Structurally complex constructions with high accuracy are considered as hierarchically complex systems. For these systems, the requirements for the manufacture of elements, for the formation of control actions, and for instrumental monitoring of system parameters are formulated. The peculiarities of applying multilevel optimization methods to technical object designs are that multiparametric optimization is necessary at all hierarchical levels. At the same time, rigorous statements and solutions of extremal problems are di cult. The problems of optimization of precision constructions in the literature are considered for an extremely narrow class of technical objects. A practical (numerical) approach to multilevel optimization of structures is proposed. In the framework of this approach, multi-parameter weakly formalizable problems are solved at various hierarchy levels with additional requirements for high accuracy of parameter monitoring. Multilevel optimization is applied to a wide class of technical objects.
In most cases precision constructions are structurally complex technical objects. To one or several elements of these designs, special requirements are imposed on the accuracy of the manufacture of elements, the requirements are imposed on the formation of control actions, the requirements are imposed on the instrument control or on the estimated evaluation of the parameters of the system. The methodology of design, analysis, optimization of structurally complex designs of technical objects presupposes the consideration of them as complex hierarchical Copyright c by the paper's authors. Copying permitted for private and academic purposes.
In: S. Belim et al. (eds.): OPTA-SCL 2018, Omsk, Russia, published at http://ceur-ws.org
systems. The development of methods for optimizing hierarchical structures is
one of the important problems of developing the theory of hierarchical
structures [
The speci city of optimization of precision designs lies in the fact that, in addition to methods of proper optimization, additional tools are used to improve the accuracy of the results. These tools are technical solutions of structures, specialized software and algorithms, additional mathematical models in the subject area, as well as control systems.
In [
Obtaining numerical estimates of the deformed state of precision structures is one of the tools used in the design calculations of precision structures In this case, the accuracy, low level of errors of such estimates is one of the most important factors for ensuring precision. In this paper, we propose an approach to multi-level optimization of precision designs based on the application of their nite element models in conjunction with an additional procedure for inverse analysis of errors in numerical solutions. 2
The Concept of Multi-Level Optimization of Precision Constructions The following concept of multi-level optimization of precision constructions is proposed. This treatment of the process is developed on the assumption that the parameters of precision (a quantity, to the accuracy of determining which special requirements are imposed) are the displacements (absolute deformations) of individual elements of the system expressed in natural terms (units of length) or the integral indices (dispersion, mean-square deviation of displacements). The principal characteristic of the concept is that a special procedure for the analysis of a computational error is included in the decision contour when searching for the optimal variant, the evaluation of which is taken into account in the inequalities expressing the constraints. Then in the general case the constraints take the form [p] pcalc
perr pcalc + perr or [p]; (1) (2) where p is the precision parameter; pcalc is the calculated value of the precision parameter; perr - evaluation of the computational error of pcalc; [p] the maximum (allowed) value of the parameter.
The concept assumes 1) multi-level decomposition of the design with the selection of nodes (groups of elements) according to functional and technological features: the unit includes elements that share some function of the system;
2) selection of the parameter (parameters) of precision and the assignment of its (their) limit values;
3) formulation of local optimization problems (selection of design variables, objective functions, constraints) and obtaining Pareto-optimal solutions for each node on the basis of nite element analysis of its deformed state; 4) global optimization of the entire system in the collection; 5) evaluation and accounting for equation (1) of computational errors in determining the precision parameter at all optimization levels.
Within the framework of the concept of various hierarchical levels, the solution of multiparameter weakly formalizable optimization problems is realized. When comparing constructive options from the positions of the theory of decision-making, we are dealing with the problem of ordering objects by preference in the case of multicriteria choice on a nite set of alternatives. This problem is solved using the hierarchy analysis method, which is based on the use of comparative object scores by criteria using paired comparisons. The Method of Inverse Analysis of Numerical Solutions Errors Let us consider the resolving system of linear algebraic equations (SLAE) arising from nite element analysis
Ax = b;
(3)
where A is the square sparse matrix of coe cients (sti ness matrix); b is the
vector of the right side (load vector). In this paper, we propose modi ed methods
for a posteriori error analysis [
Accumulation of the error in the numerical solution of algebraic equations is the total e ect of roundings made at individual steps of the computational process. Therefore, for a priori estimation of the total e ect of rounding errors in numerical methods of linear algebra, one can use a scheme of so-called inverse analysis.
