Comp Geo 02¶
BETA CAE
1.
/ > ... Generative AI is currently displacing large numbers of human artists—for example, industrial product designers, graphic designers, and photographers—and will inevitably also displace many junior software developers. This raises issues of intellectual-property provenance. At present, when ChatGPT o4 answers questions in mathematical theory, it appends the books and academic papers it referenced; however, when generating code it does not attribute the code authors it drew upon, and other generative AIs likewise do not credit the human works they referenced when producing images or short videos.
/ > The author has long studied generative AI; for image, video, and 3D-model generation, AI systems are capable of tracing provenance and identifying which human works inspired the current content.
/ > However, no AI company offers provenance functionality; instead they half-deceive, half-market by claiming that AI-generated works are independent of human intellectual property. This is especially severe in code generation. Generative AI relies on so-called reinforcement learning—improving system performance through interaction with human users. Thus, if a researcher invents a new algorithm and teaches it to the AI via reinforcement learning, the ownership of that invention cannot be safeguarded.
/ > Some of the author's students work on large models at top IT companies in North America. They often complain to the author about being exploited by their companies, developing new AI models under intense pressure, and deeply feel they are "digging their own graves." At present, Microsoft has tied its compiler to Copilot, so programmers' foundational coding skills will inevitably deteriorate rapidly, leaving them dependent on AI to program. In this way, the IT industry will inevitably be monopolized by AI giants. Preventing AI monopolies and safeguarding human dignity and rights is becoming increasingly urgent. It is not that there are no technical means to safeguard this; rather, what is needed is foresight—establishing corresponding laws and institutions.
https://mp.weixin.qq.com/s/YwsMI-FRZrxLfFL1tRZKEA
2.
/ > The global obstructions encountered when extending local geometric constructions to a global setting are often characterized, in modern geometry and topology, by some cohomology theory.
/ > ... The computation of the Riemann–Roch theorem has long remained an open problem. As society's demand for industrial software has grown, we have come to recognize that the theoretical foundation of structured mesh generation lies in the Riemann–Roch theorem. That is, the quadrilateral mesh of a surface is a global meromorphic section of a certain specific holomorphic line bundle, and the space of quadrilateral meshes satisfying particular singularity conditions is characterized by the Riemann–Roch theorem. Finding a basis for this space has become a key problem in computational conformal geometry, and naturally also a key problem in CAD and CAE. The traditional algebraic-geometry approach represents a Riemann surface as an algebraic curve (an algebraic variety, i.e., the zero set of a bivariate polynomial equation), and then computes a specific ideal in the polynomial ring by algebraic methods. In real engineering practice, the input surfaces are surfaces of physical entities—mostly surfaces obtained by 3D scanning or spline surfaces from CAD design. There is currently no mature algorithm to convert these actual surfaces into algebraic curves, and even if such a conversion were possible, its numerical error would be hard to control. Therefore, the algorithm we developed performs numerical computation directly on triangular meshes based on geometric-analysis methods, involving complex topological operations and the solution of geometric partial differential equations, and the results can be applied directly to industrial geometric models.
In the invention of the algorithm, it was primarily human logical thinking and geometric intuition that played the decisive role.
Of course, in the implementation of the algorithm, we used AI to write some low-level basic numerical routines, which greatly improved development efficiency.
/ > In the office of one of BETA CAE's founders, the author saw Professor Tom Hughes's three-volume magnum opus "Computational Mechanics," and this founder also admired the isogeometric analysis method founded by Professor Hughes. Hughes himself has stated publicly many times that the theory and engineering algorithms of isogeometric analysis are relatively complete, and that the true bottleneck lies in the generation of structured hexahedral meshes. Several technical directors at BETA CAE deeply agreed, uniformly holding that hexahedral-mesh algorithms are crucial. The academic community has long disputed this:
/ > Many schools of thought hold that structured hexahedral meshes are not very necessary, and that unstructured tetrahedral meshes are sufficiently practical. Meanwhile, academia publishes many papers each year on hexahedral mesh generation, yet none meet the practical requirements of industry.
/ > They were all very interested in the hexahedral mesh generation algorithm that the author's team developed based on surface foliation theory, and considered this direction promising. At the same time, they have begun exploring topology optimization, applying this technique to automobile weight reduction. They mainly use the traditional SIMP method, optimizing the material distribution density to achieve design goals; however, the resulting designs lack clear boundary surfaces and require denoising. In particular, they need mesh-to-spline-surface conversion technology, which requires globally structured quadrilateral mesh generation on surfaces—something their current technology cannot achieve.
