Applications (2)

1. Applications (2) Lecture 4

2. CA in engineering • The CA has opened a totally new filed of scientific investigations in engineering sciences which is based on • powerful methods for the analysis of multiparametric solutions and • the characterization of solutions of equations which model physical or industrial processes. • Symbolic data processing has steadily become a powerful tool for engineers. • Without symbolic utilities, effective mathematical problem solving in a theoretical as well applied environment is almost unthinkable of, nowadays. • The first CAS available for engineers was implemented on IBM-machines in the Mid ’70s, based on PL/1. • In 1979 the usefulness of Macsyma was demonstrated for the symbolic vector and dyadic analysis. • In 1987 the large deformations of a cylindrical liquid-filled membrane was analyzed under the action of a viscous shear flow; using Reduce for a fifth order series solution of the free-boundary-value problem led to results which described much better the physical behavior of this system than the numerical analysis could do.

3. CA in engineering • Modern material research • relies on simulation of physical processes which can only be described by extremely complicated models in order to achieve the required accuracy. • The complexity of the model equations makes the use of suitable utilities a necessity. • Symbolic programs enable one to derive, swiftly and without making errors, systems of differential equations for particular materials models, and convert them to difference form. • Practical questions as e.g. the lubrication of a four-parameter Oldroyd fluid in the slider bearing was successfully solved using a CAS. • Another example for the combination of an algebraic and numerical procedure to to solve a stability problem for the Taylor-Couette system was given in 1997: the time-independent exact solution in parametric form was found for a polymer solution by CA solving an ordinary differential of first order and fifths degree. • Transforming equations with respect to different coordinate systems chosen for particular problems. • In mechanics, one often has to analyze structures composed of a great number of individual parts which in turn are supply connected among each other, and are agitated by external forces. • For example, this is the case when studying a sequence of robotic motions, where one has to derive and solve systems of differential equations whose number can easily be in the order of several thousands. Nowadays, CAS generate those gigantic systems of equations, making extensive use of symbolic manipulation of data. • Software packages such as Axiom, Macsyma, Maple, Mathematica or Reduce are utilized also to avoid tedious investigations – so called parameter studies – for most of the time.

4. CA in engineering • For practical purposes, it is usually sufficient to reduce large systems of equations to a smaller number of equations, and then to analyze the qualitative properties of the solutions of the reduced system. • A typical example for this procedure is given by the methods in perturbation theory of ordinary differential equations. • Although the employment of computer algebra is still not as wide-spread as it would deserve, given its versatility and potential for applications, it is currently successfully used in several areas, such as: • robotics • nonlinear dynamics • elastomechanics • fluid dynamics • visco-elastic materials • rarefied gas dynamics • aerodynamics • control theory • real time systems.

5. CA in engineering CA can be integrated into algorithms for solving problems from areas of engineering in many different ways: • on the lowest level, it is used to perform basic operations of real and complex calculus, e.g. symbolic differentiation, integration, summation, or computing limits; • on a higher level, assists investigating linear systems of differential equations which arise e.g. in modelling nonlinear, mechanical systems by linearization; • application by initial and boundary value problems of partial differential equations whose analytic solutions provide more insights for parameter studies than numeric computations could do; this technique is widely used in elasto-mechanics; • solve certain classes of differential equations; • when numerical schemes become complicated an investigation of the numerical stability of the algorithm can be carried out using a CAS.

6. CA in engineering • For a project engineer, e.g., dealing with combustion, problems from flow dynamics and thermodynamics which can be treated by nonlinear field theory, are of special interest. • Particularly in computational flow dynamics, CA is used with increasing tendency, especially for deriving difference approximations to given PDEs. • First, differential equations, given by physical models, are analyzed for point transforms which leave the equations invariant (method feasible, e.g., for equations from gas dynamics). • In a next step, the first differential approximation, a differential relation which mirrors the effects of discretization, is derived from the corresponding difference equations. • If the discrete equations admit the same invariant transforms as the original differential equations then the system of difference equations is called an invariant scheme, and one is assured that the symmetry properties of the given equations carry over to solutions of the approximations. • As a practical application, where one has to conclude the stability or instability of the result from numeric computations, deriving detonation and flame fronts in combustion chambers of piston engines is of prime importance. • For an engineer, this requires a priori knowledge of the validity of the results, i.e., he has to be certain that no artificial instabilities are introduced by numerics. • CA programs provide assistance to swiftly obtain correct results, e.g. in dimensional analysis involving many independent variables during the design phase of an experiment. Determining all dimension-less variables is at the beginning of any lab-experiments.

