JP3884294B2 - Design support apparatus and program - Google Patents

Description

【００２０】
応答曲面近似式を作成する過程においては、各変数が応答に与える影響度を多項式の成分に直交分解して評価する方法を用いることで分散分析を行なうことが可能である。ここで、影響度とは各変数の変動に伴う応答の変化の程度を表している。この方法により影響度は、応答曲面近似式を作成する過程で１次、２次などのように分解された特性に対して求めることができる。
【００２１】
このように当該ＳＴＥＰ０によって、対象とする現象あるいはシステムにおける設計空間内を領域分割し、それぞれの分割領域についての（部分的な）初期の応答曲面近似式が作成される。そしてこの初期の応答曲面近似式から、サンプリング点が観測データ（値）として算出され、後の処理（応答曲面モデルの更新：ＳＴＥＰ２）にて用いることができるようになる。なお、当該ＳＴＥＰ０により作成された初期の応答曲面近似式は、次に説明する新たな応答曲面モデルの想定にも利用され得る。
【００２２】
（ＳＴＥＰ１：各種の応答曲面モデルの想定）
非線形性を伴う複雑かつ大規模な現象やシステムであっても、変数変換により作成された多変数関数を新たな変数として再定義することにより、これらを線形１次式の応答曲面モデルとして近似的に表現できる場合が多いと考えられる。しかしながら、このような変数の再定義の仕方は多数存在するため、たとえある次数以下に限定した場合であっても、新たな変数となる多変数関数を有する応答曲面モデルは多数存在し得る。そのため、より高精度な応答曲面近似式あるいはより広い領域における応答曲面近似式を構築するにあたり、応答曲面モデルの想定を効率的に行い得る実用に則した方法が必要である。
【００２３】
設計変数データベース２０には、各設計変数が格納されるが、各設計変数は線形化可能な非線形関数、例えば指数関数、べき乗関数、有理関数、対数関数、ロジスティック関数などに対し事前に線形化を施しておいたものとしても良い。応答曲面モデルデータベース３０には、想定され得る各設計変数の多変数関数の集合が格納されている。
【００２４】
ＳＴＥＰ１では、この応答曲面モデルデータベース３０を参照し、複数の異なる応答曲面モデルを想定する。すなわち、応答曲面モデルデータベース３０から多変数関数を選択して新たな変数として再定義し、それらを、再定義した各変数の係数に関し線形１次式となるように組み合わせることにより想定する。
【００２５】
尚、応答曲面モデルの想定においては、特性値の単位系を考慮した次元解析により、多変数関数の選択を絞り込むようにしてもよい。または、応答曲面モデルを、初期の応答曲面近似式（設計変数の主効果のみを考慮）に、多変数関数を新たに再定義してなる１次の各種変数項を追加することによって想定してもよい。
【００２６】
本実施形態では、別領域において構築した応答曲面近似式から算出したサンプリング点あるいは新たに得られた数値実験点を観測データとして、ベイズ理論（参考文献(3)〜(7)）に基づくカルマンフィルタによりアップデート（ＳＴＥＰ２）しつつ、逐次、多重仮説検定による適合確率の指標をもとに各応答曲面モデルを算定する（ＳＴＥＰ３）。
【００２７】
変数変換により作成された多変数関数を新たな変数Ｘ_iとして再定義し、これを用いて想定した線形一次式の応答曲面モデルは、一般に、以下のように表現される。
【００２８】
【数２】

【００２９】
ただし、式中のＮは変数の数、aは未知パラメータを意味している。
【００３０】
ＳＴＥＰ２における応答曲面近似式のアップデートにおいては、別領域（他の部分領域）の応答曲面近似式をベースに算出されたサンプリング点、あるいは、新たに得られた数値実験点を観測値として用いる。ステップｋにおいて、観測値Ｙ₁，・・・，Ｙ_Nが得られた場合、次の観測方程式（３）を構成できる。
【００３１】
【数３】