If the inverse analysis scheme is applied to evaluate the solution of SLAE, then this scheme is as follows. The uh SLA solution, calculated by the direct method M, does not satisfy the original system (3), but can be represented as an exact solution of the perturbed system (A +
A) u = b + b
The quality of the direct method is estimated by the best a priori estimate
[
Such "best" A and b are called respectively the matrix and the vector of the equivalent perturbation for the method M: If the estimates for A and b were obtained, then theoretically the error of the approximate solution uh can be estimated by inequality u
uh kuk 1 cond(A) kA 1k k
Ak k
Ak + k bk : kAk kbk
As already noted, the estimate of the norm of the inverse matrix A 1 is rarely known, and the main reason for this inequality is the ability to compare the quality of di erent methods. Below is a view of some typical estimates for the matrix A. For methods that use orthogonal transforms and oating-point arithmetic (the coe cients of the system A; b are assumed to be real numbers) k
AkE f (n) kAkE ":
puter, kAkE = the relative accuracy of arithmetic operations in a com
1 Pi;j ai2j 2 is the Euclidean matrix norm, f (n) a function of (4) (5) the form Cnk, where n is the order of the system. The exact values of the constant C and the exponent k are determined by such details of the computational process as the rounding method used, and the type of operation of accumulation of scalar products. In the case of methods of Gauss type, the right-hand side of the estimate (5) also contains the factor g(A), which re ects the possibility of growth of elements of the matrix A at intermediate steps of the method compared to the original level (such growth is absent in orthogonal methods ). To reduce the value of g(A), di erent methods of selecting the leading element are used, which prevent the elements of the matrix from increasing.
Let us describe the methods of a posteriori analysis of SLAE errors based on the inverse analysis of errors. Let there be given two SLAEs having one matrix of coe cients A (rigidity matrix), the rst system
Au = binit
Az = bnew; has the vector binit of the right-hand sides, for the second system we form the vector of right-hand sides bnew in such a way that the resulting system has a known solution. Let us single out two variants of the formation of systems for which inverse error analysis is applicable. In the rst variant, solving the system (6) by the numerical method M , we nd the solution unum, then substituting this solution vector in the left-hand side of the system (6), calculate the vector bnew = Aunum. Thus, we construct a system of linear algebraic equations of the form (7), the exact solution of which is the vector of the numerical solution of the system (6) unum, as can be seen by simple substitution. The norm of the di erence vector kunum znumk kunum uk, that is, the vector of the di erence unum znum approximates the error vector of the numerical method K).
