​Keizo Ishii (Representative Director of Quint Co., Ltd.), who was invited as a lecturer at the career path seminar “Creating Markets: Software Development in Unexplored Areas” held on October 11, 2019, gave a lecture on mathematics. We asked students and faculty members who majored in

Continuation of (1)​

バイクのパーツ軽量化で採算クリア

1989年にトポロジー最適化の商用ソフトウェアOPTISHAPE（オプティシェイプ）を作り、日本中のいろいろな会社に魅力を説明して回りました。手厳しいことを言われて落ち込むこともありましたが、1500年代の戦国時代に布教に来た宣教師フランシスコ・ザビエルの心境で、「諦めてはいけない」と気持ちを奮い立たせて、次の会社に向かいました。

And then we arrived at Tochigi, a Japanese automaker that once dominated the F1 race at the end of the 1980s with two genius drivers, Ayrton Senna and Alain Prost. That person bought it saying, "I don't know if it's useful, but it's interesting anyway!" It was my first contract.

ただひとつ心配だったことはOPTISHAPEがそれなりの価格でしたので、買うという決断をした方が責任を問われてクビになっていないかと。それが怖くてなかなか連絡できなかったのですが、半年後におそるおそる電話をしてみると、「あー、あの件は既にペイしたので心配ないですよー」と明るく言ってくださって、半年間の悶々とした思いがすっと消えて、思わず嬉し涙が出ました。

So what did that automobile company do? It was a world-class motorcycle manufacturer. There's an aluminum part like a mold ruler. It is a shape with a hole in it.

それをOPTISHAPEでトポロジー最適化を試みた結果、穴のパターンが変わり、パーツの剛性を確保しつつ27％の軽量化に成功しました。アルミは鉄の4、5倍値段が高く、軽量化できたパーツを100万個生産して浮いた材料費だけでソフトウェアの購入費はすぐに消化できたと言われました。「やっぱり頭のいい人はすぐに応用を考えるのだな！」と感激したことを覚えています。

天才芸術家は『テルマエ』主人公と同じ？

「天才は人の見えないものが見える」とよく言われます。ローマの観光名所カンピドリオ広場にミケランジェロが描いた直径10mを超える大きな円の絵があることをご存知でしょうか。私はトポロジー最適化でこの形が作れないか、考えてみました。

Fix the red center of this beige disk and apply a force to rotate it in the circumferential direction, like a clock at 12:01:02:03. At the same time, remove 3/4 of the mass of the beige part so that it can resist this force, and calculate the 1 quantile on the PC so that the remaining 1/4 of the material can be used to make the shape. comes out. Moreover, the number of sides is exactly the same. I was shocked by this. After all, the genius Michelangelo, what you see is different.

この話をある社外のセミナーで対談相手にお話ししたところ、「石井さん、それは違います」と言われました。僕が映画好きであることから日本映画の『テルマエ・ロマエ』を引き合いに出して、「ミケランジェロはこの映画の主人公と同じです。きっと現代に来てOPTISHAPEの計算画像を見て、自分の時代に帰ってからこれを作ったのですよ」と、そう言って譲らないんです（笑）。お世辞だとしても嬉しかったです。

その方は、八重洲ブックセンターのビジネス書売上No.1に輝いたこともあり、さすがに発想は面白いなと感心して、「そのアイデア、僕も使っていいですか？」と聞いてお許しが出たので、今日ここでお話ししています。

数学の知見に裏打ちされた形状最適化

In addition to topology optimization, which is a feature of OPTISHAPE, I will also talk about shape optimization, which is another feature of OPTISHAPE. A simple explanation of the difference between shape and topology is that structures with different numbers of holes (number of surfaces) have different topologies. Cubes like dice, rectangular parallelepipeds, and spheres have the same topological shape, but the topology changes only when they become donut-like shapes.

However, Professor Hideyuki Azegami, currently a professor at Nagoya University, has been conducting research for many years on how to obtain the desired performance by changing only the surface shape and using conventional manufacturing methods without changing the topology. While researching shape optimization for 30 years, Professor Azegami has devoted himself to verifying whether each question is mathematically correct and whether it converges.

OPTISHAPE makes use of that knowledge, and it's amazing that "things that are properly backed up by mathematics won't go bankrupt." With this technology, we have never lost in a common benchmark test against the world's strongest opponents. OPTISHAPE always wins. That's amazing.

Textbooks on mechanics of materials include a "three-point bending problem" in which both ends of a bar are clamped and a load is applied to the center. If we make this a little more difficult by making a structure with round holes, and try to solve this from the viewpoint of shape optimization, we will use the symmetry condition to create a model of only half of the rod, and then make a model of a cylinder for the rod. Assume that there is a hole and that the orange part of the cross section (see the figure below) should not be cut. You can cut other parts. If you give various conditions from here and optimize it, the result is that the round hole in the rod becomes a triangular truss shape.

Normally, overseas software cannot reach this level, but this is also an example of OPTISHAPE's ability to derive solutions depending on ingenuity.

