Understanding Finite Element Analysis

Key Results for Finite Element Vibration Modal Analysis Beam Natural Frequencies Converge at Order 4. In this post, I use insights into the finite element method to answer the practical question of the optimal order of convergence of beam bending vibration with mesh refinement. Taking advantage of symmetry and antisymmetry, closed-form solutions for both frequencies and mode shapes are obtained for beam finite elements with consistent mass. Asymptotic frequency error rates are verified numerically and shown to be of order 4 in element size as the number of elements is increased. What’s Inside:  1. Basics of Vibration Modal Analysis  2. Derivation of the exact beam stiffness matrix using three methods         - The first is not shown in finite element books (or any books that I am aware of)         - The last is shown in structural analysis books.   3. Derivation of Consistent Mass Matrix using fundamental principles of virtual work and d’Alembert’s mass inertia  4. Derivation of analytical benchmark solution        - Frequency-dependent bending wave speed           - Explanation of 1/L^2 scaling  5. Closed-Form Finite Element Modal Analysis  6. Study of Frequency Error with Mesh Refinement  7. Summary of standard error estimates for frequency and mode shapes  8. Validation of Convergence Rates and Monotonic Convergence  9. Discussion for practical modal analysis 10. Conclusions and key takeaways 11. Exercises included Feel free to reuse the figures for your classes or posts; tag me so I can see how you extend the discussion! Enjoy! P.S. Learned something new? Tell me which part.

Many of our #finiteelementanalysis packages are black boxes. Often, they have manuals that contain vague references to how certain elements are implemented. They might give you a reference...they might not. I've been often disappointed when I'm trying to understand how a feature should behave and find that the documentation is sparse or non-existent. So when we created #fenix, we made sure to document some of the more...nuanced portions of the solver. Take, for example, our element formulations. Every single one is documented...and not just with a vague reference, but with a full wiki page detailing how the element is formulated, the various matrices and variables, shape functions and their derivatives, and more! You also get any references we used when implementing the element. Here's an example: https://lnkd.in/g75Pa7qi I think this level of insight into the guts of a solver is incredibly important. Not only can it increase trust, but it also allows those who are interested to understand each type of element much better. How can you know whether a particular element is suitable for your problem if you don't have access to detailed information about said element? Our documentation was generated in such a way that you should be able to implement a formulation yourself with sufficient knowledge of coding. No need to sift through dense journal articles that are written for mathematicians (unless you want to...and often it can be helpful). And they said you'll never use calculus again! 😁 Fenix can be found here: https://lnkd.in/gDvGjJnq #structuralengineering #structuraldesign #csharp #parametricdesign #structuralanalysis

🚀 MechCADemy Notes: Undamped Modal Analysis: From Theory to Finite Element Practice (simple, clear, powerful (Enjoy!)) This new document takes a comprehensive look at undamped modal analysis, bridging the gap between theoretical foundations and practical FEA application. 📘 Inside the document, you’ll find: - Why is modal analysis needed? - Step-by-step computation of natural frequencies and mode shapes - Description of pre-stress modal analysis - Derivation of lumped and consistent mass matrices - Physical interpretation of modal response - Detailed explanations of participation factors and effective mass - Insights into how different types of excitation activate different modes - Several examples with detailed solutions - Connections to real-world engineering scenarios, software tools like ANSYS, and code compliance requirements 🧠 Pro Tip: One of the best ways to learn and understand buckling is through modal analysis! Because eigenvalue buckling is mathematically formulated almost identically to undamped modal analysis — just with a geometric stiffness matrix replacing the mass matrix. This deep connection helps build intuition for stability, critical loads, and mode shapes associated with buckling. This release is part of an exciting new initiative called MechCADemy Notes — a growing collection of downloadable, high-quality learning documents that complement MechCADemy videos. 🧠 The first MechCADemy Note covered Nonlinear FEA, introducing large deformations and the Newton–Raphson method. 🎯 The long-term vision is to create a powerful synergy between these notes and the MechCADemy YouTube videos — turning each topic into a rich, multi-format learning experience. 📺 Check out my YouTube primer on Finite Element Analysis here:  https://lnkd.in/gV_p_WeT 🙌 Please support MechCADemy by: ✅ Subscribing to the channel ✅ Sharing this with your students, colleagues, or friends ✅ Commenting with your feedback and suggestions 🌞 Summer will be a busy and exciting time for MechCADemy — more videos and more notes on Finite Element Analysis, CAD, Design and Manufacturing, Mechanics of Materials, and Education. 🔖 #MechCADemy #MechCADemyNotes #ModalAnalysis #FEA #FiniteElementAnalysis #NonlinearFEA #MechanicalEngineering #StructuralDynamics #StructuralEngineering #CivilEngineering #NuclearEngineering #TransportationEngineering #EigenvalueProblem #ModeShapes #NaturalFrequencies #MassMatrix #HarmonicResponse #DynamicResponse #DynamicExcitation #EffectiveMass #ParticipationFactor #FEsoftware #ANSYS #Abaqus #COMSOL #CAD #CAM #Simulation #EngineeringEducation #EngineeringPedagogy #STEMeducation #EdTech #DigitalEngineering #EngineeringDesign #AcademicResources #StudentSuccess #SummerLearning #Vibrations #OnlineLearning #MechanicalDesign #EngineeringTheory #EngineeringSimulation #TeachingWithTechnology #PreStressModalAnalysis #GeometricStiffness #MaterialStiffness #Buckling #ShapeFunctions #DropTest #Impact #AIinEducation #CodesAndStandards