7. Critical Load Computations for Jet Engines • In manufacturing of aircraft turbines, there is a considerable difference in temperature between the exterior and the interior during the hardening process. • Tempering exerts forces onto the connection of the parts. • In the most basic, one-dimensional case, it is just the classic critical load problem. • Of special interest is to conduct critical load computations which could be maximized with respect to a given class of functions modelling the joints. • Such a class is defined by a set of basis functions, where a general function is represented by a linear combination with appropriately many parameters. • Axiom was used in 1995 to solve a such problem.

8. Audio Signal Processing • Audio signal processing arose as one of the oldest research fields within the engineering society. • Meanwhile it has become an interdisciplinary topic with deeper connections to mathematics, computer science, psychoacoustics, among others. • Delicate open research problems emerged from application areas like digital audio broadcasting, multimedia, the Internet, audio-on-demand, as well as broadband communication. • Fundamental constructs are described properly using algebraic structures. • Among those structures are polynomials, power series and matrices over certain fields or rings modelling signals, filters, and linear systems. • Many systems are described by highly structured linear transformations which facilitates the design of efficient algorithms. • Among those transforms are Fourier, cosine, and (in connection with wavelets) block Toeplitz transforms. • The theory and design of filters and filter banks relies on methods for the treatment of polynomials and formal power series, like fast multiplication or factorization. • In multidimensional case, Groebner basis techniques are used. • Efficient Fourier transform algorithms facilitate fast convolution and hence fast filtering for real-time applications and implementation on digital signal processors. • Filter design algorithms as well as other signal processing applications frequently require high precision outputs.

9. Audio Signal Processing • Many software packages well suited for audio signal processing are now available. • The widely used Matlab package for numerical signal processing and visualization should be mention; a plug-in to this package, the symbolic toolbox, contains the kernel software of Maple, allowing symbolic calculations within Matlab. • Several toolboxes supporting the design of wavelets and filter banks, for example the Wavelab package and the FBT-tools have been implemented.

10. Robotics • Robotics integrates a very large spectrum of technical sciences ranging from electrical and mechanical engineering to mathematics and computer science. • The primary area of robotic-specific applications of algorithms from computer algebra is robot kinematics. • Kinematics investigates the relationship between the joints (variables) of arbitrary mechanisms and the pose (position and orientation) of its links (bodies) in space. • For a given robot, the so-called inverse kinematics problem (IKP) requires to determine all sets of joint values that take the robot hand to a given pose in space. • A similar problem is the direct kinematics problem (DKP) of so-called generalizes Stewart platforms, i.e. flight-simulator-type mechanisms consisting of a stationary and a mobile platform which are joined via a passive ball-joints and actuated telescope legs. • The DKP asks for all mobile platform poses that can be attained for a given set of leg lengths. • Both problems require to solve non-linear systems of sine-cosine equations (some of the variables being trigonometric functions) which relate joint-variables, pose-parameters and constant mechanism-dimensions.

11. Robotics • Industry demands all solutions of the IKP for any given hand pose within several milliseconds because otherwise the robot cannot be driven fast an smoothly along arbitrary desired trajectories. • This speed requirement prohibits the use of iterative algorithms. • Thus, kinematic systems of equations must be solved symbolically, i.e. in terms of implicit or even explicit solution formulae that reduce computations at run-time to a minimum • However, even simple manipulator geometries are far too complex for a direct application of state-of-the-art symbolic solution techniques like the Groebner basis computation. • Specific kinematical strategies for solving the IKP symbolically were developed and they work far better than elaborate computer algebra techniques for elementary manipulator classes but fail for difficult classes. • Based on a combination of traditional and new kinematic techniques large numbers of simple unspecialized elimination ideals are compiled and searched by elementary artificial intelligence methods and the optimal one is solved symbolically by specific adaptations of multivariate resultant algorithms to systems of sine-cosine polynomials with large number of formal parameters.