ブレーキキャリパーの振動制御例

In the field of engineering, there is a long-standing research topic of "controlling natural vibrations." This is a difficult task, as there are many natural frequencies that can be used as indicators, and if you change the shape to increase one natural frequency, the values of other frequencies will decrease, just like whack-a-mole. Things don't go as planned. It is said that it is really hard to control just three.

という前提を踏まえて、これから紹介する例は「14個の固有振動数を全てある値にしてほしい」というクレイジーなリクエストに応えた事例です（笑）。

A car company came to us saying, "Our brake calipers make a very unpleasant sound when you apply full braking from a very high speed." When I tested it on a foreign car under the same conditions, it made almost no noise, and the designer thought that the first factor that came to mind was the brake caliper.

The brake caliper has a hydraulic piston and a large brake disc, and the shape of the disk disc, its surroundings, and the part where the round piston reciprocates cannot be manipulated. But the outside can be tweaked, which is exactly the problem of shape optimization.

我々が取り組んだことはまず、嫌な音がしなかった海外の車のブレーキキャリパーの加振実験から得られた固有振動数をいただき（表左側のexperiment）、それから、キャリパーの有限要素モデルを作り、固有振動数を計算し（表右側のFEM Analysis）、計測値に近づけることを目標としました。

As a result, the difference between all 14 natural frequencies and the measured values was less than 0.1%. I was able to take it. I just turned it around 15 times, and while the shape changed during that time, the numbers finally matched perfectly.

This can be achieved very easily by properly calculating the eigenvalues of the model in which OPTISHAPE formulates the dynamic phenomenon and the sensitivity, which is the ratio of changes in the evaluation function to minute changes in the design variables. Because there is Using a high-performance PC, the following results can be obtained in just 17 minutes. This is a story that can only be done because the reasoning is sound.

VOXEL（ボクセル）モデルの開発

Since we are a venture company, we mainly do things that other people do not do. The development of image-based modeling, structural analysis, and measurement software VOXELCON is one of them. VOXEL is a cube, a coined word that combines volume and pixel.

開発のきっかけはこうです。アメリカのビッグスリーの一社のピックアップトラックに使われているトランスミッションケースは、実は日本の広島の会社が設計していました。そのトランスミッションケースを切削加工するときに亀裂が入り、「設計ミスではないか？」とクレームが入り、同社のCAEグループに解析依頼が来たそうです。

In order to investigate the cause, we tried to analyze it, but in the end, we found that it took a huge amount of man-hours just to create an FEM model, and the person in charge was at a loss. Therefore, Professor Noboru Kikuchi of the University of Michigan, who was mentioned earlier, proposed that "If X-rays are applied with a CT, images of the complicated parts inside the transmission case will also appear." Using the VOXEL finite element method program newly developed at the University of Michigan, we applied a load and calculated the stress, and found nothing unusual.

When the results were fed back to the client and verified by the manufacturing department, it turned out to be a careless mistake that had not been processed according to the manual at the manufacturing site.

そこからです、これは面白いから製品化しようと。しかもただ製品化するだけでは面白くないのでマルチスケール解析をHomogenization method（均質化法）で行うソフトウェアも一緒にVOXELCONの中に入れました。今、このソフトに関していろいろなところから問い合わせや注文が来ています。最初の開発からほぼ四半世紀が経っていますが。（笑）

ベンチャー企業が斬新な商品を作ると25年や30年は我慢しないとなかなか花が開きませんが、花が開き始めると必ず世界の巨大な資本が参入してくるので結構大変なんですよ（笑）。

「くいんとセミナー」で最先端技術を紹介

我々の会社は今18人ですが、スタート当初は5人しかいませんでした。こんな小さい会社を世界に知ってもらうには、「よそではやってないことをやらなければ」と考えました。

Therefore, what we focused on was the deepening of the user's understanding of the various aspects inherent in CAE—the underlying theory and the equations that make up it. When using software, we tend to use such things as black boxes, but they are actually very important. Focusing on this, we decided to hold a unique seminar once a year inviting leading researchers in the relevant field from around the world.

これはもう、当初は全部会社の持ち出しですが、続けているとだんだんお客様が増えてきました。最近よく言われるのは「こういうセミナーって考えてみると、日本中どこの会社もやっていないよね」。それはそうです。Topology OptimizationやImage Based CAE、先ほどの畔上先生のH1 勾配法、それからHomogenization methodによるマルチスケール解析、これらの技術は全部、我々が国内でいち早く紹介してきたトピックス。「くいんとセミナー」で紹介した技術が15年後20年後に世界のスタンダードになっていきました。

Whether it's Adaptive Mesh Refinement or Topology Optimization, world powerhouses and ventures will enter one after another and compete for the market. The market for the homogenization method has already grown tremendously.

このようにいつも先頭を切ってやってしまうとビジネス的には窮地に陥るのですが、私はそれで世界が広がったのならよしとしようと考えるようにしています。まあ、「正直、負け惜しみ」ですが、そういう風に考えています。

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