【００３２】
ただし、ν_kはガウス性の白色雑音を示している。
【００３３】
ここで、パラメータ（a₁,a₂・・・,a₉）を状態量とおくと、式（４）の状態方程式が得られる。
【００３４】
【数４】

【００３５】
係数ベクトルａの事前分布をｆ（ａ）とする。得られたサンプリング点（Ｙ_ｋ，Ｃ_ｋ）をもとに、式（６）で表されるｆ（ａ｜Ｙ_ｋ）をベイズ理論を用いて推定することを考える。
【００３６】
【数５】

【００３７】
このとき、ａkの期待値は、有効推定（最小分散推定値）としての性質をもつ。状態量の期待値と分散を効率良く求めるアルゴリズムとして、本実施形態ではカルマンフィルタのアルゴリズム（参考文献(8)(9)）を用いる。
【００３８】
（ＳＴＥＰ２：カルマンフィルタによる応答曲面近似式のアップデート）
カルマンフィルタは最小２乗推定法の１つである。システム内の不確定量と観測の不確定量を含む状態方程式（８）と観測方程式（９）により、対象とするシステムを表現できる場合を考える。このとき、ステップｋにおける状態量推定は式（１０）、カルマンゲインは式（１１）、誤差共分散行列は式（１３）により算出される。
【００３９】
【数６】

【００４０】
ただし、ａ（ｋ）は状態ベクトル、ｙ（ｋ）は観測ベクトル、Ａ及びＣは行列、ｗ（ｋ）はシステム雑音、ｖ（ｋ）は観測雑音、ｋは離散時間系である。また、時間間隔をΔｔとすると、ｋΔｔが観測時刻であり、行列ＡおよびＣはモデル化により決定される。観測値ｙ（ｋ）が得られるごとにａ（ｋ）の推定値が求められる。初期の状態推定ベクトルの誤差共分散行列を仮定することにより、カルマンフィルタのアルゴリズム（参考文献（３））を用いてパラメータ（ａ₁，ａ₂・・・，ａ₉）の推定値を求め、応答曲面近似式をアップデートすることが可能である。
【００４１】
複数の応答曲面近似式からサンプリング点を算出する過程においては、ステップｋにおいて、各近似式から算出するサンプリング点の内訳つまり観測値ベクトルの要素の内訳をコントロールするよう実施形態を構成してもよい。例えば、応答曲面近似式の精度を高めたい領域付近において、サンプリング点内訳比率を高めることが好ましい。あるいは、全ステップにおける各近似式から算出するサンプリング数の内訳比率をコントロールすることも好ましい。この場合において、交差確認法（Cross-Validation法）やＢｏｏｓｔｒａｐ法などの再サンプリング手法を利用することが好ましい。
【００４２】
（ＳＴＥＰ３：ベイズの定理を用いた多重仮設検定）
ベイズの定理を用いた多重仮設検定法は、プラントなどの異常兆候判定法として福田・清水らにより提案・適用され有効性が示されている（参考文献(3)(4)(6)(7)）。本実施形態では、ベイズの定理に基づく多重仮説検定法により、応答曲面モデルの適合判定を行う。具体的には、応答曲面近似式がアップデート（ＳＴＥＰ２）されるごとに多重仮説検定を行い、各応答曲面モデルの有意性を算定する。
【００４３】
ベイズの定理は式（１６）のように定式化されている。
【００４４】
【数７】

