This is a sample blog post to demonstrate the formatting and features available on this blog. It covers everything from basic prose to code blocks, mathematical notation, and embedded media.

Writing and Prose

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“The purpose of computation is insight, not numbers.” — Richard Hamming

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Code Blocks

Here’s an example of a Python function with syntax highlighting:

import numpy as np

def topology_optimize(domain, volume_fraction=0.5, penalty=3.0):
    """Run SIMP-based topology optimization on a 2D domain."""
    density = np.ones(domain.shape) * volume_fraction
    for iteration in range(200):
        ke = compute_element_stiffness(density, penalty)
        u = solve_fem(ke, domain.loads, domain.bcs)
        sensitivity = compute_sensitivity(u, ke, density, penalty)
        density = optimality_criteria_update(density, sensitivity, volume_fraction)
    return density

And some inline code: the density array stores per-element material distribution values between 0 and 1.

Mathematics

The SIMP (Solid Isotropic Material with Penalization) method relates element stiffness to density:

$$E_e = E_0 \cdot \rho_e^p$$

where $E_e$ is the element Young’s modulus, $E_0$ is the base material modulus, $\rho_e$ is the element density, and $p$ is the penalization factor (typically $p = 3$).

The optimization problem is formulated as:

$$\min_{\rho} \quad c(\rho) = \mathbf{U}^T \mathbf{K} \mathbf{U} = \sum_{e=1}^{N} (\rho_e)^p \mathbf{u}_e^T \mathbf{k}_0 \mathbf{u}_e$$

subject to:

$$\frac{V(\rho)}{V_0} = f, \quad 0 < \rho_{\min} \leq \rho_e \leq 1$$

Images

Here’s an example of an inline image (using an existing project image):

Compliant mechanism design

Lists

Things I find interesting:

  • Topology optimization and generative design
  • Reinforcement learning for engineering applications
  • Nonlinear finite element analysis
  • Diffusion models for structural design

Some useful references:

  1. Bendsøe & Sigmund (2003) — the foundational text on topology optimization
  2. TopOpt group at DTU — excellent educational resources
  3. FEniCSx documentation — the FEA framework used in my solvers

This is a sample post. Delete it when you publish your first real article.