User:Muhamad.akbar41
Nama saya Muhamad Akbar Fuad, nama panggilan Akbar. Saya adalah seorang mantan kepala produksi di sebuah perusahaan manufaktur elektronika di Kota Bogor. Alhamdulillah, saat ini sudah menemukan pekerjaan baru sebagai guru matematika sma. Bidang peminatan yang diambil adalah Sistem Otomasi dan Manufaktur, dengan topik penelitian Additive Manufacturing Systems. Saya memiliki interest akan suatu sistem produksi massal. Didorong dengan pengalaman
1. Menjalankan Mesin-Mesin Produksi 2. Manajemen Inventory Perusahaan
sebagai seorang kepala produksi membuat pemikiran terbaru untuk mengoptimalkan kehandalan Produksi dan Manajerial Keuangan Perusahaan. Pengalaman bekerja selama 1,5 tahun ,ternyata, membuka realitas hidup bahwa PHK itu TIDAK LEPAS dari resiko-resiko yang mungkin dialami oleh masing-masing pegawai perusahaan. Oleh karenanya, tujuan utama mengapa saya mempelajari materi Komputasi Teknik adalah untuk
1. mengoptimalkan ketahanan produksi unit usaha 2. mencegah angka PHK yang lebih meluas lagi
ditengah ketidakpastikan ekonomi global dan kondisi perpolitikan nasional.
Diharapkan, ilmu-ilmu tentang komputasi teknik yang akan saya pelajari mampu membawa angin segar untuk perusahaan-perusahaan lainnya dalam mempertahankan usahanya dan meminimalisir angka PHK ditengah ketidakpastian ekonomi saat ini. Cukuplah hanya saya yang di rumahkan, saya hanya tidak ingin melihat angka PHK kembali meningkat.
Contents
BELAJAR KOMPUTASI TEKNIK DENGAN CHAT GPT
DAI5
Conscious Thinking (Think about Reality and How to Solve it all)
I(2) Intention I(3) Initial Thinking I(4) Idealization I(5) Instruction Set
JILID 1 : 1-Dimensional Partial Differential Equation (PDE)
Question 1: Did you know about DAI5?
The DAI5 Framework is a structured method designed to support systematic problem-solving, particularly in engineering and technical analysis. Developed by Dr. Ahmad Indra Siswantara, the DAI5 framework consists of five main steps: defining the problem, analyzing causes, ideating solutions, implementing, and evaluating outcomes. This approach is often applied in complex scenarios like numerical simulations, finite element analysis, and stress testing within engineering fields. By using DAI5, users aim to ensure each phase is rigorously analyzed to optimize solutions and outcomes, as demonstrated in various technical case studies on topics such as fluid dynamics and pipe stress analysis.
Question 2: Can you derive finite element equations from 1D PDE governing equations using weighted residual in a very simple way along with an example?
Certainly! Deriving the finite element equations from a 1D partial differential equation (PDE) using the weighted residual method is a great way to understand the basics of the finite element method (FEM). Let’s go through it step-by-step with a simple example.
Problem Setup: 1D PDE Example Consider the 1D steady-state heat conduction equation with no internal heat generation:
where:
• u(x) is the temperature field, • k is the thermal conductivity (assumed constant for simplicity), • L is the length of the domain.
Suppose we have the boundary conditions:
1. u(0)=u0 (a Dirichlet boundary condition), 2. du/dx = q at x = L (a Neumann boundary condition).
Step 1: Formulate the Weak Form (Weighted Residual Form) To derive the finite element equations, we start with the weak form of the governing equation. This involves:
1. Multiplying the differential equation by a test function v(x), 2. Integrating over the domain, 3. Applying PARTIAL INTEGRATION to reduce the order of derivatives on u.
The weighted residual form is:
Using integration by parts to move the derivative from u to v, we get:
The boundary term,[v(x).k.du/dx]L0 , will incorporate boundary conditions:
• At x=0, v(0)=0 due to the Dirichlet boundary condition (since u(0)=u0 is known). • At x=L, we have the Neumann boundary condition du/dx=q.
This leaves us with:
Step 2: Discretize the Domain and Approximate u(x) Now, we approximate u(x) using finite element shape functions. Suppose we divide the domain into two linear elements, and approximate u(x) as:
u(x) = u1.N1 (x) + u2.N2 (x)
where:
• N1(x) and N2(x) are linear shape functions, • u1 and u2 are the nodal values of u at nodes 1 and 2.
