Numerical Analysis

THE ART OF MODELLING

Modelling

• Physical Model
  • a true reproduction of a thing (prototype)
  • reproduction in scale
  • is a dowel version of reality
  • presence of ‘differential’ in materials, size etc.
  • Numerous expensive uses in materials…
• Animal Model
  • Used in medicine field for pharmaceutical study
  • The results can’t always be apply to humanca
• Conceptual Model
  • farther before mathematical models and after mathematical models
  • descriptive (no equations, only description)
• Mathematical Model
  • with motivated number
  • to obtain security criteria
• Data Driven Models
  • Not needing deriving equations, a sort of exact equations
  • Attempt to treat discrete data and obtain a trend (indications)
• Mechanism Driven Models
  • Require way to describe a structure
  • Starting from:
    • beam neutrality formul
  • to:
    • given by Lowe-Hurrey equation

CHAPTER 3: ELEMENTS OF COMPUTATIONAL METHODS

ELEMENTARY FOUNDATIONS

Physical Problem

Computation of a deform structure of a lamina.

(Starting from a real problem)

N.B.: no equations at all

Defining:

• k ∈ ℝ
• P = set of problems
• V = set of solutions

In respect logic…

… Pass a solution which is called:

Xp = {temperature, pressure, deformation}

The Art of Modelling

Physical Model

• Is a reproduction of a thing (reduction)
• Reproduces in scale
• Is a down version of reality
• Presence of differences in materials they are made of
• No proof decision on materials

Conceptual Model

• Stays before mathematical model and after experimental description
• No equation, only description

Mathematical Model

• With mentioned number
• To obtain security criteria

Animal Model

• Used in medicine field for pharmaceutical study
• The results can't always be apply to humans

Data Driven Models

• No need of any equation, a sort of expert systems
• Allows to treat data only and obtain a trend (indicator)

Mechanism Driven Models

• Require way to describe a situation
• Starts from: beam with boundary condition to:
• Given by linear-elastic equation

Chapter 3: Elements of Computational Methods

Elementary Foundations

Physical Problem: Computation of the algebraic structure of a formula

(Start from a real problem)

• Has a solution which is exact: X
• X can contain many other variables which are pertinent to the description of the physical problem
• X = {temperature, pressure, deformation}

N.B.: no equation at all

Definimo:

• W ∈ R
• P: set of parameters
• V: set of vectors
• Vector Spaces
• W ∉ V, W is a vector calculation with specific constraints

In context of def..., truncation errors

In context of approx., disabled derivatives...

*Scalar Multiplication:

α∈Kω∈W(α,ω)⟶α⋅ω∈W

*Vector Addition:

ρ, μ ∈ W(ρ+μ) ∈ W

Ex: give context W=ℒ²

We are able to sum the two elements.We can express ω as a linear combinationof (ℓ₂,ℓ₁)

Properties:

1. ω+ν=ν+ω    ∀ω,ν∈W
2. ω+(ν+μ)=(ω+ν)+μ   ∀ω,ν,μ∈W
3. ∃0_W∈W     ρ+0=ρ
4. ∀ω∈W  ∃−ω∈W     ω+−ω=0
5. ω∈W  α,β∈K     α(βω)=(αβ)ω
6. 1⋅ω=ω    ∀ω∈W
7. α(ω+ν)=αω+αν    ∀ω,ν∈W
8. (α+β)ω=αω+&