12. Robotics • Methods of CA play an important role in the so-called piano movers or collision avoidance problem where a robot of certain geometry is supposed to move a payload with given shape through an environment containing a number of known obstacles without collision. • All involved objects are usually specified by semi-algebraic sets. • In the CA solutions again Groebner basis are involved. • A field which is closely related to kinematics is mechanism synthesis. • In the classical form of the problem a set of desired precision points and the kinematical structure of a mechanism are given and dimensions of the mechanism have to be calculated such that the mechanism tool is driven through the precision points by movement of some actuated input-joint. • This field poses a similar scope of CA problems as kinematics. • An example of CA applications to mechanism-synthesis is the design of so-called cam-follower mechanism for peeling clams.

13. Computer Aided Design and Modelling • CAD/CAM systems: means of increasing the efficiency of the design process • Advantages: reduction of lead times, quality improvements and cost reduction by saving time spent implementing engineering changes in the design process • From a mathematical point of view almost all the CAD/CAM problems are related to the manipulation of geometric objects into the two or three dimensional space, mainly curves and surfaces and combinations of both. • These geometric entities are usually presented through polynomials via their implicit of parametric representation. • There are four different problem classes to be considered where computer algebra and computer aided geometric design meet: • the implicitization (or variable-elimination) questions, • the intersection of parametric curves and surfaces, • design with exact arithmetic and • the algebraically guided tracing of algebraic curves and surfaces implicitly defined. • Other problems that can be theoretically dialed by CA methods are: • Geometric formats conversion (from rational to polynomial parametrizations); • use hybrid (symbolic-numeric) methods for polynomial system solving arising in computer aided geometric design; • parametric computer aided geometric design.

14. Implicitization and variable-elimination questions • One of the main problems arising in the manipulation of parametric curves and surfaces in CAGD is the finding of efficient algorithms for computing the implicit equation of curves and surfaces parametrized by rational functions. • This is due, for example, to the fact that, if for tracing the considered curve and surface the parametric representation is the most convenient, to decide in an efficient way the position of a point with respect to the curve or surface considered, the implicit equation is desired. • Other problems similar to the implicitization problem and where the solution is obtained by eliminating from the initial equations some variables are: • computation of offset curves and surfaces, • computation of constant-radius blending surfaces, • computation of the convolution of two plane curves or surfaces, • computation of the convolution common tangent of two plane curves, • computation of the inversion formula for parametric surfaces. • These geometrical operations are often used when generating the boundary of a configuration space obstacles, in order to construct collision free motion paths for translating objects.

15. Implicitization and variable-elimination questions • The implicitization of, e.g., a rational surface, is difficult to achieve by applying directly resultants or Groebner bases because, • first, it is usually a very costly algebraic operation and, • second, the coefficients of the polynomials in the parametrization are usually floating-point real numbers. • These difficulties can be overcome in two ways: • by using multivariate resultants, • or by taking into account that, in general, a concrete object to model is made by several hundreds (or thousands) of small patches, all of them sharing the same algebraic structures. • In the last case, for a such object a database is constructed containing the implicit equation of every class of patch appearing in its definition. • This database must also contains the inversion formulae. • In the case of implicitatization of rational curves and surfaces can be used a method that uses the moving line ideal basis for planar rational curves, respectively the moving surfaces. • The inverse problem to the one considered here, given a curve or surface by its implicit equation, to determine if it could be presented in a rational way, can be solved in the case of planar curves (the curve admits a rational parametrization if and only if its genus is zero) • For surfaces is more complicated but also a genus computation gives the answer, at least for several kind of surfaces.

16. Algebraically guided tracing of algebraic curves and surfaces implicitly defined • Many important problems in CAGD are reduced to the computation of the graph of a planar algebraic curve presented implicitly. • The problem of computing the graph (even topologically) of a planar algebraic curve defined implicitly has received a special attention from CA since it has been responsible of many advances regarding • subresultants, • real root counting, • infinitesimal computations, • etc. • The problem of tracing surfaces implicitly defined is considerable more complicated since, • first, their topology structure is no so easy as the one for curves • and, moreover, the degrees of the univariate polynomials to deal with are extremely big.

17. Intersection of parametric curves and surfaces in CAGD using CA • The problem of the intersection between curves and surfaces is very important in CAGD and is usually reduced to the resolution of a polynomial system of equations, one of the most studied problems in CA. • The intersection of two planar curves without common components given parametrically can be determined by • merely computing the implicit equation of the first curve • and then substituting the parametrization of the second one into this implicit equations: • the problem is solved by solving an univariate polynomial equations. • The intersection when the two curves are given by their implicit equations can be determined by a similar argument.

