generated, Pareto navigator is computationally efficient because the computations are performed in the polyhedral approximation and for that reason function evaluations of the actual objective functions are not needed. Thus, the method is well suited especially for problems with computationally costly functions. Furthermore, thanks to the visualization technique used, the method is applicable also for problems with three or more objective functions, and in fact it is best suited for such problems. After introducing the method in more detail, we illustrate it and the underlying ideas with an example.     

Multi-objective scheduling problems: Determination of pruned Pareto sets

There are often multiple competing objectives for industrial scheduling and production planning problems. Two practical methods are presented to efficiently identify promising solutions from among a Pareto optimal set for multi-objective scheduling problems. Generally, multi-objective optimization problems can be solved by combining the objectives into a single objective using equivalent cost conversions, utility theory, etc., or by determination of a Pareto optimal set. Pareto optimal sets or representative subsets can be found by using a multi-objective genetic algorithm or by other means. Then, in practice, the decision maker ultimately has to select one solution from this set for system implementation. However, the Pareto optimal set is often large and cumbersome, making the post-Pareto analysis phase potentially difficult, especially as the number of objectives increase. Our research involves the post Pareto analysis phase, and two methods are presented to filter the Pareto optimal set to determine a subset of promising or desirable solutions. The first method is pruning using non-numerical objective function ranking preferences. The second approach involves pruning by using data clustering. The k-means algorithm is used to find clusters of similar solutions in the Pareto optimal set. The clustered data allows the decision maker to have just k general solutions from which to choose. These methods are general, and they are demonstrated using two multi-objective problems involving the scheduling of the bottleneck operation of a printed wiring board manufacturing line and a more general scheduling problem.     

Multi-objective genetic algorithm for solving N-version program design problem

N-version programming (NVP) is a programming approach for constructing fault tolerant software systems. Generally, an optimization model utilized in NVP selects the optimal set of versions for each module to maximize the system reliability and to constrain the total cost to remain within a given budget. In such a model, while the number of versions included in the obtained solution is generally reduced, the budget restriction may be so rigid that it may fail to find the optimal solution. In order to ameliorate this problem, this paper proposes a novel bi-objective optimization model that maximizes the system reliability and minimizes the system total cost for designing N-version software systems. When solving multi-objective optimization problem, it is crucial to find Pareto solutions. It is, however, not easy to obtain them. In this paper, we propose a novel bi-objective optimization model that obtains many Pareto solutions efficiently.We formulate the optimal design problem of NVP as a bi-objective 0–1 nonlinear integer programming problem. In order to overcome this problem, we propose a Multi-objective genetic algorithm (MOGA), which is a powerful, though time-consuming, method to solve multi-objective optimization problems. When implementing genetic algorithm (GA), the use of an appropriate genetic representation scheme is one of the most important issues to obtain good performance. We employ random-key representation in our MOGA to find many Pareto solutions spaced as evenly as possible along the Pareto frontier. To pursue improve further performance, we introduce elitism, the Pareto-insertion and the Pareto-deletion operations based on distance between Pareto solutions in the selection process.The proposed MOGA obtains many Pareto solutions along the Pareto frontier evenly. The user of the MOGA can select the best compromise solution among the candidates by controlling the balance between the system reliability and the total cost.     

The melting points of ultralong paraffins and their homologues

Because of the recent availability of the melting points of several ultralong normal paraffins, the melting behavior of normal paraffins has been investigated. Taking the melting point of polyethylene to represent the melting point of an ultralong paraffin, a new function has been established to represent the melting points of alkanes from the carbon number 32 onwards. Adopting the same value for the limiting melting point of an ultralong paraffin, equations are derived for the melting points of several homologous series.Nomenclature a A constant to be determined - b A constant to be determined - m Number of methylene groups in the molecule - n Number of carbon atoms in the molecule - n ^* Number of carbon atom above which Eq. (6) is applicable - T ₀ Temperature constant, K - T _c Critical temperature, K - T _c Critical temperature of an ultralong normal paraffin, K - T _b Normal boiling point, K - T _b Normal boiling point of an ultralong normal paraffin, K - T _m Melting point, K - T _m Melting point of an ultralong paraffin, K - Standard deviation {[(T _m (obsd)–T _m (calc)]²/(No. points–No. parameters)}^0.5     

Some recent results on the stability of linear time varying systems

Some theorems derived recently by the authors on the stability of multidimensional linear time varying systems are reported in this paper. To begin with, criteria based on Liapunov’s direct method are stated. These are followed by conditions on the asymptotic behaviour and boundedness of solutions. Finally,L ₂ andL ∞ stabilities of these systems are discussed. In conclusion, mention is made of some of the problems in aerospace engineering to which these theorems have been applied.     

