WELCOME ALL !!
Dr Airil Yasreen Mohd Yassin is offering a free online Finite Element Method (FEM) course designed to build strong foundations in numerical methods and computational mechanics. Starting on 3 June, classes will be held every Wednesday, 9:00–11:00 PM, over approximately 12 weeks. The course begins from first principles, making it suitable for beginners, and follows the team’s FEM textbooks. It also provides a pathway to more advanced topics, including meshless methods, NURBS, and the Numerical Derivation Method (NDM).
1️⃣ For those who have not registered yet
Please complete your registration using the following link:
[https://forms.gle/ArsRx4a35TpcSkum7]
2️⃣ Join the WhatsApp Group
Please join the WhatsApp group for updates and discussions using the following link:
[https://chat.whatsapp.com/DqS4h7eJdVJ8LazYPR4sM5?s=cl&p=i&ilr=4]
3️⃣ Download the Red Book & Blue Book
The PDF books can be downloaded from the following link:
[https://github.com/msnm-official/fem_books]
WEEK 01
Introduction to numerical methods
03 JUNE 2026
We have learned the basics of numerical methods and their application in solving mechanics problems governed by differential equations. Since computers and programming languages primarily operate with matrix systems, these governing equations must be transformed into matrix form. This can be achieved using methods such as the collocation method and the weighted residual method. The collocation method satisfies the governing equation at specific points, whereas the weighted residual method minimizes the overall error across the domain. As a result, the weighted residual method is less likely to overlook local regions with high gradients within the domain, leading to improved accuracy and a more representative solution.
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WEEK 02
Galerkin WRM approach & FEM
10 JUNE 2026
We successfully learned the Galerkin Weighted Residual Method (WRM) and its application in the Finite Element Method (FEM). The session emphasized the derivation of the weak form through integration by parts (IBP), where natural boundary conditions appear naturally within the formulation. We also explored the importance of the Kronecker delta property of finite element shape functions, which allows essential boundary conditions to be imposed directly. In addition, we studied the formulation of elemental equations and their assembly into a global system, providing a strong foundation for accurate and efficient numerical solutions to engineering problems.
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WEEK 03
Higher-Order FEM for Bar Element + Beam Problem
17 JUNE 2026
We first discussed the formulation of the bar problem and considered its p-refinement using quadratic elements. We then learned how the Finite Element Method (FEM) can be naturally extended from bar elements to beam problems using the same discretization philosophy, element formulation procedure, and matrix assembly process. By introducing beam elements with both displacement and rotational degrees of freedom, we observed that the resulting system of matrices closely resembles those of bar elements. This demonstrates the power and versatility of FEM as a unified framework for solving diverse engineering problems while maintaining a strong connection to the underlying mechanics and physics.
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WEEK 04
Derivation of Differential Equations – 1D and 2D Applications
24 JUNE 2026
Now we have learned how differential equations are derived from engineering and physical systems. Students will derive Ordinary Differential Equations (ODEs) for bar and beam problems, as well as Partial Differential Equations (PDEs) for two-dimensional heat transfer phenomena. As consistently emphasized in previous classes, understanding the underlying physics and mechanics is essential for developing accurate mathematical models and becoming a true engineer. Furthermore, this foundation is crucial for pursuing advanced studies in mechanics and contributing to new discoveries and innovations in academic research.
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WEEK 05
Extension of FEM to 2D Heat Transfer Problems
01 JULY 2026
We extended the Finite Element Method (FEM) from the previously developed 1D formulations to two-dimensional heat transfer problems, following a very similar procedure to the 1D case but now involving two spatial directions. We have applied weighted residual methods and the Galerkin approach to derive the weak form of the steady-state heat conduction equation in 2D. Through integration by parts, higher-order derivatives were reduced and natural boundary conditions in the form of heat flux naturally emerged. With variation in two axes, the governing PDE was compacted into a matrix system through spatial discretization. We have discretized the thermal field into finite elements, transforming the PDE into a system of algebraic equations. This led to the global conductivity matrix, which governed heat diffusion and enabled prediction of temperature distribution in 2D geometries.
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WEEK 06
From Heat to Stress: FEM for 2D Plane Stress
08 JULY 2026
In the session, we moved from heat transfer into solid mechanics by extending FEM to two-dimensional plane stress analysis, where the unknowns became displacement fields in the x and y directions describing structural deformation under loading. We introduced how stress and strain are distributed within a material and how linear elasticity governs their relationship. While the physical meaning changed, the same FEM framework—weak form, discretization, and matrix assembly—remained the core idea, now leading to a stiffness matrix formulation. As consistently emphasized, FEM served as a unified computational framework for solving a wide range of engineering physics problems once the formulation was understood.
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WEEK 07
Application to Eigenvalue Problems: Buckling
15 JULY 2026
In this session, we extended the Finite Element Method (FEM) to structural buckling analysis by introducing the concept of eigenvalue problems in engineering mechanics. We derived the governing equations for buckling and formulated the generalized eigenvalue equation using the structural stiffness matrix together with the geometric stiffness matrix. Through this formulation, students learned how FEM predicts critical buckling loads and the corresponding buckling mode shapes of structures under compressive loading. This session demonstrated how FEM can be applied not only to static analyses but also to stability problems, providing valuable insight into the safe and efficient design of engineering structures.
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WEEK 08
Application to Eigenvalue Problems: Free Vibrations
22 JULY 2026
Now, we have extended the Finite Element Method (FEM) from buckling to vibration analysis by first deriving the general dynamic equilibrium equation with a time-dependent external force, leading to the forced vibration formulation. To determine the natural frequencies and mode shapes, we removed all external forces, including Neumann boundary conditions, reducing the system to the free vibration eigenvalue problem using the global stiffness and mass matrices. This demonstrated that the same FEM framework applies naturally to dynamic analysis. Next week, we will revisit the complete dynamic equation and study forced vibration under time-dependent external loading.
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Recorded Video (Part 1)
Recorded Video (Part 2)
WEEK 09
Application to Dynamic Problems: Forced Vibrations
SCHEDULED FOR 29 JULY 2026
Google meet link: meet.google.com/aeu-rsmu-vua
In this session, we will extend the free vibration formulation to the complete forced vibration problem by incorporating time-dependent external loading into the dynamic equilibrium equation. Unlike the free vibration eigenvalue problem, the governing equations will no longer admit a direct eigenvalue solution and must instead be solved in the time domain. We will introduce the Finite Difference Method (FDM) to discretize the second-order time derivatives, enabling the displacement, velocity, and acceleration responses to be computed step by step. This session will demonstrate how the Finite Element Method (FEM) for spatial discretization is naturally combined with finite difference time integration to predict the dynamic response of structures subjected to realistic time-varying loads…
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Note (Week 09)
Recorded Video (Part 1)
Recorded Video (Part 2)