Finite Element Method
from First Principles

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.

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.

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.

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.

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.

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.

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.

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.

In this session, the application of computational tools and artificial intelligence-assisted programming in solving Finite Element Method (FEM) problems was explored. The concept of “vibe coding” was introduced through a step-by-step demonstration of using ChatGPT and Google Colab to develop Python codes for FEM beam analysis, including problem formulation, code implementation, and visualization of numerical results. The session highlighted how modern computational workflows can enhance engineering simulations while maintaining the importance of fundamental mechanics and numerical methods. The integration of FEM knowledge, programming skills, and AI-assisted tools was emphasized as an essential competency for future engineers, researchers, academicians, and industry professionals.

In this session, we extended 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 no longer admitted a direct eigenvalue solution and were instead solved in the time domain. We introduced 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. The session demonstrated how the Finite Element Method (FEM) for spatial discretization could be naturally combined with finite difference time integration to predict the dynamic response of structures subjected to realistic time-varying loads. We also discussed the phenomenon of resonance, highlighting how the structural response can become significantly amplified when the excitation frequency approaches the structure’s natural frequency, underscoring the importance of dynamic analysis in engineering design.

In this session, we extended the Finite Element Method (FEM) from linear to nonlinear problems by introducing two fundamental nonlinear solution strategies: direct substitution (Picard iteration) and the Newton–Raphson method. We derived both approaches step by step and examined how iterative procedures were used to obtain converged solutions when the governing equations became nonlinear. We also highlighted the close connection between nonlinear FEM solvers and optimization techniques such as gradient descent used in artificial intelligence and neural networks. This demonstrated how mathematics and physics provided a common foundation across engineering and computer science, enabling multidisciplinary connections between computational mechanics and AI. The session reinforced the importance of understanding first principles and mathematical fundamentals as a basis for developing new computational methods, research, and engineering innovations.

In this session, we further explored the formulation of nonlinear problems in the Finite Element Method (FEM), beginning with nonlinear differential equations in which the unknown variables may interact with themselves or their derivatives. Building upon the Newton–Raphson method introduced previously, we formulated the residual equation and used Taylor expansion to obtain iterative corrections to the degrees of freedom, where convergence is achieved as the residual approaches zero. The session highlighted the importance of strong mathematical foundations, particularly tensor algebra and matrix calculus, for deriving and understanding advanced nonlinear FEM formulations such as geometric nonlinearity and fluid–structure interaction. Moving forward, we will continue strengthening these foundations through numerical integration before progressing toward frame analysis in the final part of the course.

In this session, we continued strengthening the mathematical foundations required for advanced FEM formulations by focusing on numerical integration. The session covered the basic concept of Gaussian quadrature, in which functions were evaluated at discrete Gauss points and combined with appropriate weighting factors. Particular attention was also given to coordinate transformation from the physical domain to the natural coordinate system, the use of shape functions for geometric mapping, and the role of the Jacobian matrix in transforming derivatives and differential area elements. This session provided an important bridge between the theoretical derivation of element matrices and their practical computational implementation, while preparing the group for a more detailed treatment of two-dimensional numerical integration and plane stress formulation in the following sessions.

In this session, we introduced the governing equations for two-dimensional incompressible Navier–Stokes flow. Starting from the general three-dimensional compressible formulation, several assumptions were introduced, including incompressible flow, isothermal conditions, and negligible variation in the z-direction.
We also introduced the Eulerian description of fluid motion and the material derivative, which combines local and convective acceleration. The mass-conservation equation was then derived using a differential control element, followed by the x- and y-momentum equations based on Newton’s second law. Stress components were expressed using pressure, deviatoric stress, and Newtonian constitutive relations, leading to the final set of 2D incompressible Navier–Stokes equations. The session also highlighted the importance of engineering assumptions through comparisons with Euler–Bernoulli and Timoshenko beam theories. In the next session, we will introduce Physics-Informed Neural Networks (PINNs) before returning to finite element analysis..