Optimization of Physics-Mechanical Parameters of Hardening Complex Metal Coatings: Regression-Tensor Approach
A nonlinear multidimensional regression-tensor model is constructed and investigated to the end of grounding (necessary and sufficient conditions are implied) of an optimal multifactor physics-chemical process of hardening metal coatings. A robust-adaptive strategy of rational forming the goal functional of physics-mechanical quality of metal working is proposed. The results obtained may form a methodological ground in constructing the systems of computer-aided design, technologies of hardening surfaces of complex composite fabricated metal products of the basis of complex tribological tests.
Keywords: Tribological tests; Regression-Tensor Model; Hardening Metal Coatings
Development of methods of hardening the working surfaces of cutting machines presumes complex physics-chemical processes (PCP). So, essential still are the issues of formalization/processing the respective mathematical models. In the given context, regression models (linear ones, nonlinear ones, including matrix ones, where the important class of systems is represented by regression-tensor systems are in demand [1-7]. On the one hand, from the viewpoint of properties, these systems are quite close to polynomial ones, which presume a rather detailed analytical description on the basis of (i) tensor calculus, (ii) strong differentiability of vector mappings and (iii) the theory of extreme problems [2,7,8]. On the other hand, these systems acquire an important role in nonlinear modeling multifactor tribological properties of synthesized metal coatings, in particular, in prognostic description of surface nano-dimensional structures [9,10].
Below considered are the problems stated in the conclusions of paper, while the objective is not the formal precision conclusions but rather the clarity of the conceptions in development of tribological problems [5,11]. Hence the issue of forming the functional of metal coating physics-mechanical properties for the mode of hardening is solved in this context. Determined are the strong analytical interpretations of the interconnected conditions, which define an optimal mode of the PCP given by nonlinear constraints and providing for adequacy of the PCP model to the data of tribological tests [12,13]. So, solved is the problem of multicriterial identification (by the least squares method (LSM)) of coordinates of covariant tensors in the PCP equation as a multidimensional regression with a minimum tensor norm.
Let R be a field of real numbers, Rn be an n-dimensional vector space over R with the Euclidean norm ||.||Rn,
be a vector column with the elements
be a space of all n×m-matrices with the elements from R. Next, let us denote by
the space of all covariant tensors of k-th valence (of real poly-linear forms fk,m:
→ R) with the tensor norm
, where ti…j are coefficients (coordinates of tensor fk,m, whose values are given with respect to the standard (natural) orthonormalized basis in the Euclidean space Rm .
be a vector of varied physics-chemical predictors of regression of PCP with a fixed beginning in
(support mode of hardening),
is a vector of qualitative indicators of PCP . As far as the given problem statement is concerned, consider a multidimensional nonlinear “input-output” system described by the following vector-tensor k-valent equation of multifactor regression
, vector function
belongs to the class
are invariant tensors of zero valence (tribological quality indicators of the scrutinized physics-chemical process in the supportmode
of its physics-chemical predictors) [7,11].
Remark 1: The description of PCP in terms of the regression system (1) is adequate on account of Proposition 2 related to continuous dependence of the solution of the PCP differential equation on the initial-boundary conditions and on the parameters [5,8,15].
The problem of an a posteriori regression-tensor modeling of an optimum PCP has been stated and investigated in detail in for a bivalent model (1). Furthermore, in the analytical solutions of three positions of the problem have been obtained :
1) for a fixed index k, a given predictor
, i.e. an open neighborhood of vector
defined are the analytical conditions, under which the vector function
of PCP quality indicators satisfies system (1);
2) constructed is an algorithm of identification of coordinates of symmetric tensors
in the mathematical model of PCP (1) on the basis of bicriterial LSM-problem (2) (parametric LSM-identification of a multidimensional regression-tensor system (1) with the minimal tensor norm) :
are vectors of experimental factor predictors of PCP (
is a “reaction” to the “variation”
with respect to the “mode”
, what is determined by condition (1*)),q is the number of tribological experiments with PCP; under such a present problem statement an approach proposed in may be applied [17,18].