/ > BETA CAE has a great many long-term, stable customers. They told the author that the automotive design and production cycle is about four years, and every new model is derived from an old model through local adjustments and optimizations. To ensure safety, each iteration is relatively conservative, and every improvement requires extensive simulation. New models must pass very stringent standard validations; physical experiments are extremely expensive, requiring several real cars to be destroyed, so numerical simulation is indispensable. For this reason, every automaker has trained many simulation engineers, and the various parameter designs and operating-condition designs of simulation software are very complex, requiring years of accumulation; therefore, automakers will not easily switch simulation software unless absolutely necessary. And each automaker's ANSA licensing cost reaches several hundred thousand dollars, so BETA's revenue is very stable.
https://mp.weixin.qq.com/s/YwsMI-FRZrxLfFL1tRZKEA
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3.
The Lattice Boltzmann Method (LBM) in CFD | SimWiki | SimScale https://www.simscale.com/docs/simwiki/cfd-computational-fluid-dynamics/lattice-boltzmann-method-lbm/
/ > Unlike the traditional Navier-Stokes-based methods, which may experience limitations when addressing complex geometries like porous materials, LBM shines. Specifically, LBM's intrinsic discrete framework provides it with the ability to efficiently simulate flows within intricate structures, making it a favorable choice for a range of complex simulations.
/ > NUMECA was also founded by a professor specializing in turbines, and for decades has focused exclusively on turbine simulation. Many airlines adopt their technical solutions, with annual fees reaching several hundred thousand dollars. The author had in-depth exchanges with NUMECA's engineers, who have profound insights and rich experience on many issues. Academia rather favors the Lattice Boltzmann Method (LBM) for computational fluid dynamics (CFD); the author asked for their views. They believe that although LBM is touted as good at handling fluid mixing, its computational efficiency is low and it is immature for handling complex cases involving phase changes. They are reserved about topology optimization; the core reason is that turbine blades bear very large stresses, and current metal 3D-printing technology cannot produce blades that meet strength requirements, so topology-optimized designs cannot be manufactured. They are also reserved about whether AI technology can replace human programmers, because turbine simulation requires substantial physical intuition and mathematical foundation; AI can help preliminarily predict results, but cannot replace precise simulation. They have written multiple versions of CFD algorithms, initially based on structured hexahedral meshes, later on unstructured and hybrid meshes. They believe structured meshes still offer the highest computational efficiency and accuracy, but generating globally structured meshes remains the most difficult problem, forcing them to settle for hybrid meshes. Similarly, their mesh-generation algorithm is based on spatial octree decomposition, and currently only hybrid-mesh generation is available—they cannot obtain globally structured meshes. They believe there is great application value in converting an initial dirty-geometry, trimmed-NURBS-based CAD model into a globally watertight CAD model; the algorithm for unfolding curled surfaces is critical; and the generation algorithm for all-hexahedral meshes has always been the core challenge of the turbine simulation industry. The author introduced to them surface mesh generation based on Ricci flow, surface quadrilateral mesh generation based on the characteristic class of holomorphic line bundles, and hexahedral mesh generation based on surface foliation theory, all of which elicited strong interest and a desire for collaboration.
/ > As the "Holy Grail" problem in mesh generation, hexahedral mesh generation has intrinsic difficulty. Although there are now many empirical algorithms, its rigorous theoretical foundation has yet to be established. Many fundamental questions remain to be answered: for example, what equation does the singular set of a hexahedral mesh satisfy? The author believes that some theories and methods recently developed in low-dimensional topology can help understand this problem more deeply. For example, Thurston once proposed the Virtual Haken Conjecture: a finite cover of a hyperbolic 3-manifold is Haken, i.e., there exists an essential embedded 2-dimensional surface inside the 3-manifold. An essential surface satisfies the following condition: if a loop on the surface cannot be contracted to a point, then it also cannot be contracted to a point inside the 3-manifold. Hyperbolic 3-manifolds contain many immersed essential surfaces, and the essence of the Virtual Haken Conjecture is that there exists a finite cover of the 3-manifold that unfolds the curled, self-intersecting essential surfaces into embedded surfaces. Mathematicians have proven the Virtual Haken Conjecture, and the theory and methods developed for it provide essential help for understanding the curling and intersection of surfaces inside 3-manifolds.