18. Intersection of parametric curves and surfaces in CAGD using CA • The intersection of a rational surface (or a surface given by its implicit equations) and a rational curve into a 3-space, when the implicit equation of the surface is available, is reduced to the computation of the parametrization of the curve into the implicit equation of the surface. • If the implicit equation of the considered rational surface is not available, then, by equating both parametrizations one must solve a polynomial system of 3 equations and 3 unknowns. • If using CA, the only used technique is that based on multivariate resultants and mainly in Dixon’s formulation for eliminating two of the unknowns from the system. • The intersection of two rational surfaces is reduced to the tracing of a implicitly defined plane algebraic curve plus a lifting process if the implicit equation of one of the surfaces is available. • If no implicit equation is available, then one has to solve a polynomial system of 3 equations and 4 unknowns, and computer algebra can be used to eliminate two of the unknowns from the system. • If the two surfaces are presented by their implicit equation then the system to solve has only 2 equations and 3 unknowns.

19. Computer aided geometric design with exact arithmetic • It is assumed that all the input objects are known exactly and with their representation involving only rational numbers or algebraic numbers exactly presented. • In this case all the problems are reduced to the resolution of a polynomial system of equations or to the elimination of one or several variables • For these problem the using of Groebner bases or resultants are special suited.

20. Computer Algebra in Chemistry • One of the basic tasks in chemistry consists in exploring the relationship between reaction kinetics of molecular structures and the spatial arrangements of their atoms or molecular sub-structures respectively. • This leads to a number of problems with CA appeal. • One of these problems is the design of reasonable branch and bound methods for constructing all connected multigraphs for a given sum formula, where the degree of a vertex corresponds to the number of valences of an atom. • Closely related to this isomerism problem, yet much more intricate, is the stereoisomerism problem pertaining to the area of algorithmic real algebraic geometry. • A paradigm is the determination of all configurations of cyclo-hexane and cyclo-heptane; restrictions with respect to bonding distances and angles are known for certain atomic bonds, and the problem is to find ways of computing the space of all configurations satisfying these constraints. • The problem can be transformed into that of finding all real solutions of a system o polynomial equations and inequalities, or the number of its connected components, respectively. • Currently, the most important field of application is probably the determination of protein structure from 2D-NMR data.

21. CA in Crystallography • Other interesting problems arise when symmetry properties of the molecular structures under investigation (including crystals) are either known or can be presumed to exist. • In this case, many questions address combinatorial group theory. • It starts with the classification of all three-dimensional crystallographic group. • The classification of all four-dimensional crystallographic groups (important for the study of incommensurable crystals, and possible also for quasi-crystals), and of all five and six-dimensional crystallographic groups could not have been accomplished without development and implementation of methods from computational group theory. • Other problems relate to the theory of tillings, and to orbitfold theory as well as the theory of so-called Delaney symbols. • Their study requires algorithmic methods from both combinatorial group theory and algorithmic topology. • Advances in this area should have considerable potential for the classification of the geometry of crystalline structures, • e.g., the derivation of criteria for distinguishing geometrically (if possible) high temperature supra-conductors from other crystals. • Examples for application of CA to chemical equilibrium computations are also available

22. Chemical Reaction Systems • The dynamics of chemical reaction systems is often described by an explicit system of first order ordinary differential equations for the concentrations of the given species. • The right hand side is pure polynomial, typically of degree two, and most of the time consists of one or tow terms in each row. • Of special interest are stationary states of such systems, which lead to a polynomial system of equations. • For small or medium systems (e.g. about 40 species), the solutions can be algebraically determined by a combination of combinatorial methods, factorization, and Groebner bases techniques. • In contrast to numerical approximation methods, algebraic solutions provide a summary of all possible stationary states. • For small systems, the solutions can be computed as functions of the reaction constants (which will be symbolic in this case).

23. CA in education • There is no doubt that sooner or later CAS or hand-held CA tools will be used by everybody in the same way as numeric calculators are used today. • Whereas calculators brought more numeric computation into the classroom, CAS enable the use of more symbolic computation. • The usage of CA in the classroom is increasing worldwide, and slowly these steps are institutionalized and Math curricula are adapted accordingly. • Several hand-held products incorporating computer algebra tools are now available from Casio and Texas Instruments: TI-89, TI-92, Casio CFX-9970G, Casio Algebra FX2.0. • It is a fact that hand-held scientific calculators have significantly changed the high school and university mathematics curriculum around the world in the past 25 years. • For example, many topics that dealt with paper and pencil computation involving transcedental functions have been deleted. • Many sections and even some chapters in textbooks dealing with paper-and-pencil computation methods became obsolete and disappeared from the curriculum, because hand-held scientific calculators provided better ways to compute than paper-and-pencil methods. • The same thing (obsolescence) will soon happen with paper-and-pencil symbolic algebraic manipulations common today because of student use of inexpensive hand-held CAS that now exist and soon will proliferate.

