What Are Numerical Simulations and How Do They Work?

Numerical simulations use computer calculations from a specific mathematical model to imitate physical processes. They function as virtual experiments, allowing for the prediction and analysis of how complex systems behave under various conditions. This approach is useful for scenarios that are too difficult, expensive, or hazardous to test in the real world. A simulation acts as an advanced “what-if” machine that relies on mathematical equations, allowing engineers to test countless variables like the stress on a bridge or airflow over a wing. The goal is to identify design flaws and optimize performance before a physical prototype is built.

The Core Components of a Simulation

At the core of every simulation is a mathematical model, a set of equations describing the system’s governing principles. For instance, the behavior of air flowing over a car is described by the Navier-Stokes equations in computational fluid dynamics (CFD). These equations act as the “rules” the computer follows to replicate real-world physics, such as how fluids move or heat transfers.

Since computers cannot solve these equations for an infinite number of points, a process called discretization is used. This breaks down the object’s geometry into a finite number of small pieces, creating a “mesh” or “grid.” Similar to how a digital image is formed from pixels, the computer performs calculations on each cell of this mesh to approximate the behavior of the entire system.

To solve the equations on the grid, the computer uses numerical methods and algorithms. These are step-by-step computational recipes that find an approximate solution. Techniques like the Finite Element Method (FEM) transform the model’s complex differential equations into algebraic equations a computer can solve, allowing it to calculate variables like pressure, velocity, and temperature.

The Simulation Workflow

Executing a numerical simulation follows a structured, three-stage workflow. The first stage is preprocessing, which involves defining the object’s geometry, often from computer-aided design (CAD) software, and generating the mesh. During this phase, material properties are assigned, and boundary conditions are set to define how the object will be tested.

The second stage is solving, where the computer performs the intensive calculations. The solver applies the chosen numerical methods to the mathematical model, working through the problem iteratively until a stable solution is reached. This stage can be computationally demanding and may require high-performance computing systems.

The final stage is post-processing, where the raw numerical data is translated into understandable formats. The output includes color-coded plots, graphs, and animations that reveal patterns, such as stress distribution or airflow. This visual analysis allows for informed decisions about design optimization.

Applications Across Industries

Numerical simulations are a tool used across a range of industries to enable innovation and improve safety.

• Engineering and Manufacturing: Simulations are used for aerodynamic design and virtual crash tests for automobiles. Simulating a car crash allows engineers to analyze impact on the vehicle and its occupants, refining safety features like airbags and crumple zones without destroying costly physical prototypes, leading to safer vehicles developed in less time.
• Weather and Climate Science: This field relies on numerical models to forecast daily weather and project long-term climate change. These complex models run on supercomputers to predict temperature, wind speed, and precipitation, which informs agriculture, travel, and emergency preparedness.
• Medicine and Biology: Simulations offer insights into the human body and disease. They can model blood flow to predict artery blockages or show how a new drug might interact with proteins, speeding up the development of new medicines by identifying promising candidates before extensive lab testing.
• Entertainment: The entertainment industry uses simulations to create realistic visual effects. The physics of an explosion or the movement of water are simulated to generate lifelike animations that would be impractical to create by hand, adding a layer of realism to the final product.

Understanding Simulation Accuracy and Limitations

Numerical simulations are approximations of reality, and their accuracy is subject to potential errors. Modeling errors can occur because the mathematical equations are simplifications of complex physics. Numerical errors can arise from the discretization process and the finite precision of computer calculations, where a finer mesh can yield a more accurate but computationally expensive result.

To build confidence in simulation results, a two-part process of verification and validation is used. Verification checks that the computer code correctly solves the mathematical equations of the model, asking, “Are we solving the equations correctly?” This is often done by comparing the output to known solutions for simpler problems.

Validation addresses whether the model accurately represents the real world, asking, “Are we solving the correct equations?” This step involves comparing the simulation’s predictions to data from physical experiments. If the results align, the model is considered validated for that scenario.