3) under the given predicting vector
and ε(ω,v)≡0 for the case of bivalent regression-tensor model (1) we have obtained an analytical solution corresponding to “v-optimization” of the quadratic function of varied predicting factors for the scrutinized PCP with respect to its support mode ω(a “support vector” of predicting factors):
where vector function
has the coordinate representation corresponding to the identifies model (1)_(2); ri>0 are the weighting coefficients, which reflect a relative priority of some of tribological characteristics wi,
of PCP physics-mechanical properties.
Problem statement (on the basis of the conclusion from) . It is required to determine the necessary conditions in the solution of problem (3) when k = 3 (finding the stationary points in (3) for the 3-valent model (1)), and complement this determination with finding sufficient conditions of “v-optimization”, i.e. provision for the “elliptic character” of functional F critical points at the expense of dependence of spectral characteristics of its Hessian on variations of vector r := col
,with respect to some “initial” positions
Consider the case of equations of multidimensional regression with the tensor structure of valence k = 3; solving problem (2) when k = 3 represents a trivial modification of the proof of Proposition 3 . Under such a problem statement, the system of equations (1) may be represented in the vector-matrix-tensor form
(in this case, we assume that each Bi is an upper triangular matrix), from now on, the upper prime index ` denotes the operation of transposition of either a vector or a matrix; the vector function ε(ω,.):
satisfies (due to condition (1*) and Proposition 2 the following analytical estimate .
When k = 3, functional F is twice continuously differentiable, what guarantees the equality of mixed derivatives∂
therefore, the principal result of solving problem (3) (due to Theorem 3 and Theorem 7.2.5. for the 3-valent model (4) presumes the following proposition [8,14].
Proposition 1: Let
, where each Bi is a matrix of system (4) and, furthermore, consider the following vector function
Hence the stationary points
of problem (3) is solutions of equation
in this case, the sufficient condition of the statement that point v* of the space of predicting factors provides for “maximum quality of PCP” of the form
represents the following requirement: v* as a critical point of functional F(v) must be of special elliptic type. This is precisely the same as the statement that
det [bij]p < 0, p=1,…,m,
are main submatrices of Hessian G(r) at point
or similarly to state that the characteristic numbers λp of matrix G(r) satisfy the condition
Corollary 1. When k = 2, Hessian G(r) of functional F is
furthermore, when rank G(r) = m, the solution of equation (5) is unique and has the form
Obviously, (5) represents an intersection of m quadrics so, if conditions (6) (or, similarly, (7)) are not satisfied, then the critical point (/points) (5) is hyperbolic (saddle) one. Therefore, existence of a saddle point guaranteed by the replacement of < in (6) or (7) with > at least in one (not in all) inequalities (see e.g. (16) ). Replacement of inequality < with the reflexive < induces in v* the structure of stationary parabolic point of functional F(.); in this case, rank G(r)
It is clear that coordinate adjustment of vector r is one of the factors, which influence the geometry of F(.) at the critical point v*. This determines the statement of the problem of “adaptive correction”
for (3). Analysis of adaptive correction is conducted below.
In this section we consider the following problem. It is necessary to use the regression-tensor model (4) as the basis and construct a numerical procedure of choosing the vector of weighting coefficients
, which would provide for an elliptic character of the fixed stationary point v* (a solution of equation (5)) of the goal functional
while proceeding from the assumption that algebraic (spectral) conditions (7) are satisfied.
Remark 2: Despite the fact of algebraic equivalence (6)~(7), an attempt of usage of expansion of determinants (6) in constructing an adaptive correction
is almost inevitably condemned to failure because there is a large number of terms present in such an expansion.
