Visualizing Propeller Design with Algebraic Scissors and Pythagorean Blades

My paper "Multigrid-inspired, selective mass scaling scheme for explicit dynamic analysis" has been published in Computer Methods in Applied Mechanics and Engineering. The paper describes an approach for increasing the critical timestep of explicit solvers without modifying the lower, physically meaningful dynamic modes of the computational model. The key to this end is the establishment of an Auxiliary Coarse Mesh (ACM) fully containing the actual finite element mesh of the explicit computation. By decomposing the nodal displacement vector of the actual mesh into two parts, i.e. an ACM part and a "finer-mesh correction" part, and then applying mass scaling only for the latter, it is possible to increase the critical timestep (so that it is controlled by the ACM element size), without scaling the part of the kinetic energy corresponding to the lowest modes of dynamic response (the latter can be captured by the ACM).  The paper can be accessed for free for a limited time using the following link: https://lnkd.in/d2s-7kAr The capability of the Multigrid-based mass scaling (MBMS) approach is verified in the paper with one-dimensional and two-dimensional analyses. We have also implemented the method in our research code FEMultiPhys, and have begun using it for more "challenging" simulations with very good results. An example animation attached, involves analysis of a reinforced concrete wall under lateral load. It is noted that, while the use of conventional mass scaling (to increase the timestep by a factor of 12) completely distorts the load-displacement response (and damage patterns) in the analysis, the MBMS scheme enables speedups by factors as high as 24, without any significant distortion of the global response and damage patterns.

Deep Warning to CAD Engineers & AI Developers: Trigonometry Isn’t as Simple as You Think Most engineers know trigonometry from textbooks; most mathematicians treat it as abstract formulas. But here’s the catch: the real engineering depth exposes critical gaps that traditional mathematics misses. Sanjoy Nath’s Geometrifying Trigonometry (SNGT) reframes trigonometry not as static equations, but as dynamic geometric operators acting over line segments, triangles, and structures. Every operation rotation, projection, or angle calculation is non-commutative. The order matters. Ambiguity multiplies at each stage, and naive numeric substitutions fail to capture the system’s true behavior. For CAD programmers, AI researchers, and engineers building geometry engines or Godel-Cohen cycle simulations, this is a warning: Drawing & Cloning: Each geometric object carries a history of transformations; replicating or cloning it requires precise operator tracking. Operator Applications: Operators like A, B, C, D (and beyond) are not simple rotations/scales—they encode reference-based transformations, reflections, and conditional orientations. Arithmetic for Geometry: Using standard arithmetic blindly leads to misinterpretations; SNGT introduces operator-aware arithmetic that ensures geometric consistency. AI Implementations: Training models to generate or validate designs requires encoding these operators as matrices or transformations, else the AI will produce invalid geometries. Ignoring these nuances is not just a minor bug it propagates structural and computational errors in simulations, CAD designs, and automation systems. Engineers, CAD programmers, AI researchers: think beyond classic trigonometry. Consider the operational logic, the non-commutativity, and the multi-stage ambiguities. SNGT doesn’t just compute angles—it validates the geometry, physics, and logic simultaneously, generating robust cycles of validation like the Godel-Cohen cycle. In short: don’t treat triangles as NON numbers. Treat them as NON COMMUTATIVE operators. Treat trigonometry as a NON COMMUTATIVE OPERATOR ALGEBRA LIKE system. #EngineeringMathematics #CADProgramming #AIinEngineering #GeometrifyingTrigonometry #SanjoyNath #GodelCohenCycle #GeometricOperators #NonCommutativeMath #DeepEngineering #StructuralDesign #AIforCAD #TrigonometryReimagined

🚀 Day 54 of Learning with GeeksforGeeks ✒️ Course name: Engineering Mathematics 🔗 Course link: https://lnkd.in/e-4XD_-i 💻 Topic: Multivariable Calculus - 2 📖 No. of articles: 1 ❓ No of questions(quiz): 6 🎯 Marks obtained: 25 Exploring the beauty of Mathematics in Motion! 🔄📐 The article delved into the Surface Area of Solids of Revolution, a fascinating concept where a curve rotates around an axis to form a 3D shape — and we calculate its surface area through integration. This topic deepened my understanding of: 🔹 Surface area formulas for rotations around X-axis and Y-axis 🔹 Different coordinate forms — Cartesian, Parametric & Polar 🔹 Real-world applications in Mechanical, Civil & Aerospace Engineering 🔹 How calculus connects theory with practical design and geometry Learning how mathematics shapes the real world — from aircraft design to structural engineering — makes the process all the more exciting! 🚀 #skillupwithgfg #nationskillup #Mathematics #Calculus #Engineering #STEM #LearningEveryday #ProblemSolving