【００４５】
ただし、Ｐ（Ｅ_i）は事象Ｅ_iの事前確率、Ｐ（Ｅ_i｜Ａ）は事象Ｅ_iの事後確率（事象Ａのもとで特定の事象Ｅ_iが生起する確率）、Ｐ（Ａ｜Ｅ_i）は事象Ｅ_iのもとで事象Ａの起こる確率である。この式から、事象Ａという条件下で特定のＥ_iが生起するたびに、事象Ｅ_iの事後確率Ｐ（Ｅ_i｜Ａ）を更新すると、逐次的に正しい確率に漸近していく。すなわち、はじめは過去の経験に基づく判断や僅かな観測データに基づいてシステムの状態を推定せざるを得なかった場合であっても、新たなデータを得るたびに事後確率を修正して、正しい確率へ近づけることができる。ベイズの定理を多重仮説検定に応用し、カルマンフィルタのアルゴリズムによりアップデートされた応答曲面近似式の残差（観測値と推定値の差）から、観測システムの状態を逐次的に推定できる。
【００４６】
カルマンフィルタから算出される残差は、状態方程式と観測方程式から記述されるモデルが、対象とする現象あるいはシステムと一致している場合、平均０の正規分布に従う。応答曲面モデルiに対するカルマンフィルタから出力される残差γ(k)は、平均０、共分散Ｖ_i（ｋ）の正規分布Ｎ（0,Ｖ_i（ｋ））に従う。残差γ(k)がγ+dγの間に出現する確率は式（１７）のように表現できる。
【００４７】
【数８】

【００４８】
この確率は式（１６）のＰ（Ａ｜Ｅ_i）に相当する。
【００４９】
ただし、ｍはｙの次元、Ｖ_i（ｋ）＝Ｅ（γ_i ^T（ｋ）γ_i（ｋ））である。
【００５０】
故に、事後確率Ｐ_i（ｋ）（＝Ｐ（Ｅ_i｜Ａ））は式（１８）のように表現できる。
【００５１】
【数９】