Necessary and sufficient conditions of solving problem (3) may be obtained only in exceptional cases. As a rule, for such problem statements, the general problem turms out to be NP-hard. Below, we intend to discuss an approach to solving this problem for the functional F(.). Such an approach is grounded on the provisions of the theory describing localizations and perturbations of eigenvalues of the matrix from . Transformation of conditions (7) to the so called problem of quadratic stability represents another efficient technique. The problem of quadratic stability is usually reduced to constructing the Lyapunov function in the affine family of matrices under the assumption that this family, in turn, is functionally (due to the second formula in (3)) dependent on coordinates of vector
Let there be given an initial vector
of weighting coefficients from (3). For example, the process of goal-oriented choosing of r0 may be realized on account of equality of its coordinates r0i,
to the values of some (given) functions
in the “auxiliary problems” related to forecasting PCP quality with respect to some indicators wi, 1 ≤ i ≤ n. Due to Corollary 2, for a bivalent model of regression (1) this situation is formalized in the following proposition .
Proposition 2: When k = 2, the initial weighting coefficient vector ()0010 , ,nrcolrr=… with the coordinates
has the following analytical representation
Remark 3: The statement “when k = 2” is not the key one because the given construction of vector r0 may be used also in the case of the 3-valent (with respect to the predictors) form of regression-tensor model (4); obviously, in this case, r0 may be “corrected”, on account of the condition of normalization
Next, denote by
a critical point of functional F(.) (a fixed solution of equation (5)) in the position, when r=r0; by
denote the Hessian of functional F(.) computed at point v0; now let
Hence, in case of varying of vector r according to the condition
the parametric family of Hessians G(r) from Proposition 1 is defined by the following affine matrix manifold of the form
Hessians (8) are symmetric matrices .
In case of an arbitrary matrix, the only description its eigenvalues presumes that these are solutions of its characteristic equation. Obtaining eigenvalues for the Hessian G(r) may also be (due to the Courant–Fischer Theorem, characterized as solving an optimization problem . The sphere of possible interpretations of the Courant–Fischer Theorem includes the speculations of Weyl’s Theorem on the relations between the eigenvalues of Hessian G0 and any Hessian from the manifold
This allows one to understand the following “variation” sense of the robust-adaptive constructions needed for correction of
. On account of the constructions introduced above, the potential of robust-adaptive adjustment of functional which provides for satisfaction of inequality (7) at the critical point (in case of varying
), contains Proposition 3. A modification of Theorem 6.3.12 on account of Theorem 4.1.3, which takes account of symmetric structure of Hessians (8), is implied .
be eigenpairs of Hessian and G0
Hence the characteristic numbers
of Hessian G(r), where
System (9) gives the possibility to assess how sensitive the eigennumbers of Hessians (8) to variations of the weighting coefficients
are. Obviously, this analysis is approximate (it is valid only for small
; see also the formulas of the perturbation theory, what, on account of Corollary 1, is expressed in the following Corollary .
Corollary 2: If k = 2, n = m,
is a vector of eigenvalues of Hessian matrix
are eigenvectors corresponding to them,
is the vector of characteristic numbers of Hessian G(r), which are standard with respect to criterion (7),
is an m×m-matrix with elements
it is possible to expect that eigenvalues of G(r) are equal to standard ones
Remark 4: Since system (9) is valid for a small value of
the following issue remains open. It is not clear whether the iteration process of the form
Constructed due to Corollaries 1, 2 on account that
Converges, when the initial divergence
is substantial or not. Obviously, due to structure of functional (3), it is necessary to verify conditions
on each iteration step “j” for the coordinate vector
Now, while treating the situation in the context of Remark 4, consider the result of computing the upper estimate for the relative perturbation
be a matrix norm in
is a unit matrix; for example, the Frobenius matrix norm is .
And the spectral (induced) matrix norm .
Thus, when turning back to Corollary 2, we have (due to the prototype of system (9)):
With det B ≠ 0. Suppose, vector
(in particular, at the expense of term
of system (9)) and matrix B transforms into B + D. In such a problem statement, the vector of adaptive adjustment Δr obtains (due to modification of Corollary 2) increment θ, and assumes the form
, which satisfies the following linear algebraic equation:
, model perturbations of a “desired change” of the vector of eigen numbers
, as well as the inaccuracy of parametric assessment of matrix B (note, if
[20, p. 197]).
Details of the approach to computing the upper estimate of relative perturbation
are formulated in Corollary 3 (technical details may be found in .