The Department of Geospatial Science and Technology, Delhi Technological University (Formerly DCE), organized a two-hour hands-on workshop in collaboration with DesignTech Systems Pvt. Ltd., a leading provider of CAD 🧩, CAE 🧮, PLM ⚙️, and 3D Printing 🖨️ technologies, and an official MathWorks reseller for MATLAB & Simulink 💻. Participants — students and Ph.D. scholars from Geoinformatics and Geospatial Science 🌏 — explored: 🔹 Fundamentals of MATLAB and its Live Script interface 🧠 🔹 Data visualization and image processing applications 📊🛰️ 🔹 Machine Learning and classification using both code and no-code tools 🤖 🔹 Self-paced learning through MATLAB Onramp courses 🎓 The department extends heartfelt thanks to ARMAN ANSARI, Application Engineer, DesignTech Systems Pvt. Ltd., for delivering an insightful and engaging session 👏. The workshop empowered participants to leverage MATLAB for geospatial data analysis, visualization, and automation, bridging computational learning with real-world applications. 🔖 #MATLAB #Simulink #DesignTech #MathWorks #DGST #DTU #GeospatialScience #Geoinformatics #RemoteSensing #ImageProcessing #MachineLearning #DataVisualization #AI #Automation #Innovation #STEM #HigherEducation #Workshop #SpatialAnalytics #Research #DTUEvents #LearningByDoing #TechnologyIntegration

🔥 Can we model battery temperature accurately without complex 3D simulation? A new paper from Mark Blyth and Alastair Hales (University of Bristol, The Faraday Institution) presents a powerful new way to capture battery thermal behaviour without resorting to full 3D models and expensive software licenses. 🧩 The challenge - Standard ECMs (equivalent circuit-based models) often assume a uniform temperature, but real cells develop gradients that affect voltage, lifetime, and safety. Discretised thermal models can capture this, but parameterising them correctly is complex and time-consuming. ⚙️ The breakthrough - This study reverses the usual process. Instead of measuring thermal parameters directly, it trusts the electrical model first and fits thermal behaviour using voltage data. The result is an effective heat transfer coefficient that captures how a real, non-uniform cell behaves. This is a step towards using only voltage data to capture complex battery behaviour. 🔍 Why it matters - Predicting temperature is one of the most critical aspects of battery design and modelling, influencing lifetime, safety, and fast charging. This method bridges detailed physics and engineering practicality, enabling complex thermal simulations directly in Python or MATLAB Simulink. 👉 Read the paper: Thermal parameters fitted from electrical data enable lumped models of heterogeneous battery cells by predicting their effective temperature.  https://lnkd.in/eFp5FxJm This works was done as part of the Multi-scale modelling project, and ties closely to our ongoing collaboration with University of Bristol on Battery Model Validation Standards, funded through the Faraday Institution

After mastering Forward Kinematics, Students feel invincible! "I can calculate where my robot will be for any joint configuration! This is easy!" Then I give them the inverse problem: "I want the robot's hand HERE. What joint angles achieve that?" Silence. Because Inverse Kinematics (IK) is everything Forward Kinematics (FK) isn't: ✅ FK has one solution. IK might have zero, one, or infinite solutions. ✅ FK is straightforward calculation. IK requires solving nonlinear equations. ✅ FK always works. IK can fail for unreachable points. ✅ FK is fast. IK can be computationally expensive. This is where robotics students learn that real engineering isn't about having answers - it's about navigating problems with no guaranteed solutions. We approach IK systematically: ✅ Understanding the Problem Space - Why can the same end position come from multiple joint configurations? What does "elbow up" vs "elbow down" mean? When is a position unreachable? ✅ Analytical Solutions - For simple geometries (2-3 joints), we can derive closed-form solutions. Students work through the trigonometry, understanding the geometric relationships. ✅ Numerical Methods - For complex robots, we use optimization approaches. Jacobian matrices. Gradient descent. Iterative refinement. The math gets sophisticated, but the concept is accessible: adjust joints until you're close enough. ✅ Singularities and Edge Cases - What happens near workspace boundaries? When multiple solutions exist, how do you choose? What about joint limits? I watch the same pattern every cohort: Students who rushed through FK struggle badly with IK. They don't have the geometric intuition. They can't visualize why a solution doesn't exist. But students who deeply understood FK? They approach IK with geometric reasoning. "This position is outside the workspace because even with all joints fully extended, we can't reach that far." They sketch solutions before coding them!

Segment 3: Applied Physics Panel Discussion Amelia: Captain Mitchell, from a test pilot perspective, how does propeller geometry affect aircraft handling? Captain Mitchell: Every blade’s angle, pitch, and rotational velocity is crucial. In F-15 Eagles, even though propellers aren’t used in jets, the same principle applies to rotor systems and turbine intake vanes. Pythagorean-based thrust calculations ensure balanced vectors, reducing torque-induced roll and improving maneuverability during high-G turns. ⸻ Dr. Helen Carter: Adding to that, applied physics of blade interaction with air involves: 1. Pressure differentials (Bernoulli principle) 2. Lift vectors and resultant thrust 3. Energy dissipation through blade material elasticity When designing blades, we use geometric equations to model: • Vortex formation • Drag minimization • Positive additive currents (“+C”) in airflow ⸻ H.J. Miller: Mathematically, propellers are a dynamic system of rotating vectors. • Algebraic methods are sufficient for low-speed training aircraft • Pythagorean and trigonometric analysis is required for high-speed, multi-blade turbines • Optimization involves solving differential equations for torque, RPM, and airflow efficiency ⸻ Dr. James Hannah: We also benchmark fuel efficiency and structural stress using these geometric models. For example, increasing the number of blades distributes torque, which reduces vibration and improves long-term structural stability — directly predicted by vector geometry and Pythagorean equations. ⸻ Segment 4: Educational Insights Amelia: How can we integrate this science into pilot and aircraft engineering training? RSA: We need to teach students both: • Algebraic reasoning — understanding basic thrust, lift, and energy vectors • Geometric/Pythagorean reasoning — for advanced turbine and high-speed aircraft modeling By blending linguistic mechanics (how components interact) with applied physics, trainees learn to anticipate airflow patterns, torque effects, and mechanical efficiency. Captain Mitchell: Exactly. Our Rider Training School simulators now include propeller vector modeling, so pilots see not just lift, but the invisible “C” currents in real time.

