ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ Εισαγωγή

1. ΕΙΣΑΓΩΓΗ Η ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ και ΤΑ ΥΠΟΛΟΓΙΣΤΙΚΑ ΜΑΘΗΜΑΤΙΚΑ είναι σχεδόν ταυτόσημες έννοιες. Αποτελούν τον κλάδο των Εφαρμοσμένων Μαθηματικών που ασχολείται με τη «διακριτοποίηση» και την εύρεση προσεγγιστικών λύσεων Μαθηματικών προβλημάτων των οποίων η αναλυτική λύση είναι αδύνατον να βρεθεί αναλυτικά ή σχεδόν ακατόρθωτη. Το διακριτό πρόβλημα που προκύπτει ονομάζεται Αριθμητική Μέθοδος.

Τι είναι η Αριθμητική Ανάλυση; Είναι Επιστήμη: Ασχολείται με μεθόδους επίλυσης μαθηματικών προβλημάτων με χρήση αριθμητικών πράξεων (με Η/Υ) καθώς και με την ανάλυση των σφαλμάτων στην προσέγγιση των λύσεων. Είναι Τέχνη: Αφορά στην επιλογή εκείνης της μεθόδου που είναι πιο «κατάλληλη» για την επίλυση ενός συγκεκριμένου προβλήματος.

Το θεωρητικό μέρος της Αριθμητικής Ανάλυσης περιλαμβάνει την κατασκευή αλγορίθμων - ανάλυση, μελέτη της ακρίβειας και της ευστάθειας - , δηλαδή, την ανάλυση και εύρεση των πιθανών σφαλμάτων τους. Το εφαρμοσμένο μέρος αφορά τον προγραμματισμό των αλγορίθμων σε μια γλώσσα προγραμματισμού με το βέλτιστο τρόπο, δηλαδή, με όσο το δυνατό λιγότερο υπολογιστικό χρόνο (CPU) και απαιτούμενο χώρο μνήμης (RAM). Το θεωρητικό και το εφαρμοσμένο μέρος είναι, συνήθως, αλληλένδετα. Η ανάπτυξη των υπολογιστικών συστημάτων καθιστά απαραίτητη και επιτακτική την εκμάθηση αριθμητικών μεθόδων για την επίλυση προβλημάτων επιστημονικών εφαρμογών.

Συνεχείς διαδικασίες  Διακριτές διαδικασίες Άπειρες διαδικασίες  Πεπερασμένες διαδικασίες Στόχος : Η προσεγγιστική επίλυση προβλημάτων που συναντώνται στις επιστήμες και την τεχνολογία, σε εφικτό υπολογιστικό χρόνο και με το μικρότερο σφάλμα.

A Small Example The Difference in Numerical Computing is the numbers A computation of π

Simple iteration:

Result of 15 digit computation 2 2.828427124746190 3 3.061467458920719 4 3.121445152258053 5 3.136548490545941 6 3.140331156954739 7 3.141277250932757 8 3.141513801144145 9 3.141572940367883 10 3.141587725279961 11 3.141591421504635 12 3.141592345611077 13 3.141592576545004 14 3.141592633463248 15 3.141592654807589 16 3.141592645321215 17 3.141592607375720 18 3.141592910939673 19 3.141594125195191 20 3.141596553704820 21 3.141596553704820 22 3.141674265021758 23 3.141829681889202 24 3.142451272494134 25 3.142451272494134 26 3.162277660168380 27 3.162277660168380 28 3.464101615137754 29 4.000000000000000 30 0.000000000000000 31 0.000000000000000 Result of 15 digit computation Red digits are correct Black and green digits are incorrect

Result of 15 digit computation 2 2.828427124746190 3 3.061467458920719 4 3.121445152258053 5 3.136548490545941 6 3.140331156954739 7 3.141277250932757 8 3.141513801144145 9 3.141572940367883 10 3.141587725279961 11 3.141591421504635 12 3.141592345611077 13 3.141592576545004 14 3.141592633463248 15 3.141592654807589 16 3.141592645321215 17 3.141592607375720 18 3.141592910939673 19 3.141594125195191 20 3.141596553704820 21 3.141596553704820 22 3.141674265021758 23 3.141829681889202 24 3.142451272494134 25 3.142451272494134 26 3.162277660168380 27 3.162277660168380 28 3.464101615137754 29 4.000000000000000 30 0.000000000000000 31 0.000000000000000 . . . Result of 15 digit computation Red digits are correct Black and green digits are incorrect π = 0 ?