Numerical study of geometric constraint and cohesive parameters in steady-state viscoelastic crack growth

This paper presents a finite element study of cohesive crack growth in a thin infinite viscoelastic strip to investigate the effects of viscoelastic properties, strip height, and cohesive model parameters on the crack growth resistance. The results of the study show that the dependence of the fracture energy on the viscoelastic properties for the strip problem is similar to that obtained for the infinite body problem even when the cohesive zone length is large compared to the height of the strip. The fracture energy also depends on the crack speed v through the dimensionless parameter v τ/L _∞ where L _∞ is the characteristic length of the cohesive zone and τ is the characteristic relaxation time of the bulk material. This relationship confirms that at least two properties of the fracture process must be prescribed accurately to model viscoelastic crack growth. In contrast, the fracture energy and crack speed are insensitive to the strip height even in situations where the growth of the dissipation zone is severely constrained by the strip boundaries. We observe that at high speeds, where the fracture energy asymptotically approaches the maximum value, the material surrounding the cohesive zone is in the rubbery (equilibrium) state and not the glassy state.     

A computationally efficient metamodeling approach for expensive multiobjective optimization

This paper explores a new metamodeling framework that may collapse the computational explosion that characterizes the modeling of complex systems under a multiobjective and/or multidisciplinary setting. Under the new framework, a pseudo response surface is constructed for each design objective for each discipline. This pseudo response surface has the unique property of being highly accurate in Pareto optimal regions, while it is intentionally allowed to be inaccurate in other regions. In short, the response surface for each design objective is accurate only where it matters. Because the pseudo response surface is allowed to be inaccurate in other regions of the design space, the computational cost of constructing it is dramatically reduced. An important distinguishing feature of the new framework is that the response surfaces for all the design objectives are constructed simultaneously in a mutually dependent fashion, in a way that identifies Pareto regions for the multiobjective problem. The new framework supports the puzzling notion that it is possible to obtain more accuracy and radically more design space exploration capability, while actually reducing the computation effort. This counterintuitive metamodeling paradigm shift holds the potential for identifying highly competitive products and systems that are well beyond today’s state of the art.     

MONSS: A multi-objective nonlinear simplex search approach

This article presents a novel methodology for dealing with continuous box-constrained multi-objective optimization problems (MOPs). The proposed algorithm adopts a nonlinear simplex search scheme in order to obtain multiple elements of the Pareto optimal set. The search is directed by a well-distributed set of weight vectors, each of which defines a scalarization problem that is solved by deforming a simplex according to the movements described by Nelder and Mead's method. Considering an MOP with n decision variables, the simplex is constructed using n+1 solutions which minimize different scalarization problems defined by n+1 neighbor weight vectors. All solutions found in the search are used to update a set of solutions considered to be the minima for each separate problem. In this way, the proposed algorithm collectively obtains multiple trade-offs among the different conflicting objectives, while maintaining a proper representation of the Pareto optimal front. In this article, it is shown that a well-designed strategy using just mathematical programming techniques can be competitive with respect to the state-of-the-art multi-objective evolutionary algorithms against which it was compared.     

On the elasticT-term of a main crack induced by near tip microcracks

In the expansion form of the stress field near a crack tip, the nonsingular and constant stress acting parallel to the crack plane is called the elasticT-term, which is very important in fracture analysis. The elasticT-term of a main crack induced by near tip microcracks is analyzed. The original problem considered consists of a main crack accompanied by near tip microcracks, which are loaded remotely by the first three terms of the crack tip stress field expansion parameterized by stress intensity factorsK _I ,K _II , and the nonsingular stress, i.e., theT-term,. With the principle of superposition, the problem can be reduced to a system of Fredholm integral equations which can be solved numerically. After obtaining the solution, the induced elasticT-term is evaluated. Several typical numerical examples are given and the results are shown in Figures, from which some useful discussion and conclusions are obtained.     

Semi-recorrido condicionado (Expresion asintotica de la r-esperanza condicionada)

Summary In a probability space (Ω, σ,P), for α⊂σ a sub-σ-field, in general the best approximation inL ^∞ by elements ofL ^∞(α) has not an unique solution. For the election between these, we prove the convergenceP-almost surely of the conditionalr-means, whenr→∞, to one solution, which we call conditional mid-range. This is characterized for each ω∈Ω by the mid-range, of one regular conditional distributionQ (ω, •):

     

Development of a flow field structure in the vicinity of a circular cylinder in the presence of a laminar-turbulent transition

The Reynolds equations and the two-parameter differentialq-Ω model of turbulence are used to investigate a flow past a circular cylinder with an isothermal surface (temperature factorT _w0 = 0.5) at the Mach numberM _∞ = 5 in the range of Reynolds numbers Re = 10⁴-10⁸. It is demonstrated that the turbulization of flow leads to a shift of the separation point downstream, a reduction and stabilization of the separation zone length, a decrease in the maximum velocity in the separation zone, and an increase in the heat flux at the rear stagnation point compared with its value at the forward stagnation point.     