CFD Learning Roadmap Getting started with CFD and feeling overwhelmed? Here is a simple learning roadmap to guide you! 1. Begin with the fundamentals of fluid mechanics. Understand flow behavior, velocity, pressure, viscosity, and types of flows. 2. Learn the difference between Eulerian vs. Lagrangian descriptions of motion and get a solid conceptual feel for fluid acceleration. 3. Build intuition around the Navier–Stokes equations. Don’t worry about solving them at first, just understand what they represent. 4. Study partial differential equations and the role of boundary and initial conditions in fluid problems. 5. Explore numerical methods and how they help us approximate solutions of governing equations. 6. Start with Finite Difference Method (FDM) and solve simple numerical problems to grasp discretization using pen and paper. 7. Move on to Finite Volume Method (FVM) — the backbone of most industrial CFD solvers. 8. Download a beginner-friendly CFD software (student versions are free) and start experimenting. 9. Focus on simple 2D laminar cases to build confidence; avoid jumping straight into complex simulations. 10. Follow beginner tutorials, recreate benchmark examples, and learn step-by-step. 11. Gradually introduce turbulence. Understand RANS, LES basics, and when to use them. 12. Learn proper meshing practices and perform mesh-independence studies. 13. Understand convergence, monitor residuals, and learn what “good results” look like. 14. Study verification & validation — trust your simulation only after you test it. 15. Stay active in CFD forums and communities. Learning accelerates when you ask and discuss. Don’t rush. CFD mastery takes time. #cfd_fundamentals, #cfd

Physics Simulation–Enhanced Augmented Reality (PSE-AR) This project explores how integrating physics simulation into augmented reality can reshape architectural education. In many design studios, gravity and material behavior are treated as abstract ideas. PSE-AR bridges this gap by linking 3D modeling with real-time physical simulation, allowing geometry to move, balance, and respond as it would in the real world. Through this approach, students learn to design with gravity instead of against it. Every modeling decision becomes a conversation between intention and physical force, helping students understand how stability, rhythm, and assembly emerge through interaction rather than control. PSE-AR encourages a new way of thinking about architecture—not as a collection of fixed forms but as a living process shaped by feedback, motion, and adaptation. It combines visual intuition with computational reasoning, turning digital models into experiments that reveal how matter behaves under natural laws. By merging simulation and design, the project promotes a deeper awareness of how structure and environment are connected. It offers a framework for learning that values exploration, responsiveness, and collaboration between imagination and the physical world.

Non-reciprocal three-dimensional mechanical metamaterials by Qingxiang Ji, Jinliang Wang, Brahim Lemkalli, Gwenn Ulliac, Changguo Wang, Sébastien Guenneau, and Muamer Kadic Journal: Journal of the Mechanics and Physics of Solids (2025) DOI: 10.1016/j.jmps.2025.106403 🔍 Key Ideas The paper introduces 3D mechanical metamaterials that break reciprocity, meaning their mechanical response differs when loads are applied in opposite directions — a property not seen in conventional materials. Highlights from the paper: ✅ 3D Non-reciprocal design: The authors design and fabricate (numerically and/or experimentally) metamaterials whose internal geometry leads to direction-dependent elastic behavior. 🧩 Analytical model for asymmetry: A new analytical framework is presented to explain asymmetric deformation, describing how the geometry leads to different stiffness or strain responses depending on load direction. ⚙️ Directional elasticity: The elastic moduli (like Young’s modulus or shear modulus) are non-symmetric with respect to direction — a hallmark of non-reciprocal elasticity. 🎚️ Tunable band gaps: The material’s phononic band structure (wave propagation characteristics) changes when the loading direction is switched — allowing for mechanical or acoustic tuning without altering the structure itself. 🌍 Scientific Context Traditional materials obey Maxwell-Betti reciprocity, meaning the deformation response is symmetric with respect to the direction of applied forces. Non-reciprocal metamaterials violate this symmetry, offering new ways to control wave propagation, energy transmission, and mechanical isolation. Extending these effects to 3D structures (as opposed to 1D or 2D) is a major step forward for applications in vibration control, soft robotics, and adaptive structures. 📘 Potential Applications Directional shock absorption or impact isolation Mechanical diodes that allow deformation in one direction only Tunable phononic crystals or vibration filters Programmable mechanical logic systems