Where’s the problem? is calculated as zero

with the algebraically identical expression Let’s replace with the algebraically identical expression

New iteration: results in …

2 2.828427124746190 3 3.061467458920719 4 3.121445152258053 5 3.136548490545941 6 3.140331156954739 7 3.141277250932757 8 3.141513801144145 9 3.141572940367883 10 3.141587725279961 11 3.141591421504635 12 3.141592345611077 13 3.141592576545004 14 3.141592633463248 15 3.141592654807589 16 3.141592645321215 17 3.141592607375720 18 3.141592910939673 19 3.141594125195191 20 3.141596553704820 21 3.141596553704820 22 3.141674265021758 23 3.141829681889202 24 3.142451272494134 25 3.142451272494134 26 3.162277660168380 27 3.162277660168380 28 3.464101615137754 29 4.000000000000000 30 0.000000000000000 31 0.000000000000000 2 2.828427124746190 3 3.061467458920719 4 3.121445152258053 5 3.136548490545940 6 3.140331156954753 7 3.141277250932773 8 3.141513801144301 9 3.141572940367091 10 3.141587725277160 11 3.141591421511200 12 3.141592345570118 13 3.141592576584872 14 3.141592634338563 15 3.141592648776985 16 3.141592652386591 17 3.141592653288992 18 3.141592653514593 19 3.141592653570993 20 3.141592653585093 21 3.141592653588618 22 3.141592653589499 23 3.141592653589719 24 3.141592653589774 25 3.141592653589788 26 3.141592653589792 27 3.141592653589793 28 3.141592653589793 29 3.141592653589793 30 3.141592653589793 31 3.141592653589793 π correct to all digits

The result of this computation affects The ability of the next plane you fly to stay in the air The integrity of the next bridge you cross The path of a missile that isn’t intended to strike you

Numerical Disasters Patriot system hit by SCUD missile position predicted from time and velocity the system up-time in 1/10 of a second was converted to seconds using 24bit precision (by multiplying with 1/10) 1/10 has non-terminating binary expansion after 100h, the error accumulated to 0.34s the SCUD travels 1600 m/s so it travels >500m in this time Ariane 5 a 64bit FP number containing the horizontal velocity was converted to 16bit signed integer range overflow followed

Application areas of numerical analysis Petroleum modeling Atomic energy – including weapons Weather modeling Other modeling such as aircraft and automobile Computer graphics & computer vision Simulation for prototyping Circuit design Mechanical design CAD/CAM

Algorithm areas of numerical analysis Linear Equations Nonlinear equations - single and systems Optimization Data Fitting - interpolation and approximation Integration Differential Equations - ordinary and partial

Why You Need to Learn Numerical Methods? Numerical methods are extremely powerful problem-solving tools. During your career, you may often need to use commercial computer programs (canned programs) that involve numerical methods. You need to know the basic theory of numerical methods in order to be a better user. You will often encounter problems that cannot be solved by existing canned programs; you must write your own program of numerical methods. Numerical methods are an efficient vehicle for learning to use computers. Numerical methods provide a good opportunity for you to reinforce your understanding of mathematics. You need that in your life as an engineer or a scientist.