24. CA in education • What changed was not the to-do but the how-to do the to-do. • The same content topics will be continue to be teach, but one should expect the methods use to-do or to-apply the topics will change. • Integrating CA into the curriculum means the same topics can be taught in less time so more time can be devoted to new mathematics, better mathematics, understanding, proof, problem solving etc. • What is needed today in the future is a school and university mathematics curriculum that takes advantage of CA technology to assist students in gaining mathematical understanding, in becoming powerful and thoughtful thinkers, communicators, and problem solvers. • There should be a balanced approach to the use of CA technology in mathematics teaching and learning. • Recruitment of students to engineering degrees in has been falling steadily over the past decade. • Among the many reasons for this phenomenon is the perception among school students that mathematics and science are hard and unglamorous subjects of study. • Students are now beginning their undergraduate studies lacking much of the mathematical ability, thinking and confidence which earlier cohorts have displayed. • An approach is to broaden the curriculum by developing undergraduate engineering degree courses that are focused more on design, communications or technology management rather than the more traditional and mathematically more demanding areas of mechanical or electrical engineering. • Fortunately the great majority of today’s new undergraduates possess significant IT skills. • Students can spend several hours in a laboratory for the mathematical methods modules. • In lectures laptops and overhead viewscreens can be used for demonstration with CA tools. • The strategy of identify the relevant mathematics, progress the solution, seek assistance of a CAS, progress the solution, is continued until a satisfactory solution is obtained.

25. CA in education • Computer algebra has been less readily accepted into the mathematics degree programme because the need to develop a different view of mathematics and the associated knowledge and skills and the more traditional curriculum. • Computer allowed a much wide class of examples to be investigated, giving students a more solid grounding in their area and sometimes allowing them to make discoveries that earlier researchers had overlooked. • CA systems allow: • active involvement of students in learning; • experimentation as a means of understanding mathematical concepts; • visualization of mathematical processes; • access of students to real-world problems. • The main concerns raised by opponents of computer-based calculus courses are gadgetry over intellect, proof-abuse and lack of necessary hand skills. • Although there were rebuttals to each of these concerns by proponents of computer-based calculus, CAS are still far from being in universal use for calculus introduction.

26. CA in education • CAS are beginning to be used for introducing students to a new set of mathematical concepts, e.g. visualization of very large and very small numbers, fractal dimension, cellular automata-generated music and other topics formerly outside the mainstream pre-college curriculum. • CA is not the object to be taught but serves as an aid in the process of teaching and learning mathematics. • Thus the focus entirely lies on mathematics, and the CAS are mainly a tool. • The importance of visualization can hardly be overestimated in general cognitive skill acquisition and problem solving processes. • Pictures activate mental processes such as the perception of spatial relationships, intuitive comprehension of complex processes, or the observation of patterns and therefore, aid the process of understanding. • Looking to a picture, we use it as a vehicle of thinking, but intend to understand processes and behaviors of the real world. • The useful aspects of visualization are the translation from representations which are more abstract to those which are less abstract. • In particular in mathematics we deal with abstract structures, which visual representation helps to enlighten. • The objects are not primarily geometric figures, but arbitrary mathematical objects such as infinite sequences, complex functions, or conformal mappings. • Computer algebra systems provide the necessary algorithms needed to compute mathematics visualizations. • Examples in this direction are the projects Illustrated Mathematics and Analysis Alive.

27. CA in education • Illustrated Mathematics offers a comprehensive collection of graphics and animations for topics in mathematics at the high-school and undergraduate and graduate college level. • The collection provided as Mathematica notebooks is organized by mathematical topics (ranging from basics such as sequences and series and end up with complex functions and minimal surfaces) and • allow users to experiment by seeing the effects of changing parameters on the objects they are studying. • Analysis Alive addresses students of mathematics as well as of sciences and engineers. • Electronic documents are presented in form of Maple worksheets.