How do you solve complex mechanics problems with no analytical solutions?

1 Simplify the problem

One of the first steps to solve a complex mechanics problem is to simplify it by making some reasonable assumptions, approximations, or simplifications. For example, you can neglect some small or negligible effects, use idealized models or boundary conditions, apply scaling or dimensional analysis, or reduce the number of variables or parameters. Simplifying the problem can help you identify the essential features and mechanisms, as well as reduce the computational cost and complexity.

2 Use numerical methods

Another strategy to solve a complex mechanics problem is to use numerical methods, which are algorithms that use numerical calculations to approximate the solution. Numerical methods can handle problems that are too difficult or impossible to solve analytically, such as nonlinear differential equations, integral equations, or optimization problems. Some common numerical methods in mechanics are finite difference, finite element, finite volume, boundary element, and meshfree methods. To use numerical methods, you need to discretize the problem domain, formulate the governing equations, choose an appropriate solver, and verify and validate the results.

3 Apply physical principles

A third strategy to solve a complex mechanics problem is to apply physical principles, which are fundamental laws or rules that govern the behavior of physical systems. Physical principles can help you understand the underlying physics and logic of the problem, as well as provide some guidance or constraints for the solution. Some examples of physical principles in mechanics are conservation of mass, momentum, and energy, equilibrium and compatibility conditions, constitutive relations, and symmetry and invariance properties.

4 Seek empirical or experimental data

A fourth strategy to solve a complex mechanics problem is to seek empirical or experimental data, which are observations or measurements of the physical system or phenomenon. Empirical or experimental data can help you test your assumptions, hypotheses, or predictions, as well as calibrate or validate your models or methods. Some sources of empirical or experimental data in mechanics are literature reviews, databases, laboratory experiments, field tests, and simulations.

5 Compare and contrast different solutions

A fifth strategy to solve a complex mechanics problem is to compare and contrast different solutions, which are possible answers or outcomes of the problem. Comparing and contrasting different solutions can help you evaluate the advantages and disadvantages, strengths and weaknesses, and similarities and differences of each solution. Some criteria for comparing and contrasting different solutions in mechanics are accuracy, precision, efficiency, robustness, reliability, and feasibility.

6 Communicate and document your solution

A sixth strategy to solve a complex mechanics problem is to communicate and document your solution, which are ways of presenting and explaining your solution to others. Communicating and documenting your solution can help you share your findings, insights, and recommendations, as well as receive feedback, suggestions, and critiques from others. Some methods of communicating and documenting your solution in mechanics are reports, papers, presentations, posters, and code.

7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?