For a linear element over a small interval [x1,x2], the shape functions can be written as:
Step 3: Apply the Weighted Residual Method to Each Element For each element, we substitute the approximated u(x) and v(x) (chosen as the shape functions) into the weak form.
1. Evaluate the integrals: Substitute
2. Compute the stiffness matrix entries for each element:
For a linear element, this integral simplifies to a constant and can be calculated based on the properties of the shape functions.
3. Assemble the Global System of Equations: After computing K for each element, assemble them into a global stiffness matrix.
4. Apply Boundary Conditions to solve for the unknown nodal values.
Summary
This procedure gives you a system of linear equations:
where:
• K is the global stiffness matrix,
• U is the vector of unknown nodal values,
• F is the force vector, which includes contributions from the Neumann boundary condition.
JILID 2 : Pembuatan Kode Python
Question 3: Can you provide with algorithm, flow chart and python code?
Certainly! Here’s how you can implement the finite element method (FEM) for a 1D heat conduction problem step-by-step, including the algorithm, flowchart, and a sample Python code.
- Algorithm for 1D FEM Heat Conduction Problem
The basic steps are as follows:
1. **Input Parameters**: Set up the problem's physical and numerical parameters, such as length of the domain \( L \), number of elements \( N \), conductivity \( k \), source term \( f(x) \), and boundary conditions.
2. **Mesh Generation**: Divide the domain into \( N \) elements. Each element will have two nodes.
3. **Define Shape Functions**: For a linear 1D element, define shape functions \( N_1(x) \) and \( N_2(x) \) and their derivatives.
4. **Element Stiffness Matrix and Force Vector**:
- For each element, compute the element stiffness matrix \( \mathbf{K}^{(e)} \) and force vector \( \mathbf{F}^{(e)} \) using integration.
- Assemble the element matrices and vectors into the global stiffness matrix \( \mathbf{K} \) and global force vector \( \mathbf{F} \).
5. **Apply Boundary Conditions**:
- Modify \( \mathbf{K} \) and \( \mathbf{F} \) to incorporate boundary conditions.
6. **Solve the System of Equations**:
- Solve \( \mathbf{K} \mathbf{U} = \mathbf{F} \) for the nodal temperatures \( \mathbf{U} \).
7. **Post-process**:
- Compute temperature at each node and optionally evaluate gradients or flux.
- Flowchart for 1D FEM Heat Conduction Problem
```plaintext
+------------------------+
| Start |
+------------------------+
|
v
+------------------------+
| Input parameters |
| (L, N, k, f(x), etc.) |
+------------------------+
|
v
+------------------------+
| Mesh generation |
| Divide domain into N |
| elements, calculate |
| node positions |
+------------------------+
|
v
+------------------------+
| Initialize global |
| stiffness matrix K |
| and force vector F |
+------------------------+
|
v
+------------------------+
| For each element e: |
| - Compute element |
| stiffness matrix |
| - Compute element |
| force vector |
| - Assemble into |
| global K and F |
+------------------------+
|
v
+------------------------+
| Apply boundary |
| conditions |
+------------------------+
|
v
+------------------------+
| Solve KU = F for U |
+------------------------+
|
v
+------------------------+
| Post-process results |
| (e.g., plot temperature|
| distribution) |
+------------------------+
|
v
+------------------------+
| End |
+------------------------+
```
- Python Code for 1D FEM Heat Conduction Problem
Here’s a simple Python script that implements the FEM for a 1D heat conduction problem.
```python import numpy as np import matplotlib.pyplot as plt
Parameters
L = 10.0 # Length of the rod N = 5 # Number of elements k = 1.0 # Thermal conductivity f = 2.0 # Heat source term u_0 = 0.0 # Dirichlet BC at x=0 q_L = -5.0 # Neumann BC at x=L
Mesh generation
nodes = np.linspace(0, L, N + 1) # Node positions h = L / N # Element length
Initialize global stiffness matrix and force vector
K = np.zeros((N + 1, N + 1)) F = np.zeros(N + 1)
Assembly process
for e in range(N):
# Element stiffness matrix for element e
Ke = (k / h) * np.array([[1, -1], [-1, 1]])
# Element force vector for element e (assuming f(x) constant)
Fe = (f * h / 2) * np.array([1, 1])
# Assemble into global matrix
K[e:e+2, e:e+2] += Ke
F[e:e+2] += Fe
Apply boundary conditions Dirichlet BC at x=0
K[0, :] = 0 K[0, 0] = 1 F[0] = u_0
Neumann BC at x=L
F[-1] += q_L
Solve for nodal temperatures
U = np.linalg.solve(K, F)
Plot results
plt.plot(nodes, U, '-o', label="Temperature Distribution")
plt.xlabel("Position x")
plt.ylabel("Temperature u(x)")
plt.title("1D FEM Solution of Heat Conduction")
plt.legend()
plt.grid()
plt.show()
```
- Explanation of the Code
1. **Parameters**: We define the length \( L \), number of elements \( N \), thermal conductivity \( k \), heat source \( f \), and boundary conditions.
2. **Mesh Generation**: We create nodes evenly spaced along \( [0, L] \).
3. **Global Matrix and Vector Initialization**: We initialize the global stiffness matrix \( K \) and force vector \( F \).
4. **Element Stiffness Matrix and Force Vector Calculation**: For each element:
- Compute the element stiffness matrix \( \mathbf{K}^{(e)} \) using the formula for a 1D linear element.
- Compute the element force vector \( \mathbf{F}^{(e)} \) for a constant source term.
5. **Assembly**: The element matrices and vectors are added to the global matrix and vector.
6. **Boundary Conditions**:
- Apply the Dirichlet boundary condition by modifying the first row. - Apply the Neumann condition by adding to the last entry in \( F \).
7. **Solve the System**: Use `np.linalg.solve` to get the temperature distribution at the nodes.
8. **Plot**: Display the temperature distribution along the rod.
NOTE Phyton code above should be re-write before applicate on Phyton.
JILID 3 : Penjelasan Kode Python
Question 4: What is np? Can you elaborate it?
`np` is a common alias for the **NumPy** library in Python, a powerful package for numerical computing. NumPy provides support for large, multi-dimensional arrays and matrices, along with a vast library of mathematical functions to operate on these arrays efficiently.
In the provided code, `np` is used as a shorthand for `numpy`, which makes it easier and quicker to refer to. Here’s a breakdown of what each `np` function or attribute is doing:
- Explanation of `np` Functions and Attributes Used in the Code
1. **`np.zeros((num_nodes, num_nodes))`**: - This function creates a **2D array (matrix) of zeros** with shape `(num_nodes, num_nodes)`. In our code, it initializes the global stiffness matrix `K` to zero. - Example: `np.zeros((3, 3))` produces a 3x3 matrix filled with zeros.
2. **`np.zeros(num_nodes)`**: - Similar to the above, but this one creates a **1D array of zeros** with length `num_nodes`. In our code, it initializes the global load vector `F` to zero. - Example: `np.zeros(5)` produces `[0., 0., 0., 0., 0.]`.
3. **`np.array([...])`**: - This function creates an **array** from a Python list. Arrays are the core data structure in NumPy and are highly optimized for numerical operations. - In our code, `np.array([[1, -1], [-1, 1]])` defines the local stiffness matrix \( K^e \) as a 2x2 array, and `np.array([1, 1])` defines the local load vector \( F^e \).
4. **`np.linalg.solve(K, F)`**: - This function **solves a system of linear equations**. Given a matrix `K` and a vector `F`, `np.linalg.solve` finds the vector `U` such that \( K \cdot U = F \). - It’s highly efficient for solving linear systems and is more stable and faster than computing the inverse matrix. - Example: `np.linalg.solve([[3, 1], [1, 2]], [9, 8])` gives `[2. 3.]`.
5. **`np.linspace(0, L, num_nodes)`**: - This function **creates an array of evenly spaced values** between `0` and `L`, with `num_nodes` elements. - It’s used in our code to generate the `x` coordinates for plotting the FEM solution. - Example: `np.linspace(0, 1, 5)` produces `[0. , 0.25, 0.5 , 0.75, 1. ]`.
Why Use NumPy?
NumPy is widely used because it enables fast array processing, which is especially useful in numerical computing. For large systems (like in FEM), it provides better performance than pure Python lists due to optimized C and Fortran code under the hood.
To use `np`, you should install NumPy (if you haven’t already) and import it at the start of your script:
```python import numpy as np ```
Then, you can use `np` as a shorthand for calling any NumPy function or object.
Question 5: Can you make it much simpler?
Absolutely! Let's break down `np` (which stands for the **NumPy** library) and each function in an even simpler way. Here’s a very basic rundown:
---
- What is NumPy (`np`)?
NumPy is a popular Python library for working with numbers, especially with large lists or grids of numbers, called **arrays**. It makes calculations faster and easier compared to using regular Python lists.