Grain size correction of welding residual stress measurement in a carbon steel plate using the critical refraction of longitudinal waves

In this, a method to measure welding residual stress in butt-welded joints of carbon steel plates using longitudinal critically refracted wave (L_cr wave) is proposed. Cross-correlation was employed to calculate the difference in time of flight between L_cr wave, and the optimal step length for the measurements is discussed. To determine L_cr wave acoustoelastic coefficient of the heat affected zone (HAZ), the relationship between the L_cr wave acoustoelastic coefficient and the grain size is established. The results show that one cycle is the optimal step length for the difference in the time-of-flight calculation, and with increasing grain size increase, L_cr wave acoustoelastic coefficient decreases in the form of a power function. In addition, grain size can be determined by using amplitude of the L_cr wave, so that the measured value of welding residual stress in HAZ can be corrected. The welding residual stress in melted zone (MZ) is corrected by calibrating acoustoelastic coefficient of the MZ. The acoustoelastic coefficient of the MZ is larger than that of parent material (PM). At last, welding residual stress in the butt-weld joint is measured and corrected with the L_cr wave technique. The results are verified by the hole drilling method.     

Unsteady boundary-layer induced pressures at high Mach numbers

Summary A flat plate moving at a high Mach number and zero angle of attack is stepwise accelerated in the direction of its initial motion by a small amount. The time history of the associated flow, till the final steady state is reached, is analysed and the unsteady flow solutions determining the temporal variations in the boundary layer induced weakinteraction pressure and shear stress are numerically obtained for an insulated surface. In the analysis, the fluid is assumed to be a perfect gas. The obtained results are for a free stream Mach numberM of 10 and a given plate lengthL ^* with various values of impulsive velocity increments. It is also found that by suitable formulation of the variables and in particular by avoiding the variable of the typet ^* u ^*/x ^*, the discontinuity quoted by Stewartson and others is avoided.With 8 Figures     

Particle condensation in metallic vapour plumes

An apparatus is described for the study of aerosol particle condensation in a plume of vapour rising from a heated source. The principal advantages of this plume chamber over the heat-pulse cloud chamber previously used include the attainment of steady-state conditions at a controlled source temperature and the suppression of recirculation. The coagulation of particles and the further growth or re-evaporation that tends to occur in the outer zone of the heat-pulse equipment is discouraged, and samples taken on substrates positioned in the plume are expected to be more representative of the original nucleation and growth processes.Aerosol particles were condensed in zinc vapour emitted at various source temperatures (T ₀) with the walls at room temperature (T ). Trends occur in particle size and morphology when T ₀ is varied at fixed T which broadly resemble those in the cloud chamber work, where T is varied. Thus, in addition to the dendrites and prisms typical of low T ₀ there is an increasing proportion of spheres when T ₀ is raised above the melting point, until they eventually become the only species present. The particles condensing in the invisible fume generated at T ₀ settings below the melting point, were found to consist only of prisms and dendrites. This confirms that these particles condense directly in the solid state, unlike the spheres, which are originally liquid. The trends observed when changing T ₀ at constant T are shown to be in agreement with theoretical predictions for nucleation in a boundary layer.     

Localized meshless point collocation method for time-dependent magnetohydrodynamics flow through pipes under a variety of wall conductivity conditions

In this article a numerical solution of the time dependent, coupled system equations of magnetohydrodynamics (MHD) flow is obtained, using the strong-form local meshless point collocation (LMPC) method. The approximation of the field variables is obtained with the moving least squares (MLS) approximation. Regular and irregular nodal distributions are used. Thus, a numerical solver is developed for the unsteady coupled MHD problems, using the collocation formulation, for regular and irregular cross sections, as are the rectangular, triangular and circular. Arbitrary wall conductivity conditions are applied when a uniform magnetic field is imposed at characteristic directions relative to the flow one. Velocity and induced magnetic field across the section have been evaluated at various time intervals for several Hartmann numbers (up to 10⁵) and wall conductivities. The numerical results of the strong-form MPC method are compared with those obtained using two weak-form meshless methods, that is, the local boundary integral equation (LBIE) meshless method and the meshless local Petrov–Galerkin (MLPG) method, and with the analytical solutions, where they are available. Furthermore, the accuracy of the method is assessed in terms of the error norms L ₂ and L _∞, the number of nodes in the domain of influence and the time step length depicting the convergence rate of the method. Run time results are also presented demonstrating the efficiency and the applicability of the method for real world problems.