Numerical Analysis

The goal of numerical methods for engineers is to solve engineering problems when no closed-form, analytical solution exists. A number of methodologies exists: 

- structured programming concepts involving decision making, loops, functions, and parameter passing implemented within the MATLAB, Scilab or OpenModelica programming environment

- finite elements methods (FEM) for finding approximate solutions to boundary value problems for partial differential equations. Ex: Lisa 8.0, OpenFOAM for CFD, Elmer CSC for multiphysics, SimScale for cloud based CAE, the commercial general purpose FEM software Abaqus... 

-finite-difference methods (FDM) for solving differential equations by approximating them with difference equations, in which finite differences approximate the derivatives. Ex: Energy2D, Algodoo... 

IMPORTANT REMARKS - Numerical analysis *is not* the "bells & whistles" solution for all the engineering problem without an analytical closed form solution, but a different approach to the problem. The first step for a proper numerical model is a paper&pencil approach. Engineers and researchers should understand the solution only is their goal, while mastering the numerical tool is the goal of a skilled technician. 

The best opensource collection of Computer Aided Engineering tools is CAE Linux 2017. You can download here the latest version, and maybe use a virtual box for emulating the OS on Windows. 

Tutorials

Resources & Software

MATHEMATICS

Scilab, the popular freeware Matlab clone.

Freemat, another free Matlab clone.

wxMaxima, the free computer algebra system.

NumberEmpire, online math solver.

Keisan Calculator, online numerical differential equation solver.

ENGINEERING NUMERICAL PROBLEM SOLVING

Energy2D, free software for 2D heat transfer simulation.

Algodoo, a physics sandbox for multibody dynamics simulation is the simple 2D version of SpaceClaim Dynamics.

HotInt, an open source multibody dynamics simulation program.

SimScale, a partially free browser based finite element suite.

LT Spice IV, free software for electronic circuits simulation. 

QSapecNG, free electronic circuit solver

LISA 8.0, partially free FEM software.

list of the most popular FEM packages.

CAE Linux 2017 and a tutorial for installing it on a virtual box.

PrePoMax is a Calculix/Abaqus preprocessor. Calculix is a part of CAE Linux 2017, with a Launcher for compiling the input file (*.inp). PrePoMax provides basic FE solutions (such as linear static analysis). For additional features, it is better to edit the *.inp files with the notepad.

OpenFOAM is an opensource Computer Fluid Dynamics solver. Part of CAE Linux 2017, it can be used with the GUI HelixOS, or with the CFD workbench for FreeCAD

Abaqus Student Edition, the commercial software by Simulia available for students with academic affiliation, and a limited version for non-academics.

 

3D MODELING

FreeCAD, a popular open source parametric 3D modeler.

SolveSpace, a parametric 2D/3D CAD modeler with cinematic simulation capabilities

Fiji / ImageJ, the best scientific image processing software.

Cura by Dagoma, Netfabb Online, MeshLab, ImageToStl: all you need for 3D printing