Why use Numerical Methods? To solve problems that cannot be solved exactly

solutions! … we need grids! … Example: seismic wave propagation Seismometers Generally heterogeneous medium … we need numerical solutions! … we need grids! … And big computers … explosion

Finite Elements – Examples Virtual prototyping of engineering designs

Research Framework Motivation: Numerical investigation of cavitation phenomena occurring in turbopump inducers typical of liquid propellant rocket engines angular velocity INFLOW OUTFLOW Target : A 3D tool able to simulate complex cavitating flows in realistic geometries

Mathematical Modeling Mathematical modeling seeks to gain an understanding of science through the use of mathematical models on computers. Mathematical modeling involves teamwork

Mathematical Modeling Complements, but does not replace, theory and experimentation in scientific research. Experiment Computation Theory Is often used in place of experiments when experiments are too large, too expensive, too dangerous, or too time consuming.

Has emerged as a powerful, indispensable tool for studying a variety of problems in scientific research, product and process development, and manufacturing. Seismology Climate modeling Economics Environment Material research Drug design Manufacturing Medicine Biology Analyze - Predict

Example: Industry First jetliner to be digitally designed, "pre-assembled" on computer, eliminating need for costly, full-scale mockup. Computational modeling improved the quality of work and reduced changes, errors, and rework.

The Boeing 777 is the first jetliner to be 100 percent digitally designed using three-dimensional solids technology. Throughout the design process, the airplane was "preassembled" on the computer, eliminating the need for a costly, full-scale mock-up. The 230 000 kg plane is the biggest twin-engine aircraft ever to fly-it can carry 375 passengers 7400 km-and from its first service flight in June 1995, has been certified for extended-range twin-engine operations. Boeing invested more than $4 billion (and insiders say much more) in CAD infrastructure for the design of the Boeing 777 and reaped huge benefits from design automation. The more than 3 million parts were represented in an integrated database that allowed designers to do a complete 3D virtual mock-up of the vehicle.

Boeing based its CAD system on CATIA (short for Computer-aided Three-dimensional Interactive Application) and ELFINI (Finite Element Analysis System), both developed by Dassault Systemes of France (Dassault systems acquired ABAQUS in 2005 and ABAQUS+CATIA is known as SIMULIA) and licensed in the United States through IBM. Designers also used EPIC (Electronic Preassembly Integration on CATIA) and other digital preassembly applications developed by Boeing. Much of the same technology was used on the B-2 program. To design the 777, Boeing organized its workers into 238 cross-functional "design build teams" responsible for specific products. The teams used 2200 terminals and the computer-aided three dimensional interactive application (CATIA) system to produce a "paperless" design that allowed engineers to simulate assembly of the 777. The system worked so well that only a nose mockup (to check critical wiring) was built before assembly of the first flight vehicle which was only 0.03 mm out of alignment when the port wing was attached. Boeing also included customers and operators, down to line mechanics, to help tell them how to design the plane.

Engine Thermal Analysis Question What is the temperature distribution in the engine block? Solve Poisson Partial Differential Equation. Recent Developments Fast Integral Equation Solvers, Monte-Carlo Methods

Mathematical Modeling Process

Governed by differential equations Engineering design Physical Problem Question regarding the problem ...how large are the deformations? ...how much is the heat transfer? Mathematical model Governed by differential equations Assumptions regarding Geometry Kinematics Material law Loading Boundary conditions Etc.

Engineering design Example: A bracket Physical problem Questions: We consider here a simple example of a bracket supporting a vertical load. We need to choose a mathematical model. The choice of this model clearly depends on what phenomena are to be predicted and on the geometry, material properties, loading and support conditions of the bracket. We notice that The bracket has been fastened to a “very thick steel column”. The term “very thick” is relative to the thickness “t” and height “h” of the bracket. We translate this statement into the assumption that that the bracket is fastened to a (practically) rigid column. We also assume that the load is applied very slowly. The condition of time “very slowly” is relative to the largest natural period of the bracket: i.e., the time span over which the load W is increased from 0 to its full value is much longer than the fundamental period of the bracket. We translate this statement into meaning that we require static analysis (as opposed to a dynamic analysis). Now we are ready to develop a mathematical model of the bracket-depending on the phenomena to be predicted. Let us assume that we want to answer the following questions: What is the bending moment at section AA? What is the deflection at the pin? Questions: What is the bending moment at section AA? What is the deflection at the pin? Finite Element Procedures, K J Bathe

Engineering design Example: A bracket Mathematical model 1: beam Moment at section AA First we develop a beam mathematical model including shear deformations. We have assumed linear elastic infinitesimal deformation conditions. Hence the load must not be so large as to cause yielding of the material and/or large displacements. Let us now ask whether the mathematical model is reliable and effective. Deflection at load How reliable is this model? How effective is this model?

