Fundamental Defects of Classical Metrology

by

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

Physical Journal

of the "Collegium" All World Academy of Sciences

Munich (Germany)

12 (2012), 12

Keywords: Metrology, megascience, revolution, unimetrology, megamathematics, overmathematics, unimathematical test fundamental metasciences system, knowledge, philosophy, strategy, tactic, analysis, synthesis, object, operation, relation, criterion, conclusion, evaluation, measurement, estimation, expression, modeling, processing, symmetry, invariance, bound, level, worst case, defect, mistake, error, reserve, reliability, risk, supplement, improvement, modernization, variation, modification, correction, transformation, generalization, replacement.

Introduction

There are many separate scientific achievements of mankind but they often bring rather unsolvable problems than really improving himan life quality. One of the reasons is that the general level of earth science is clearly insufficient to adequately solve and even consider many urgent himan problems. To provide creating and developing applicable and, moreover, adequate methods, theories, and sciences, we need their testing via universal if possible, at least applicable and, moreover, adequate test metamethods, metatheories, and metasciences whose general level has to be high enough. Mathematics as universal quantitative scientific language naturally has to play here a key role.

But classical mathematics [1] with hardened systems of axioms, intentional search for contradictions and even their purposeful creation cannot (and does not want to) regard very many problems in science, engineering, and life. This generally holds when solving valuation, estimation, discrimination, control, and optimization problems as well as in particular by measuring very inhomogeneous objects and rapidly changeable processes. It is discovered [2] that classical fundamental mathematical theories, methods, and concepts [1] are insufficient for adequately solving and even considering many typical urgent problems.

Mega-overmathematics including overmathematics [2] based on its uninumbers, quantielements, quantisets, and uniquantities with quantioperations and quantirelations provides universally and adequately modeling, expressing, measuring, evaluating, and estimating general objects. This all creates the basis for many further megamathematics fundamental sciences systems developing, extending, and applying overmathematics. Among them are, in particular, science unimathematical test fundamental metasciences systems [3] which are universal.

Metrology Unimathematical Test Fundamental Metasciences System

Metrology unimathematical test fundamental metasciences system in mega-overmathematics, unimechanics, and unistrength [2] is one of such systems and can efficiently, universally and adequately strategically unimathematically test metrology. This system includes:

fundamental metascience of metrology test philosophy, strategy, and tactic including metrology test philosophy metatheory, metrology test strategy metatheory, and metrology test tactic metatheory;

fundamental metascience of metrology consideration including metrology fundamentals determination metatheory, metrology approaches determination metatheory, metrology methods determination metatheory, and metrology conclusions determination metatheory;

fundamental metascience of metrology analysis including metrology subscience analysis metatheory, metrology fundamentals analysis metatheory, metrology approaches analysis metatheory, metrology methods analysis metatheory, and metrology conclusions analysis metatheory;

fundamental metascience of metrology synthesis including metrology fundamentals synthesis metatheory, metrology approaches synthesis metatheory, metrology methods synthesis metatheory, and metrology conclusions synthesis metatheory;

fundamental metascience of metrology objects, operations, relations, and criteria including metrology object metatheory, metrology operation metatheory, metrology relation metatheory, and metrology criterion metatheory;

fundamental metascience of metrology evaluation, measurement, and estimation including metrology evaluation metatheory, metrology measurement metatheory, and metrology estimation metatheory;

fundamental metascience of metrology expression, modeling, and processing including metrology expression metatheory, metrology modeling metatheory, and metrology processing metatheory;

fundamental metascience of metrology symmetry and invariance including metrology symmetry metatheory and metrology invariance metatheory;

fundamental metascience of metrology bounds and levels including metrology bound metatheory and metrology level metatheory;

fundamental metascience of metrology directed test systems including metrology test direction metatheory and metrology test step metatheory;

fundamental metascience of metrology tolerably simplest limiting, critical, and worst cases analysis and synthesis including metrology tolerably simplest limiting cases analysis and synthesis metatheories, metrology tolerably simplest critical cases analysis and synthesis metatheories, metrology tolerably simplest worst cases analysis and synthesis metatheories, and metrology tolerably simplest limiting, critical, and worst cases counterexamples building metatheories;

fundamental metascience of metrology defects, mistakes, errors, reserves, reliability, and risk including metrology defect metatheory, metrology mistake metatheory, metrology error metatheory, metrology reserve metatheory, metrology reliability metatheory, and metrology risk metatheory;

fundamental metascience of metrology test result evaluation, measurement, estimation, and conclusion including metrology test result evaluation metatheory, metrology test result measurement metatheory, metrology test result estimation metatheory, and metrology test result conclusion metatheory;

fundamental metascience of metrology supplement, improvement, modernization, variation, modification, correction, transformation, generalization, and replacement including metrology supplement metatheory, metrology improvement metatheory, metrology modernization metatheory, metrology variation metatheory, metrology modification metatheory, metrology correction metatheory, metrology transformation metatheory, metrology generalization metatheory, and metrology replacement metatheory.