Corollary 3: Let (in addition to assumptions of Corollary 2)
Be a conventional number of matrixes B, where ||.||M is the matrix norm equal to ||.||F or ||.||S. Hence the following analytical estimate is valid 
are, respectively, the smallest eigenvalue B' and the largest eigenvalue B, then it is possible to assume that
in the latter inequality.
Remark 5: The construction of the spectral conventional number
(the conventional number obtained with the use of the spectral norm ||.||S) is transparent due to equality
When the issues of mathematical modeling of complex physics-chemical objects and processes are discussed, the following technological procedure is normally implied: (i) natural differentiation between different aspects of a definitely given PCP (as an object of mathematical investigation), (ii) description of each of the aspects on the basis of one’s own (normally, comparatively narrow and easily observable) group of mathematical assumptions, (iii) subsequent integration of the partial results obtained on account of proper specification, (iv) turning back to consideration of the complex (integrated) functioning of PCP.
Within the frames of above paradigm, the idea of the present paper presumes to develop the results of and to point to natural relationship existing between the problem of defining the domain of the matrix Hessian function values at the critical point of its goal functional of physics-mechanical quality (3) for the process of hardening metal coating, expressed by equation (1) and by vector r of weighting coefficients in (3), which reflect the “priority” between
i.e. between the modeled tribological properties of PCP . In this context, Proposition 1 and Corollary 1 show that, unlike that for the 3-valent (k = 3), in the bivalent (k = 2) model of nonlinear regression of PCP, the Hessian G(r) is invariant with respect to the position of the critical point. In this case, both of the variants (2 = k = 3) allow one to reveal the dependence
on the basis of the PCP model (1) identified in course of tribological tests with the aid of criterion (2).
Eigen values of the matrix are definitely the roots of its characteristic polynomial, so, the result of Proposition 3 is, in essence, based on the assumption that eigen values (7) are continuously r-dependent on the elements of Hessian matrix G(r) in the process of the ongoing parametric correction of the goal functional F from (3). Noteworthy, some part of information turns out to be lost, when we deal only with the characteristic polynomial, because there are many different matrices with the given characteristic polynomial. So, no wonder that stronger results obtained in modeling the Hessian’s G(r) spectrum, in particular, Proposition 3 and Corollary 2, take account of the structure of matrix G(r); the latter assume some technical simplification, which follows from the assumption that any Hessian matrix is orthogonally similar to the real diagonal matrix .
Numerical methods of finding eigen values and eigen vectors represent one of the most important divisions of the general matrix theory. The present paper, considers analysis of vector
and matrix B on account of Corollary 2, and does not touch any aspects of this complicated issue. Meanwhile, Corollary 3 suggests an upper estimate of the relative perturbation Δr obtained via the estimate of relative perturbations
, the estimate of B and the estimate of the conventional number s(B). Note, s(B) participates in assessment in all the cases, independently whether the perturbations take place only in
, only in B or in
and in B simultaneously.
In conclusion, let us discuss another approach related to adaptive correction of
which is bound up with the usage of sufficient conditions of robust stability of matrix G(r) (what is also equivalent to conditions (6), (7)). In this context, a requirement may be put forward that – in the family
and under the interval constrains imposed on variations of the coordinate vector Δr−it would be possible to construct a Lyapunov function
is a symmetric positive definite matrix; i.e. an assumption that there would exist some matrix P > 0, for which the Lyapunov matrix equation
has a solution for a given symmetric positive definite matrix
. An approach bound up with the transition to adaptive-robust quadratic stability and the methods of solving the problem with such a statement can be found in [21-24]. Owing to the abundance of computational problems described by the proposed theory and due to the great possibilities it opens for solving of multidimensional regression-tensor analysis problems, which presume various applications, this theory may now acquire an important independent value from the viewpoint of applications to the problem of synthesis of optimal coatings. It is hardly ever possible to consider all the aspects in one paper. But we are sure, that further detailed investigations in this direction will follow soon.
This work was supported by the Program “Leading Scientific Schools” (Project No. NSh-8081.2016.9) and by the Russian Foundation for Basic Research (Project No. 16-07-00201).