Engineering design Example: A bracket Mathematical model 2: plane stress Difficult to solve by hand! To measure the reliability and effectiveness of the simple beam model we consider a slightly more complicated mathematical model. This is a linear elastic two-dimensional plane-stress model. This mathematical model represents the geometry of the bracket more accurately than the beam model and assumes a two-dimensional stress situation. However, note that we have not considered the actual bolt fastening and contact conditions between the steel column and the bracket, nor have we modeled the pin carrying the load onto the bracket.

Engineering design ..General scenario.. Physical Problem Mathematical model Governed by differential equations The plane stress mathematical model is too complicated to solve by hand. We therefore solve it using a numerical technique- the finite element method. Numerical model e.g., finite element model

Engineering design PREPROCESSING 1. Create a geometric model ..General scenario.. Engineering design Finite element analysis PREPROCESSING 1. Create a geometric model 2. Develop the finite element model How do we perform FEM modeling? We first generate a solid model of the bracket using a commercial package such as Solidworks/ ProE. The next step is to develop the “finite element” model. Solid model Finite element model

Engineering design FEM analysis scheme ..General scenario.. Engineering design Finite element analysis FEM analysis scheme Step 1: Divide the problem domain into non overlapping regions (“elements”) connected to each other through special points (“nodes”) Element Node The FEM analysis scheme can be broken down into 3 steps (we will use these 3 steps throughout the course). Step 1: We divide the solid model up into non-overlapping regions called “elements”. The elements are connected to each other through special points called “nodes”. The solution, in this case the displacement field, is approximated using piece-wise polynomials in each element. This process is called “mesh generation”. This way we discretize a continuous problem, having infinite degrees of freedom to a discrete problem having a finite number of degrees of freedom. Finite element model

Engineering design FEM analysis scheme ..General scenario.. Engineering design Finite element analysis FEM analysis scheme Step 2: Describe the behavior of each element Step 3: Describe the behavior of the entire body by putting together the behavior of each of the elements (this is a process known as “assembly”) Step 2: We then describe the behavior of each element in terms of its nodes Step 3: We then describe the behavior of the entire structure by putting together the behavior of of each of these elements. In this process we are able to compute the displacements at the nodal points. Are we done?

Engineering design POSTPROCESSING Compute moment at section AA ..General scenario.. Engineering design Finite element analysis POSTPROCESSING Compute moment at section AA To compute the moment, we have to do a bit of extra work and this is known as “postprocessing”.

Engineering design Preprocessing Analysis Postprocessing ..General scenario.. Engineering design Finite element analysis Preprocessing Step 1 Analysis Step 2 Step 3 Remember that a complete FEM analysis consists of Preprocessing Analysis Postprocessing The Analysis phase alone has 3 steps (which we will be most interested in the rest of this course) Postprocessing

Engineering design Example: A bracket Mathematical model 2: plane stress FEM solution to mathematical model 2 (plane stress) Moment at section AA Deflection at load Conclusion: With respect to the questions we posed, the beam model is reliable if the required bending moment is to be predicted within 1% and the deflection is to be predicted within 20%. The beam model is also highly effective since it can be solved easily (by hand). The beam mathematical model completely neglects the stress increase due to the fillets. Hence a plane stress solution including the fillets is necessary. What if we asked: what is the maximum stress in the bracket? would the beam model be of any use?

Example: A bracket Engineering design Summary The selection of the mathematical model depends on the response to be predicted. The most effective mathematical model is the one that delivers the answers to the questions in reliable manner with least effort. The numerical solution is only as accurate as the mathematical model.

