Numerical Analysis

Appunti Numerical Analysis

The Art of Modelling

Modelling

• Physical Model

  • a full reproduction of a thing (measure)
  • reproduction in steel
  • is a down version of reality
  • manpower difference: measure the answer
  • answer depends on materials

• Conceptual Model

  • active before the mathematical model and after simulation
  • no numbers, no equations, only descriptive

• Mathematical Model

  • with monitored numbers
  • to obtain security criteria

• Animal Model

  • used in medicines fields for experimental study
  • the results can't always apply to human case

• Data Driven Models

  • Not needing having equations, a set of some equations
  • Attractive for to treat some data and obtain a trend (indication)

• Mechanism Driven Models

  • Logical way to describe a structure
  • starting from beam with boundary conditions to:

Chapter 3: Elements of Computational Methods

Elementary Foundations

• Physical Problem

  Computation of a deformed structure of a biomimic

  (start from a real problem)

For a solution sketch is needed x_pa

x_pa = {temperature, pressure, deformation}

N.B.: no equation at all

Definitions

• K ⊂ R
• P ≥ set of input data
• V ≥ set of solutions

Vector Spaces

V₁ ∪ V₂ ... up to as a VECTOR a candid element of W

In context of interest, we need to adapt contributions

Scalar Multiplication:

∀α∈K

• (α,w)→α⋅w∈W

Vector Addition:

∀u,v∈W

• (u,v)→u+w∈W

Example:

Given a vector W: R²

We are able to sum the two elements

We can express w as a linear combination of (e₁, e₂)

Properties:

1. u+v = v+u ∀u,v∈W

COMMUTATIVITY

2. u + (v+w) = (u+v) + w ∀u,v,w∈W

ASSOCIATIVITY

3. ∃θ∈W ∀u∈W, u+θ = θ

IDENTITY ELEMENT OF VECTOR ADDITION

4. ∀u∈W ∃w∈W , u+w = θ

INVERSE ELEMENT OF VECTOR ADDITION

5. w∈W α,β∈K α(β⋅w) = (αβ)⋅w

ASSOCIATIVITY OF SCALAR MULTIPLICATION

6. 1⋅w = w ∀w∈W

IDENTITY ELEMENT OF SCALAR MULTIPLICATION

7. α(u+v) = αu+αv ∀u,v∈W ∀α∈K

DISTRIBUTIVITY OF SCALAR MULTIPLICATION WITH RESPECT TO VECTOR ADDITION

8. (α+β)w = αw+βw ∀α,β∈K ∀w∈W

DISTRIBUTIVITY OF SCALAR MULTIPLICATION WITH RESPECT TO FIELD ADDITION

Example:

W:R³ : x∈R³

x = (x₁, x₂, x₃) ↔ x ∈R³

Check other properties:

1. 2y+4y=6yx=...✓
2. ...
3. x+0=x

Example: evaluation of the derivative of a function:

y ∈ C⁴ [t₀, t₀ + ]

y = y(t)

x = x(t) = y'(t)

Aim: compute the derivative at a certain point t₀.

x = x(t₀) = y'(t₀)

We can use Taylor's expansion to approximate:

y(t₀+ h) = y(t₀) + y'(t₀) h +

with point ∈ (t₀, t₀+h) but we don't know where

x = y(t₀ + h) - y(t₀) -

This formula doesn’t give us the exact evaluation of ξ because of ξ.

→ small problem has to be in the form

F(x)=0;

x - [_{y(t₀ + ) - y(t₀) -} ]| ^{= 0}

^h

We know that it admits only one solution, but we are not able to compute it.

We can assume y''(ξ) = 0

X_a=[_{y(t₀+)-y(t₀)}]=0

X_a = [_{y(t₀+)-y(t₀)}]

[ _h ]

it's different from x, this is an APPROXIMATION

X_a = [_{y(t₀+)-y(t₀)}]

INCREMENT RATIO

→ Having both continuous and discrete formulations associated with the same physical system, we naturally lead to inquire about the quality with which x represents (approximates) x.

We need to introduce the DISCRETIZATION ERROR

E = x - X

Note that E cannot be computed because x is, in general, not available.

This doesn't constitute a serious difficulty, because we are actually interested in the convergence of the approximate solution X to the solution x.

An aspect is that:

_{lim E = 0 CONVERGENCE} → error becomes smaller and smaller when we discretize the approximated model

* Same problem, but use INCREMENT m :

C₀ + C₁ x + C_m-1 x ^m-1 (no real n points) (PARABOLA) → (real roots of x-interval = VERTICES) (mid points of each interval (xi) = CENTERS)

M = 2

M = 2 (LINEAR) M = 2 (PARABOLA)

2 VERTEX

2 VERTEX

3 NODI 3 NODI

M = 3: in and interval we will have a CUBIC FUNCTION, • gives direct and indirect roots and 2 increments

Theorem 2: Given I