Theoretical Background

This page provides the theoretical foundation for the three simulation methods available in Advanced Simulation: Structural Analysis (FEA), Fluid Dynamics (CFD), and Rapid Aerodynamics (Vortex Method).

Understanding the underlying theory helps you set up simulations correctly, interpret results with confidence, and recognize the limitations of each method.

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1. Structural Analysis — Finite Element Method (FEM)

1.1. Overview

Structural Analysis in EveryDrone is based on the Finite Element Method (FEM), a numerical technique for solving partial differential equations over complex geometries.

The core idea is to subdivide the structure into a finite number of small elements (the mesh) and solve the governing equations over each element. The solutions at element boundaries are constrained to be compatible (continuous), and the global solution is assembled from the local element solutions.

1.2. Governing Equation

Under static loading conditions, the equilibrium of a linear elastic solid is governed by:

$$\mathbf{K} \, \mathbf{u} = \mathbf{f}$$

where:

• $\mathbf{K}$ — global stiffness matrix, assembled from element stiffness matrices
• $\mathbf{u}$ — displacement vector (the unknowns to be solved)
• $\mathbf{f}$ — external force vector (applied loads and boundary conditions)

Each element's stiffness matrix $\mathbf{K}_e$ is derived from the element's geometry and material properties:

$$\mathbf{K}_e = \int_{\Omega_e} \mathbf{B}^T \mathbf{D} \mathbf{B} \, d\Omega$$

where $\mathbf{B}$ is the strain-displacement matrix and $\mathbf{D}$ is the material constitutive matrix (relating stress to strain via the material's Young's modulus and Poisson's ratio).

1.3. Stress and Strain

Once displacements $\mathbf{u}$ are obtained, strains $\boldsymbol{\varepsilon}$ and stresses $\boldsymbol{\sigma}$ are computed as:

$$\boldsymbol{\varepsilon} = \mathbf{B} \, \mathbf{u}, \qquad \boldsymbol{\sigma} = \mathbf{D} \, \boldsymbol{\varepsilon}$$

The von Mises stress is commonly used to assess whether a material is at risk of yielding:

$$\sigma_{\text{VM}} = \sqrt{\frac{(\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2}{2}}$$

where $\sigma_1, \sigma_2, \sigma_3$ are the principal stresses.

1.4. Assumptions and Limitations

Assumption	Description
Linear elasticity	Material deformation is proportional to applied load (Hooke's Law). Valid when stresses remain below the yield strength.
Small deformation	Geometry changes due to deformation are negligible. Not valid for large-deflection problems.
Static loading	Inertial (dynamic) effects are not considered. Suitable for steady-state load cases.
Isotropic material	Material properties are identical in all directions. Composite or anisotropic materials require additional modeling.

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2. Fluid Dynamics — Reynolds-Averaged Navier-Stokes (RANS)

2.1. Overview

Fluid Dynamics simulations in EveryDrone solve the Reynolds-Averaged Navier-Stokes (RANS) equations, which describe the mean behavior of a turbulent fluid flow. RANS is the industry-standard approach for external aerodynamics and is computationally tractable for engineering-scale problems.

2.2. Governing Equations

The RANS equations for incompressible flow are expressed in three-dimensional integral form as follows. Here, $\Omega$ denotes an arbitrary control volume, $\partial \Omega$ its bounding surface, $\hat{\mathbf{n}}$ the outward unit normal vector of the surface, and $S_m$ the volumetric mass source term.

Continuity (mass conservation):

$$\frac{\partial}{\partial t} \int_{\Omega} \rho \, d\Omega + \oint_{\partial \Omega} \rho \, \overline{\mathbf{u}} \cdot \hat{\mathbf{n}} \, dS = \int_{\Omega} S_m \, d\Omega$$

The first term on the left-hand side represents the rate of change of mass within the control volume, the second term is the mass flux through the bounding surface, and the right-hand side accounts for volumetric mass sources (e.g., evaporation, chemical reactions). For incompressible steady-state flow with no source ($S_m = 0$), the equation reduces to $\oint_{\partial \Omega} \overline{\mathbf{u}} \cdot \hat{\mathbf{n}} \, dS = 0$.

