Metode Komputasi Numerik

L #1

Amil Ahmad Ilham

http://www.unhas.ac.id/amil/S1TIF/MKN2020/

Administrasi Kuliah

• Bahan: • Download di http://www.unhas.ac.id/amil/S1TIF/mkn2020/

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Administrasi Kuliah

Penilaian• Quiz, Final, Tugas, Presentasi, dll.

Perbaikan Nilai:• Tidak ada (waktu) perbaikan nilai.

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• Ada pemberitahuan pada saat ujian atau sebelum ujian (Do not hesitate to let me know!)

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Administrasi Kuliah

Rules:• No mark (zero) for late assignments without prior permission or

urgent reasons.• No mark (zero) for cheated assignments or exams.• No talks via mobile phone during class.• Silent mode is OK (You can leave class if you have an urgent call.)• No FB, Twitter, SMS, etc, during class.• No “kaos oblong”.• No smoking! • Snack/candy is OK during class.

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System modelingPemodelan Sistem

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Systems Modeling(How to build credible Computerized Model of a System)

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Urgensi atau Pentingnya Pemodelan Sistem

• Dalam perancangan sistem, sistem yang akan dibangun belum ada(baru ada secara “hipotetis”). Untuk membuat prediksi, harus dibuatmodel sistem tersebut.

• Seandainya pun ada sistem yang sebenarnya, sering sangat mahal(biaya dan waktu) atau sangat berisiko tinggi bahkan berbahaya untukber-eksperimen dengan sistem yang sesungguhnya.

• Untuk suatu studi dalam bidang tertentu, tidak perlu keseluruhandetail sistem dipelajari, perlu penyederhanaan dengan model.

• Perlu meng-identifikasi ENTITAS, ATRIBUT dan AKTIVITAS yang relevandalam sistem

• Pemodelan = perumusan masalah, langkah awal dalam engineering.7

Types of systems modeling

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Analytical vs. numerical solution

• Analytical solution• provides exact solution

• Numerical solution• Approximates exact solution

• Why do we need numerical solution?• There are many mathematical models that cannot be solved

exactly.

• In many of these cases, the only alternative is to develop a numerical solution that approximates the exact solution.

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Error Definitions

• True error:

• True percent relative error:

• Approximate percent relative error:

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Problem• A parachutist of mass 68.1 kg jumps out of

a stationary hot air balloon. The drag coefficient is equal to 12.5 kg/s. Compute velocity when t = 10 seconds! The parachutist is initially at rest (v = 0 at t = 0)

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Problem Analysis• Newton Formula: F = ma

F = net force (N) or (kg m/s2)

m = mass of object (kg) a = its acceleration (m/s2)

F = FD + FU

• FD is the downward force due to gravity -> FD = mg

g = the acceleration due to gravity (9.8 m/s2)

• FU is the upward force due to air resistance -> FU = -cv c = the drag coefficient (kg/s)

v = velocity (m/s)

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Problem Analysis Newton Formula: F = ma

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Problem• A parachutist of mass 68.1 kg jumps out of a

stationary hot air balloon. The drag coefficient is equal to 12.5 kg/s. Compute velocity when t = 10 seconds! The parachutist is initially at rest (v=0 at t=0)

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We need velocity when t = 10 seconds!V(10) = ?

Solution(?)Analytic:• Find v(t)• Compute v(10)

Numeric:• Find v(t)• Compute v(10)

Analytical Solution

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Analytical Solution

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Analytical Solution

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• A parachutist of mass 68.1 kg jumps out of a stationary hot air balloon. The drag coefficient is equal to 12.5 kg/s. Compute velocity when t = 10 seconds! The parachutist is initially at rest (v=0 at t=0)

V(10) = ?? V(10) = 44.87 m/s

Exercise #1:Compute velocity (v) when t = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 seconds

Numerical Solution(Euler Method)

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• A parachutist of mass 68.1 kg jumps out of a stationary hot air balloon. The drag coefficient is equal to 12.5 kg/s. Compute velocity when t = 10 seconds! The parachutist is initially at rest (v=0 at t=0)

Numerical Solution(Euler Method)

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• A parachutist of mass 68.1 kg jumps out of a stationary hot air balloon. The drag coefficient is equal to 12.5 kg/s. Compute velocity when t = 10 seconds! The parachutist is initially at rest (v=0 at t=0)

For example Δt = 2 secondst = 0 -> v(0) = 0t = 2 -> v(2) = v(0) + [9.8 – (12.5/68.1)v(0)]2

= 0 + [9.8 – (12.5/68.1)0]2= 19.6 m/s

t = 4 -> v(4) = v(2) + [9.8 – (12.5/68.1)v(2)]2= 19.6 + [9.8 – (12.5/68.1)19.6]2= 32.00

t = 6 -> v(6) = v(4) + [9.8 – (12.5/68.1)v(4)]2= 32.00 + [9.8 – (12.5/68.1)32.00]2= 39.85

t = 8 -> v(8) = v(6) + [9.8 – (12.5/68.1)v(6)]2= 39.85 + [9.8 – (12.5/68.1)39.85]2= 44.82

t = 10 -> v(10) = v(8) + [9.8 – (12.5/68.1)v(8)]2= 44.82 + [9.8 – (12.5/68.1)44.82)]2= 47.97 m/s

Numerical Solution(Euler Method)

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• A parachutist of mass 68.1 kg jumps out of a stationary hot air balloon. The drag coefficient is equal to 12.5 kg/s. Compute velocity when t = 10 seconds! The parachutist is initially at rest (v=0 at t=0)

Exercises #2:1. Using Δt = 1 second, compute velocity when t

= 15 seconds. -> v(15) = ?2. Compute True percent relative error for t = 1,

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 seconds.