Modeling a physical problem Example: A bracket ...General scenario Modeling a physical problem Physical Problem Change physical problem Mathematical Model Improve mathematical model Numerical model Does answer make sense? No! Refine analysis YES! Design improvements Structural optimization Happy 

Verification and validation Example: A bracket Verification and validation Modeling a physical problem Physical Problem Validation Mathematical Model Verification Numerical model

Critical assessment of the FEM Reliability: For a well-posed mathematical problem the numerical technique should always, for a reasonable discretization, give a reasonable solution which must converge to the accurate solution as the discretization is refined. e.g., use of reduced integration in FEM results in an unreliable analysis procedure. Robustness: The performance of the numerical method should not be unduly sensitive to the material data, the boundary conditions, and the loading conditions used. e.g., displacement based formulation for incompressible problems in elasticity Efficiency:

Design of Yacht Appendages Validation and calibration of the numerical model by comparisons with experimental results (wind tunnel and towing tank tests) Comparative analyses of design configurations: Different geometries Different sailing attitudes Different wind and boat speeds

Sail analysis

Flow around spinnaker and mainsail velocity and flow separation streamlines

Computational Cost 15.000.000 elements, 135.000.000 unknowns Which cost to achieve the desired accuracy ? CPU Time 24 hours 30 gigabytes of RAM 32 processors RAM Memory Number of Processors (450 AMD Opteron processors, 900 Gb distributed RAM, Network)

Mathematical Background Roots of equations: concerns with finding the value of a variable that satisfies a single nonlinear equation – especial valuable in engineering design where it is often impossible to explicitly solve design equations of parameters. Systems of linear equations: a set of values is sought that simultaneously satisfies a set of linear algebraic equations. They arise in all disciplines of engineering, e.g., structure, electric circuits, fluid networks; also in curve fitting and differential equations. Optimization: determine a value or values of an independent variable that correspond to a “best” or optimal value of a function. It occurs routinely in engineering contexts. Curve fitting: to fit curves to data points. Two types: regression and interpolation. Experimental results are often of the first type. Integration: determination of the area or volume under a curve or a surface. It has many applications in engineering practice, such as …

Ordinary differential equations: very important in engineering practice, because many physical laws are couched in terms of the rate of change of a quantity rather than the magnitude of the quantity itself, such as … Partial differential equations: used to characterize engineering systems where the behavior of a physical quantity is couched in terms of the rate of change with respect to two or more independent variables. Examples: steady-state distribution of temperature of a heated plate (two spatial dimensions) or the time-variable temperature of a heated rod (time and one spatial dimension).

Παράδειγμα: Επίλυση Γραμμικού Συστήματος Εξισώσεων (1)

Παράδειγμα: Επίλυση Γραμμικού Συστήματος Εξισώσεων (2) Μέθοδοι: Τύπος του Cramer () x=A-1b (2) Άμεσες μέθοδοι: Gauss () Απλή Με μερική οδήγηση Με ολική οδήγηση Επαναληπτικές μέθοδοι Jacobi Gauss-Seidel κ.ά.

Παράδειγμα: Επίλυση Γραμμικού Συστήματος Εξισώσεων (3) Κριτήρια επιλογής μεθόδου: Γενικά: Πολυπλοκότητα (υπολογιστική, μνήμης, υλοποίησης) Ταχύτητα σύγκλισης (για επαναληπτικές μεθόδους) Ακρίβεια Ανθεκτικότητα σε αριθμητικά σφάλματα (αναπαράστασης δεδομένων και πράξεων) Εξαρτώμενα από το συγκεκριμένο πρόβλημα: Κατάσταση του πίνακα του συστήματος Πλήθος μηδενικών στοιχείων του πίνακα Διαστάσεις του προβλήματος Τυχόν συμμετρίες στον πίνακα, κ.ά.

Cramer’s Rule is Not Practical

Παράδειγμα: Επίλυση Γραμμικού Συστήματος Εξισώσεων (4) Π.χ. Επίλυση του συστήματος με 4 σημαντικά ψηφία Ακριβής λύση (με 4 ψηφία): Απλή μέθοδος Gauss: Μέθοδος Gauss με μερική οδήγηση: