Josi Fajar Mahendra

Introduction

Hi, I'm Josi Fajar Mahendra, a Magister student at Universitas Indonesia 2024 (NPM.2406375554), and bachelor's graduate from Universitas Sriwijaya, interested in energy conversion and conservation, especially in the field of fluid mechanics.

1. DAI5 Framework (Komputasi Teknik)

“DAI5 Framework” adalah kerangka berpikiryang dikembangkan oleh Dr. Ahmad Indra Siswantara, dengan menggunakan conscious thinking (kesadaran berpikir) sebagai basis kerangka berfikir. Dengan kalimat “ My Heart work to encode (menulis/membuat suatu program) and My Brain decodes(mengeksekusi program)". terdapat 4 kunci yaitu Intention, Initial Thinking (about the problem), Idealization, Instruction Set. dengan DAI5 memadukan heartware & brainware Adapun makna dari setiap kata kunci adalah sebagai berikut: 1. Intention (Niat) – Memulai setiap langkah dengan niat yang kokoh serta tujuan yang jelas yang ingin diraih. 2. Initial Thinking (Pemikiran Awal) – Melakukan peninjauan awal untuk memahami konteks atau masalah sebelum memulai tindakan.. 3. Idealization (Idealasi) – Membayangkan solusi atau pendekatan yang optimal untuk mengatasi sebuah masalah. 4. Instruction Set (Instruksi) – Menguraikan proses menjadi langkah-langkah terstruktur dan sistematis untuk memudahkan pelaksanaan.

DAI5 framework is widely applicable in engineering and technical fields, such as Finite Element Analysis (FEA), where it helps to systematically break down complex problems. By separating analysis into intentional and iterative steps, DAI5 supports efficient, detailed simulations in areas like stress, thermal, and flow analysis, making it highly suitable for structured engineering solutions.

Tugas 1

kali ini saya mencoba membuka diskusi dengan AI Chatgpt mengenai "Can you derive finite element equations from 1D PDE governing equations for a bacward facing step at turbo jet engine? "

Saya Bertanya :"hi chatgpt, longtime no see"

ChatGPT : "Hi! It has been a while—great to see you! Anything new or interesting you’re working on these days?"

Saya Bertanya : "Can you derive finite element equations from 1D PDE governing equations for a bacward facing step at turbo jet engine?"

ChatGPT : Deriving finite element equations from a 1D partial differential equation (PDE) governing flow over a backward-facing step in a turbojet engine involves several steps. Typically, this type of problem is governed by the Navier-Stokes equations in compressible or incompressible flow for boundary layers. The backward-facing step creates separation and reattachment zones due to sudden changes in geometry, which are particularly significant in fluid dynamics, especially in the design of turbojet engines.

1. Define the Governing Equations

For a 1D simplified model in a turbojet engine, consider the following PDEs for steady-state, incompressible flow:

In a backward-facing step problem, boundary layer effects dominate near the step, and there is often a reattachment point downstream where the flow stabilizes.

2. Discretize the Domain

In finite element analysis, divide the domain into small elements. For a 1D problem, divide the line along x(streamwise direction) into elements between the inlet (upstream of the step) and outlet (downstream reattachment).

3. Weak Formulation of the PDE

The weak form is achieved by multiplying the governing equation by a test function vvv and integrating over the domain:

∫Ωv(ρududx+dpdx−μd2udx2) dx=0.\int_{\Omega} v \left( \rho u \frac{du}{dx} + \frac{dp}{dx} - \mu \frac{d^2 u}{dx^2} \right) \, dx = 0.∫Ωv(ρudxdu+dxdp−μdx2d2u)dx=0. Applying integration by parts to the diffusion term μd2udx2\mu \frac{d^2 u}{dx^2}μdx2d2u, we obtain: ∫Ωv(ρududx) dx−∫Ωdpdxv dx+∫Ωμdudxdvdx dx=0.\int_{\Omega} v \left( \rho u \frac{du}{dx} \right) \, dx - \int_{\Omega} \frac{dp}{dx} v \, dx + \int_{\Omega} \mu \frac{du}{dx} \frac{dv}{dx} \, dx = 0.∫Ωv(ρudxdu)dx−∫Ωdxdpvdx+∫Ωμdxdudxdvdx=0.

Boundary conditions can be applied here, depending on the inlet and outlet specifications.

4. Finite Element Approximation

Assume an approximation for uuu as a linear combination of basis functions NiN_iNi: u≈∑iUiNi,u \approx \sum_{i} U_i N_i,u≈i∑UiNi, where UiU_iUi are nodal values and NiN_iNi are shape functions (typically linear in 1D). Substitute this approximation into the weak form, resulting in a system of equations: KU=F,\mathbf{K} \mathbf{U} = \mathbf{F},KU=F, where K\mathbf{K}K is the stiffness matrix, U\mathbf{U}U is the vector of unknowns (nodal velocities), and F\mathbf{F}F is the force vector.