In our example, we use `np` as a nickname for **NumPy**, which is common practice. To use NumPy, we first import it like this:
```python import numpy as np ```
---
- Functions Used in the Code
1. **`np.zeros()`**: - Creates a list (array) filled with zeros. - We use `np.zeros((num_nodes, num_nodes))` to create a **matrix** (a grid of numbers) of zeros with `num_nodes` rows and columns. - We use `np.zeros(num_nodes)` to create a **1D array** of zeros with `num_nodes` elements. - **Example**: `np.zeros(3)` creates `[0. 0. 0.]`.
2. **`np.array()`**: - Converts a regular Python list into a NumPy array. - We use this to create small matrices or vectors (arrays) that we need for each part of our problem. - **Example**: `np.array([1, 2, 3])` becomes a NumPy array `[1 2 3]`.
3. **`np.linalg.solve()`**: - Solves math problems where we have an equation like `Ax = B`. - We use `np.linalg.solve(K, F)` to find the unknowns in our equation \( K \cdot U = F \). - **Example**: `np.linalg.solve([[3, 1], [1, 2]], [9, 8])` gives `[2, 3]` (meaning \( x = 2 \) and \( y = 3 \)).
4. **`np.linspace()`**: - Creates a list of numbers that are evenly spaced between two values. - We use `np.linspace(0, L, num_nodes)` to create a list of `num_nodes` values between `0` and `L`. - **Example**: `np.linspace(0, 1, 5)` gives `[0. , 0.25, 0.5 , 0.75, 1. ]`.
---
Summary of How `np` Functions Fit in the Code
Here's how each of these functions helps in the code:
- **`np.zeros((num_nodes, num_nodes))`**: Sets up an empty matrix for calculations. - **`np.zeros(num_nodes)`**: Sets up an empty list for storing values. - **`np.array(...)`**: Makes sure our data fits well with NumPy. - **`np.linalg.solve(K, F)`**: Solves our equation to find `U`. - **`np.linspace(0, L, num_nodes)`**: Creates the `x` values for plotting.
---
JILID 4 : Aplikasi 1-D PDE pada Sistem Manufaktur
Question 6: Is Economical Order Quantity could be defined in Partial Differential Equation?
Yes, the Economic Order Quantity (EOQ) model can be formulated with partial differential equations (PDEs) to extend its traditional applications, especially in more complex situations, such as when inventory levels or demand rates vary continuously over time or space. Let’s look at how EOQ might connect to PDEs.
- 1. **Basic EOQ Model Recap**
The classical EOQ formula minimizes the total cost \( C(Q) \) of ordering and holding inventory: \[ Q = \sqrt{\frac{2DS}{H}} \] where: - \( D \) is the annual demand, - \( S \) is the ordering cost per order, and - \( H \) is the holding cost per unit per year.
This formula assumes constant demand, lead time, and costs, so it doesn’t usually require differential equations.
- 2. **EOQ with Variable Demand or Holding Costs**
In cases where demand, holding, or ordering costs change over time or with respect to certain conditions, you can model EOQ using differential equations. For example:
- **Time-dependent demand:** If demand \( D(t) \) varies with time, we might want to model the rate of change of inventory over time as a differential equation.
\[
\frac{dI}{dt} = -D(t) + R
\]
where \( I \) is the inventory level and \( R \) is the rate at which inventory is replenished.
- 3. **Partial Differential Equation Approach**
If we extend to situations where both time and other factors (like location in a supply chain) influence inventory levels, a **PDE** may be necessary. For example:
- **Spatiotemporal EOQ Model:** Imagine a scenario where inventory demand changes over a geographical area (space, \( x \)) and time \( t \). We could have a PDE like this:
\[
\frac{\partial I(x, t)}{\partial t} + v \frac{\partial I(x, t)}{\partial x} = -D(x, t) + R(x, t)
\]
where:
- \( I(x, t) \) represents inventory at location \( x \) and time \( t \),
- \( v \) represents a velocity factor for distribution over space (if there’s a moving vehicle, for example),
- \( D(x, t) \) and \( R(x, t) \) are the demand and replenishment rates over space and time.
Such a PDE could be used to optimize order quantities at different points along a supply chain over time.
- 4. **Solution Approach for the PDE**
To solve, you’d typically set up initial and boundary conditions based on known values for inventory at certain times and places. Analytical or numerical methods (like finite difference or finite element methods) could then yield optimal order quantities under these dynamic conditions.