Further, the second variant of the reverse error analysis changes the way of constructing the right part of the SLAE. Let us formulate the basis of this method of construction. Using estimates of the norm of the solution vector of the SLAE, it is easy to obtain the inequality kuk kbk : kAk We construct a solution z = (zk)k=1;n whose components (6) (7) or zk = kbk rk(A) ;
kAk zk = k (b)k rk(A) :
k (A)k The form of the functions ; ; rk is chosen so that after substituting these functions into the SLAE the norm of the right-hand side was either equal to the norm of the right-hand side or majorized it. For example, if the components of the solution vector are put equal 0 1 A ; then after substitution in the system (7) the components of the vector of the right-hand side of this system take the form Pn
j=1 aij bnew;k = miax jbinit;ij z @ max Pn
i j=1 aij 0 1 A : (8) Then for the systems (6), (7), the matrices of the coe cients coincide and the norms of the right parts are equal. All test cases have con rmed estimates for errors in numerical solutions of these systems ku unumk kz znumk : (9) It is possible to use other types of formulas for exact solutions of SLAE with the same coe cient matrix and the right-hand side close. The proposed method is approved for the estimation of errors of nodal displacements of discrete models of a number of highly relevant technical objects. Az = bnew: Estimation of the error of the numerical solution of the original system u unum k k unum
Numerical solution unum system Au = b of the Numerically solved by one numerical method M of the system of equations Au = b and Az = bnew, where bnew = Aunum k znum k Reverse er- A system of linear equations is con- Solved numerically by the ror analysis structed Az = bnew , method M systems of equamethod II 0 1 tions Au = b Az = bnew. zk = BB@ miamxPlaxjnj=bl1jaij ACCk, toEhrsietgiimnnuaamltiseoyrnsictaeomlf sotlhuetioenrroofr thoef bnew;k = ku unumk1 0 1 kz znumk1 :
Pn mlax jbinit;lj BB@ max Pj=1n aij CC = i j=1 aij A
k = mlax jbinit;lj i i=1;::;n; The norms of the right parts of the constructed and initial systems are equal to
The table was constructed (Table 3) comparing our two approaches to the inverse estimation of the error in the numerical solution of a system of equations with a sti ness matrix and the inverse error analysis proposed earlier in Wilkinson's papers. The semantic characteristic of the estimates for methods I and II is as follows. The more accurately we solved the original SLAE, the less the discrepancy, the closer the exact and approximate solution. In this case, the evaluation by method I will be better and is optimistic. The estimation by method II is the upper limit of all possible errors and, accordingly, is pessimistic. 4
Multi-Criteria Optimization of the Precision Design of a Large Mirror Antenna The practical application of the approach is considered on the example of searching for the optimal constructive variant of a large-sized precision re ector of a parabolic mirror antenna of terrestrial satellite communication systems. Parameters to which high requirements for accuracy of determination and provision are made, are the maximum values of absolute deformations and the standard deviation of the re ecting surface of the mirror from the ideal paraboloid. The Fig. 2. The distribution of total translational displacements (absolute deformations) in the re ecting segments of the second form factor connected to the frame by means of eight brackets under the action of wind pressure (40 m/s) and its own weigth proposed procedure for the search for a rational constructive shape of a largesized precision re ector is multilevel (in accordance with the structure of the object: mirror optimization { level 1, frame { level 2, hubs { level 3), multicriteria and based on obtaining Pareto-optimal technical solutions.
When comparing constructive options from the positions of the theory of
decision-making, we are dealing with the problem of ordering objects by
preference in the case of multicriteria choice on a nite set of alternatives. This
problem is solved using the hierarchy analysis method, which is based on the
use of comparative object scores by criteria using paired comparisons [
At levels 1-3, the e ects of air ow, solar radiation, low and high temperature of the medium, and its own weight are considered. At level 3, the e ects of sinusoidal vibration and acoustic noise are additionally taken into account.
Level 1. Optimization of the mirror is to determine the best - the shapes and sizes of segments (the method for cutting a paraboloid) and their cross sections; - variants of mechanical connections of segments. We represent the space of project parameters of the mirror in the form of a set Hm = ffVmg; fGmg; fLmgg, wherefVmg is the set of variable characteristics of the mirror; fGmg a set of objective functions for the mirror;fLmg a lot of restrictions for the mirror:
Hm = ffVmg; fGmg; fLmgg fVmg = ffTmg; fJmg; fEmgfAgg fGmg = ffm ! max; m ! min; m ! min; um ! ming; fLmg = ffm [fm]; m [ m]; mm [mm]g where Tm segments; Jm a set of parameters describing the topology of re ecting mirror set of parameters of cross-sections of mirror segments; Em set of parameters of rigidity of constructional materials of mirror segments; A the set of parameters describing the location of the support brackets of mirror segments.
The quantities fm; m; mm; um denote, respectively, the lowest eigenfrequency of free oscillations of the mirror, the maximum stresses in the cross sections of the segments, the mass of the mirror, and the maximum deformation (de ection) of the mirror. The values in square brackets correspond to the allowed values of the quantities.
As a result of multivariate numerical analysis, the features of the mechanical behavior of re ecting mirror segments are established (Fig.1), which is the basis for optimizing their geometric and rigidity characteristics, the conditions for their mechanical interaction with the framework (the required coordinates of the support brackets).
Level 2. Optimization of the frame is reduced to justifying the best - the spatial structure of the frame (number, spatial orientation of the rods and connections between them); - the shape and dimensions of the cross-sections of the rods. The space of design parameters of the frame is represented by a set Hs = ffVsg; fGsg; fLsgg, where Vs is the set of variable characteristics of the frame; Gs a set of target functions for the frame; Ls many restrictions for the skeleton:
fVsg = ffTsg; fJsg; fEsgg fGsg = ffs + m ! max; s ! min; ms=m ! min; us ! min; Fb ! maxg; fLsg = ffs+m where fTsg is the set of parameters describing the topology of the framework; fJs g a set of parameters of cross-sections of rod elements;fEs g set of parameters of rigidity of used structural materials of a skeleton; fs+m; s, ms+m, us, Fb respectively, the lowest eigenfrequency of the free vibrations of the mirror and the carcass assembly, the total mass of the carcass and mirror, the maximum deformation of the carcass and the mirror assembly, the critical strength of the buckling resistance (Euler force).
The problem of optimal frame design is solved in two stages. At the rst
stage, the substantiation of the placement of the rods in the frame volume is
carried out, i.e. "A picture of a lattice" ("pattern of a structure") of a skeleton.
This is a weakly formalized procedure, based on the coordinates of the location
of the support brackets of mirror segments obtained at the rst level, Mazhid's
theorem on structural changes [
Level 3. Optimization of the hub is to justify the best - the shape and dimensions of the panels and their cross sections; - the spatial structure of the hub (number, spatial orientation of the panels). The space of design parameters of the hub is represented by a set Hp = ffVpg; fGpg; fLpgg; where fVpg a set of variable characteristics of the hub; fGpg a set of target functions for the hub; fLpg many restrictions for the hub: fVpg = ffTpg; fJpg; fEpgg; fGpg = ffp+s+m ! max; p ! min; ! ming; fLsg = ffp+s+m [fp+s+m]; p [ p]; mp+s+m [mp+s+m]; [ ]g; where fTsg a set of parameters that describe the hub topology; fJpg set of parameters of cross-sections of hub panels; fEpg set of parameters of rigidity of applied constructional materials; fp+s+m, p, mp+s+m, are respectively the lowest eigenfrequency of the free oscillations of the re ector assembly, the maximum stresses in the hub elements, the total mass of the re ector, the standard deviation of the surface of the deformed mirror from the ideal paraboloid.
In the analysis at the third level, the construction of the mirror and the frame is known. The main requirement for the projected hub is to ensure the maximum realization of the strength and rigidity potential inherent in the design of the mirror and the frame.As a result of optimization of the hub design, the compliance (validation) of the characteristics of the re ector in the assembly is checked by the requirements imposed on the lower frequencies of free oscillations (Fig.4), the maximum displacements (Fig.5), and the stresses (Fig.6), and other parameters, which is part of the limitations when performing the optimization procedure. 5
Conclusion A multi-level technology for the practical optimization of a precision re ector has been developed. This technology consists in the sequential substantiation of the constructive forms of the mirror, frame and hub, and is characterized by the continuity and inheritance of the system of constraints and objective functions applied at di erent levels. Numerical substantiation and practical optimization of rational constructive forms of the main functional subsystems of large-sized precision re ectors are performed using multivariate multivariate computational experiments. The results are satis ed by the application of the procedure of reverse error analysis. In this case, it consists in the development of several alternative versions of nite-element models and the estimation for each variant of the errors in numerical solutions. Perr: The displacement of the structures (Fig. 5) was considered as computed values of the precision parameter Pcalc:The substitution of Pcalc and Perr in (2) ensured that the permissible values of [P ] were not exceeded by moves and guaranteed achievement of structural precision. Acknowledgement. This work was done during the complex project and was nancially supported by the Russian Federation Government (Ministry of Education and Science of the Russian Federation). Contract 02.G25.31.0147.