5. Assembly of the Stiffness Matrix and Load Vector

For each element, compute local stiffness matrix Ke\mathbf{K_e}Ke and load vector Fe\mathbf{F_e}Fe by evaluating the integrals: Convective term: ∫Ωeρududxv dx\int_{\Omega_e} \rho u \frac{du}{dx} v \, dx∫Ωeρudxduvdx, Pressure gradient: −∫Ωedpdxv dx-\int_{\Omega_e} \frac{dp}{dx} v \, dx−∫Ωedxdpvdx, Diffusion term: ∫Ωeμdudxdvdx dx\int_{\Omega_e} \mu \frac{du}{dx} \frac{dv}{dx} \, dx∫Ωeμdxdudxdvdx. These are assembled into the global matrix K\mathbf{K}K and vector F\mathbf{F}F.

6. Apply Boundary Conditions and Solve

Apply boundary conditions (inlet velocity, outlet pressure, or no-slip conditions) to reduce the system and solve for U\mathbf{U}U.

7. Post-Processing

After solving, calculate secondary quantities like vorticity, reattachment length, and wall shear stress to assess the impact of the backward-facing step on the flow characteristics within the turbojet engine.

This approach provides a basis for finite element analysis of flow over complex geometries in turbojets, where accurate predictions of recirculation and separation are essential for performance evaluation and design improvement.

DAI5 Framework

Saya Bertanya : "can you tell it using DAI5 Framework" ChatGPT : Using the DAI5 Framework for deriving finite element equations from a 1D PDE governing equations for a backward-facing step in a turbojet engine can provide a structured approach. Here’s how each step of the DAI5 Framework would be applied:

1. Intention: Define the Purpose

In this step, clarify why we’re deriving the finite element equations. The main goal here is to analyze the flow characteristics over a backward-facing step in a turbojet engine, focusing on velocity profiles, pressure distribution, and recirculation zones that result from the sudden expansion in geometry. Understanding these characteristics can lead to improved designs by minimizing drag and optimizing engine efficiency.

2. Initial Thinking: Explore Assumptions and Requirements

At this stage, brainstorm the requirements and assumptions necessary to simplify the problem. For a turbojet application:

- Governing Equations: Start with the 1D incompressible Navier-Stokes equations or continuity and momentum equations as simplified models.
- Boundary Conditions: Consider boundary conditions such as inlet velocity, outlet pressure, and no-slip wall conditions at the backward-facing step.
- Discretization Approach: Plan to divide the computational domain into elements along the streamwise xxx-direction and use a finite element approximation with shape functions for velocity and pressure.
- Assumptions: Assume steady-state flow, incompressibility, and a simplified 1D flow model to manage computational complexity.

3. Idealization: Conceptualize the Desired Outcome

In this phase, envision the ideal outcome and how the derived equations will enable analysis:

- The finite element formulation should provide an accurate representation of flow separation, reattachment zones, and recirculation caused by the backward-facing step.
- The ideal solution would capture the effects of turbulence near the step, but in this simplified model, focus on primary flow characteristics like velocity and pressure gradients along the domain.
- Aim for a stable and computationally efficient solution that can later be extended to more complex, multi-dimensional models if required.

4. Instruction Set: Develop Steps for Derivation

Here, break down the derivation process into actionable steps, which can later be implemented in software:

- Formulate the Weak Form: Multiply each governing equation by a test function and integrate over the domain. Use integration by parts to handle higher-order derivatives, which will result in terms suitable for the finite element approach.
- Select Shape Functions: Choose appropriate shape functions for the element discretization, which could be linear or quadratic in 1D. These shape functions will approximate the velocity and pressure fields.
- Assemble System of Equations: For each element, derive the local stiffness matrix and force vector by evaluating integrals of the convective, diffusive, and pressure terms. Then assemble these into the global matrix.
- Apply Boundary Conditions: Incorporate boundary conditions (e.g., velocity at the inlet and outlet conditions) to refine the system for solvability.
- Solve the System: Solve the assembled finite element equations to get the nodal values of velocity and pressure.

5. Implementation: Execute the Derivation

Finally, implement the derivation and computation:

- Write or run the derived finite element equations using a suitable computational tool (e.g., MATLAB, ANSYS, or Python with FEA libraries).
- Validate the results by comparing the velocity and pressure profiles with known solutions or benchmark cases of flow over a backward-facing step.
- Evaluate results, such as the location of flow reattachment and wall shear stress, to assess the impact of the backward-facing step on the turbojet engine’s performance.

By following these structured steps of the DAI5 Framework, the finite element derivation process becomes organized, ensuring each phase is aligned with the final goal of understanding and optimizing flow behavior over a backward-facing step.