Penilaian Kualitas Interaksi Chat GPT
Tentu saja! Berikut evaluasi interaksi kita berdasarkan pertanyaan Anda dan respons saya:
- Kekuatan dalam Interaksi Kita
1. **Pertanyaan yang Jelas dan Terfokus**: - Pertanyaan Anda spesifik dan ringkas, yang membantu saya memberikan jawaban yang tepat dan relevan. - Contoh: Bertanya tentang topik seperti "derivasi FEM 1D" atau "Kode JavaScript dan HTML untuk EOQ" menunjukkan bahwa Anda tahu apa yang Anda cari.
2. **Keterlibatan dengan Penjelasan**: - Anda terlibat dengan penuh pertimbangan dengan meminta penjelasan atau elaborasi yang lebih sederhana, yang menunjukkan keinginan untuk menyempurnakan pemahaman Anda. - Contoh: Ketika Anda meminta penjelasan yang lebih sederhana tentang FEM atau EOQ, itu membantu saya menyesuaikan respons saya agar lebih jelas dan lebih mudah digunakan.
3. **Perkembangan Logis**: - Pertanyaan Anda mengikuti urutan yang logis, dimulai dengan konsep dasar (misalnya, derivasi FEM) dan beralih ke implementasi praktis (misalnya, kode). - Perkembangan ini menunjukkan bahwa Anda sedang membangun pemahaman berlapis, yang merupakan pendekatan pembelajaran yang kuat.
---
- Area untuk Peningkatan Interaksi
1. **Informasi Konteks atau Latar Belakang**: - Meskipun pertanyaan Anda jelas, memberikan lebih banyak konteks tentang tingkat pemahaman Anda atau tujuan pertanyaan Anda dapat membantu saya menyusun jawaban dengan lebih efektif. - Misalnya: Apakah Anda ingin menerapkan FEM dalam proyek tertentu atau hanya mencoba memahaminya secara konseptual? Mengetahui hal ini akan membantu dalam menyusun respons dengan kedalaman yang sesuai.
2. **Detail Tindak Lanjut**: - Saat Anda menanyakan tindak lanjut (misalnya, "Bisakah Anda membuatnya lebih sederhana?"), menentukan bagian mana yang tidak jelas akan membantu saya memfokuskan penyederhanaan dengan lebih baik. - Misalnya, alih-alih "menyederhanakan," Anda dapat mengatakan, "Saya merasa istilah matematika membingungkan," atau "Fokus pada langkah-langkah praktis."
---
- Penilaian Saya terhadap Respons Saya
1. **Komprehensif dan Disesuaikan**: - Saya berusaha memberikan penjelasan yang mendalam, langkah demi langkah dengan contoh, algoritme, dan bahkan kode jika berlaku. - Misalnya, dalam contoh derivasi FEM dan kode EOQ, saya berusaha memberikan kejelasan sambil membahas aspek teoritis dan praktis.
2. **Fleksibel**: - Ketika Anda meminta penyederhanaan atau contoh, saya menyesuaikan pendekatan saya, menawarkan penjelasan dalam istilah yang lebih sederhana dan menggunakan contoh yang relevan.
3. **Responsivitas**: - Saya berusaha beradaptasi dengan kebutuhan Anda, beralih dari teori ke aplikasi (misalnya, kode FEM ke EOQ), dan menanggapi umpan balik Anda.
---
- Saran untuk Mengoptimalkan Interaksi di Masa Depan
1. **Nyatakan Sasaran Spesifik**: - Jika ada kasus penggunaan tertentu (misalnya, proyek akademis, pengembangan perangkat lunak), menyebutkannya akan memastikan bahwa respons selaras dengan sasaran Anda.
2. **Sorot Kedalaman Pilihan**: - Menunjukkan apakah Anda lebih suka ikhtisar tingkat tinggi, derivasi terperinci, atau implementasi praktis membantu menyesuaikan jawaban dengan lebih baik.
3. **Umpan Balik Iteratif**: - Jika ada yang tidak jelas, memberikan umpan balik yang terarah (misalnya, "Kodenya jelas, tetapi saya tidak yakin tentang rumusnya.") akan membantu menyempurnakan penjelasan dalam respons berikutnya.
---
- Kualitas Interaksi Secara Keseluruhan
- Peringkat**: 9/10
- Keterlibatan yang kuat dan pertanyaan yang jelas menjadikan ini interaksi yang produktif dan menyenangkan. Menyempurnakan konteks dan umpan balik dapat meningkatkan pertukaran lebih jauh.
Beri tahu saya jika ada hal spesifik yang ingin Anda fokuskan, atau jangan ragu untuk membagikan pemikiran Anda tentang evaluasi ini!