Momentum conservation:

$$\frac{\partial}{\partial t} \int_{\Omega} \rho \, \overline{\mathbf{u}} \, d\Omega + \oint_{\partial \Omega} \rho \, \overline{\mathbf{u}} \left( \overline{\mathbf{u}} \cdot \hat{\mathbf{n}} \right) dS = -\oint_{\partial \Omega} \overline{p} \, \hat{\mathbf{n}} \, dS + \oint_{\partial \Omega} \left( \boldsymbol{\tau} - \rho \, \overline{\mathbf{u}' \otimes \mathbf{u}'} \right) \cdot \hat{\mathbf{n}} \, dS + \int_{\Omega} \rho \, \mathbf{g} \, d\Omega$$

where the viscous stress tensor $\boldsymbol{\tau}$ for a Newtonian fluid is:

$$\boldsymbol{\tau} = \mu \left[ \nabla \overline{\mathbf{u}} + (\nabla \overline{\mathbf{u}})^T \right]$$

where:

• $\overline{\mathbf{u}}$ — time-averaged velocity vector (3D)
• $\overline{p}$ — time-averaged pressure
• $\rho$ — fluid density
• $\mu$ — dynamic viscosity ($\mu = \rho \nu$)
• $-\rho \, \overline{\mathbf{u}' \otimes \mathbf{u}'}$ — Reynolds stress tensor, representing the effect of turbulent fluctuations
• $\mathbf{g}$ — gravitational acceleration vector (body force)
• $S_m$ — mass source term (source/sink in the continuity equation)
• $\boldsymbol{\tau}$ — viscous stress tensor
• $\otimes$ — tensor product (dyadic product)

2.3. Assumptions and Limitations

Assumption	Description
Steady-state	Time-averaged flow; unsteady effects (e.g., vortex shedding) are not captured.
Incompressible flow	Valid for Mach numbers below approximately 0.3. Not suitable for transonic or supersonic flows.
RANS turbulence model	Mean-flow solution only. Cannot resolve turbulent structures directly (unlike LES or DNS).
Single solid geometry	The solver treats the imported geometry as one unified body. Multi-body interactions require manual setup.

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3. Rapid Aerodynamics — Vortex Method

3.1. Overview

The Vortex Method (also known as the Panel Method or Vortex Lattice Method in discrete form) is an inviscid, potential-flow-based aerodynamic solver. It is well-suited for unsteady aerodynamic analysis of lifting surfaces such as propellers and wings.

3.2. Governing Equation — Potential Flow

The Vortex Method is based on the assumption of irrotational, inviscid flow, for which the velocity field can be expressed as the gradient of a scalar potential $\phi$:

$$\mathbf{u} = \nabla \phi$$

Substituting into the continuity equation yields Laplace's equation:

$$\nabla^2 \phi = 0$$

This is a linear equation and can be solved via superposition of elementary solutions (sources, doublets, and vortices).

3.3. Vortex Panels

The surface of the geometry is discretized into panels. Each panel carries a vortex filament of unknown strength $\Gamma$. The boundary condition enforced is the no-penetration condition: the normal component of total velocity (freestream + induced) is zero at each panel's control point.

$$(\mathbf{u}_\infty + \mathbf{u}_\text{induced}) \cdot \hat{n} = 0$$

This yields a linear system of equations for the vortex strengths $\Gamma_i$:

$$\sum_j A_{ij} \Gamma_j = -\mathbf{u}_\infty \cdot \hat{n}_i$$

where $A_{ij}$ is the aerodynamic influence coefficient representing the normal velocity induced at panel $i$ by a unit vortex on panel $j$.

3.4. Vortex Shedding

For unsteady analysis (e.g., rotating propellers), freestream vortex shedding occurs at the trailing edge at each time step. The wake evolves over time, capturing the effect of vorticity on the aerodynamic loads. This allows effective prediction of time-varying thrust and torque of propellers.

3.5. Force Computation — Kutta-Joukowski Theorem

The lift force on each panel is computed via the Kutta-Joukowski theorem:

$$\Delta \mathbf{F} = \rho \, \mathbf{u}_\text{local} \times (\Gamma \, \Delta \mathbf{l})$$

where $\Delta \mathbf{l}$ is the bound vortex segment vector and $\mathbf{u}_\text{local}$ is the local velocity including phenomena induced by the wake.

3.6. Assumptions and Limitations

Assumption	Description
Inviscid flow	Viscous effects (skin friction, boundary layer separation) are not modeled. Drag predictions are limited to induced drag only.
Potential flow	No turbulence; flow must be attached. Results become unreliable at high angles of attack where separation occurs.
Thin lifting surfaces	Most accurate for slender wings and propeller blades. Less accurate for thick or bluff bodies.

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4. Method Comparison

	FEA	CFD (RANS)	Vortex Method
Physics	Solid mechanics	Viscous turbulent flow	Inviscid potential flow
Speed	Fast	Slow	Very fast
Accuracy	High (within linear elastic range)	High (attached flow)	Moderate (attached flow only)
Unsteady	No	No (steady RANS)	Yes
Viscous drag	N/A	Yes	No
Use case	Stress, deformation	Lift, drag, pressure	Unsteady thrust, torque

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