The metrology unimathematical test fundamental metasciences system in megamathematics [2] is universal and very efficient.

In particular, apply the metrology unimathematical test fundamental metasciences system to classical metrology.

Fundamental Defects of Classical Metrology

Even the very fundamentals of classical metrology have evident cardinal defects of principle.

Modern engineering (astronautics, aircraft-building, ship-building, deep-sea industry, power engineering, electronics, chemical industry, building, etc.) requires the optimal design and a rational control of structural elements intended for extreme exploiting conditions (variable loading, high pressure, high or low temperature, radiation, etc.). Therefore, it is necessary to precisely or at least adequately measure also very nonuniform (inhomogeneous) objects and processes. In particular, this holds for stress concentration.

But classical metrology:

uses nonuniversal physical magnitudes such as dimensional implantation doses and mechanical stresses dependent on a choice of a specific system of physical dimensions (units) for a force and a length and not numerically invariant by unit transformations,

cannot regard and estimate object discretization errors,

cannot regard and correct averaging errors,

cannot adequately set and solve general metrology problems.

The finite element method (FEM) is regarded standard in computer aided solving problems. To be commercial, its software cannot consider nonstandard features of studied objects. There are no trials of exactly satisfying the fundamental equations of balance and deformation compatibility in the volume of each finite element. Moreover, there are no attempts even to approximately estimate pseudosolution errors of these equations in this volume. Such errors are simply distributed in it without any known law. Some chosen elementary test problems of elasticity theory with exact solutions show that FEM pseudosolutions can theoretically converge to those exact solutions to those problems only namely by suitable (a priori fully unclear) object discretization with infinitely many finite elements. To provide engineer precision only, we usually need very many sufficiently small finite elements. It is possible to hope (without any guarantee) for comprehensible results only by a huge number of finite elements and huge information amount which cannot be captured and analyzed. And even such unconvincing arguments hold for those simplest fully untypical cases only but NOT for real much more complicated problems. In practically solving them, to save human work amount, one usually provides anyone accidental object discretization with too small number of finite elements and obtains anyone "black box" result without any possibility and desire to check and test it. But it has beautiful graphic interpretation also impressing unqualified customers. They simply think that nicely presented results cannot be inadequate. Adding even one new node demands full recalculation once again that is accompanied by enormous volume of handwork which cannot be assigned by programming to the computer. Experience shows that by unsuccessful (and good luck cannot be expected in advance!) object discretization into finite elements, even skilled researchers come to absolutely unusable results inconsiderately declared as the ultimate truth actually demanding blind belief. The same also holds for the FEM fundamentals such as the absolute error, the relative error, and the least square method (LSM) [1] by Legendre and Gauss ("the king of mathematics") with producing own errors and even dozens of cardinal defects of principle, and, moreover, for the very fundamentals of classical mathematics [1]. Long-term experience also shows that a computer cannot work at all how a human thinks of it, and operationwise control with calculation check is necessary but practically impossible. It is especially dangerous that the FEM creates harmful illusion as if thanks to it, almost each mathematician or engineer is capable to successfully calculate the stress and strain states of any very complicated objects even without understanding their deformation under loadings, as well as knowledge in mathematics, strength of materials, and deformable solid mechanics. Spatial imagination only seems to suffice to break an object into finite elements. Full error! To carry out responsible strength calculation even by known norms, engineers should possess analytical mentality, big and profound knowledge, the ability to creatively and actively use them, intuition, long-term experience, even a talent. The same also holds in any computer aided solving problems, e.g., in hydrodynamics. A computer is a blind powerful calculator only and cannot think and provide human understanding but quickly gives voluminously impressive and beautifully issued illusory "soluions" to any problems with a lot of failures and catastrophes. Hence the FEM alone is unreliable but can be very useful as a supplement of analytic theories and methods if they provide testing the FEM and there is result correlation. Then the FEM adds both details and beautiful graphic interpretation.

Therefore, the very fundamentals of classical metrology have a lot of obviously deep and even cardinal defects of principle. There were no metrology satisfying the complex of modern requirements for inherent unity, consistency, sufficient completeness, universality, naturalness, simplicity, and convenience for practical application.

References

[1] Encyclopaedia of Mathematics / Managing editor M. Hazewinkel. Volumes 1 to 10. Kluwer Academic Publ., Dordrecht, 1988-1994.

[2] Lev Gelimson. Elastic Mathematics. General Strength Theory. The "Collegium" All World Academy of Sciences Publishers, Munich (Germany), 2004, 496 pp.

[3] Lev Gelimson. Science Unimathematical Test Fundamental Metasciences Systems. Mathematical Journal of the “Collegium” All World Academy of Sciences, Munich (Germany), 12 (2012), 1.