【００５２】
ここでいう多重仮説検定とは、検定対象となるシステムに対していくつかのモデルを仮定し、観測中のシステムがどれに該当するかを確率論的に判定する方法であって、この方法は以下の手順からなる。
【００５３】
（１）応答曲面モデル1,応答曲面モデル2,・・・,応答曲面モデルnを設定する。
【００５４】
（２）仮説Ｈ_iすなわち「観測中のシステムは応答曲面モデルi(i=1,2,・・・,n)に適合する」をたてる。この事象をＥ_ｉとする。
【００５５】
（３）「仮説は正しいＴ(True：真) として、その仮説がＨ₁,Ｈ₂,・・・,Ｈ_nである確率Ｐ（Ｅ₁｜Ｔ），Ｐ（Ｅ₂｜Ｔ），・・・，Ｐ（Ｅ_n｜Ｔ）を計算する。
【００５６】
新たなデータが与えられるたびに上記（１）〜（３）の計算を繰り返し、最大値をとり、かつ予め設定しておいた閾（しきい）値を超える仮説を採択する。これにより、適合確率を指標にして各種応答曲面モデルを逐次算定することができる。
【００５７】
線形回帰モデルの適合性や冗長性の比較において、対象とする空間内の未知データに対しても有効に働くような汎化能力を評価するための簡便な手法として、ＲｉｓｓａｎｅｎのＭＤＬ(Minimum Description Length)や赤池のＡＩＣ(Akaike’s Information Criterion)などの情報量基準（参考文献（１０）（１１））が知られている。本実施形態では、上記ＭＤＬの値を計算することが可能なように構成される。ＭＤＬは、Ｒｉｓｓａｎｅｎにより符号化における記述長最小化(Minimal Description Length)原理として導出されたものであり、式（１９）のように定義される。
【００５８】
ＭＤＬ＝ −（最大対数尤度）＋（Ｎ／２）・ｌｏｇＰ＝ −（Ｐ／２）・ｌｏｇ（ＳＳＲ／Ｐ）＋（Ｎ／２）・ｌｏｇＰ …（１９）
ここで、Ｎはモデルの自由度、ＳＳＲは残差平方和、Ｐはサンプル数を示している。第１項からモデルの近似誤差を評価でき、第２項から自由度の大小を評価できる。これらは、近似誤差が小さく且つ自由度の小さいモデルを算定するための指標として有効である。そこで、情報量基準値を指標に、上述したＳＴＥＰ２において想定する各種応答曲面モデルを絞り込むようにしたり、あるいは、情報量基準値の指標と多重仮説検定とを併用して応答曲面モデルの算定を行なうように実施形態を構成することが好ましい。
【００５９】
以上説明した本実施形態の設計支援システムは、新たな設計空間データが得られるたびに、応答曲面空間の拡張あるいはより高精度な応答曲面近似式を作成するなどといった応答曲面近似式の更新を、実験計画や解析を実行し直す多大な労力を必要とせずに容易に行うことができ、実用的である。
【００６０】
これにより、対象とする設計空間内における各変数の相互作用項に関する有益な情報を容易に抽出することができるようになる。具体的には、設計を繰り返すごとにＣＡＥ技術を用いた解析結果が設計空間に蓄えられるにつれ、積極的に設計空間情報（例えば変数の相互作用情報）を抽出することができるようになる。これは、設計を効率的に行うためのノウハウを蓄えるという意味において極めて有効である。
【００６１】
以下、より具体的な本発明の実施例を説明する。
【００６２】
［実施例］
電子機器の軽薄短小化が加速し、搭載部品に対する小型化の要求が一段と高まっている。こうした背景のもと、高密度実装が可能なフリップチップ接続技術の開発が活発に行われている。
【００６３】
図４に示すように、圧着工法によるフリップチップは、接続用樹脂４２を介してチップ４６のスタッドバンプ４４とＴＡＢ電極４３を熱圧着し接続を行う。接続信頼性を確保するためバンプ４４の周辺を樹脂４２で固着・封止する構造をとっており、樹脂選定はフリップチップ実装の高信頼性化にとって重要なアイテムである。また、圧着工法によるフリップチップの場合、ボンディング荷重（Ｂ’ｇ荷重）ｆ１および樹脂４２の硬化収縮・応力緩和・熱収縮により最終的なバンプ反発力ｆ２つまり接続マージンが決まる。樹脂選定やＢ’ｇ荷重の設定によっては、高温・吸湿試験において樹脂の膨潤により接続オープン（電気抵抗の上昇）が発生する場合がある。長期信頼性を確保するためには、樹脂４２の特性やＴＡＢ（又は基板）４０の材料特性の影響度の明確化、およびＢ’ｇ荷重ｆ１の適正化が重要課題になる。
【００６４】
上記実施形態にて説明したＳＴＥＰ０に従い、樹脂４２の特性及びＴＡＢ４０の材料特性ならびにＢ’ｇ荷重ｆ１を設計変数とするバンプ反発力ｆ２について初期の応答曲面近似式を作成した結果を以下に示す。
【００６５】
（初期の応答曲面近似式）
高温吸湿後の金バンプ／パッド界面圧縮荷重（ｇｆ） = -96.4+0.679 X₁+0.00132 X₁ ²+4.64 X₂-0.500 X₂ ²+22.9 X₃-3.89 X₃ ²+1.16 X₄-0.00593 X₄ ²
本応答曲面近似式は、ＴＡＢ弾性率が２〜６Ｇｐａを対象に近似式を構築したものである。以下、（実施例１）基板弾性率が１２ＧＰａの場合の解析データが新たに得られた場合の応答曲面近似式の更新例、（実施例２）基板弾性率が８〜１２ＧＰａの範囲において新たに得られた応答曲面近似式をベースに算出したサンプリング点を観測データとした場合の２〜１２ＧＰａの全領域に対する応答曲面近似式の構築例をそれぞれ示す。
【００６６】
［実施例１］新たな数値実験点による更新例
図５に示すように、実施例１は、基板弾性率が１２ＧＰａの場合の解析データが新たに６個ほど（他の変数については一様乱数値にて設定）得られた場合について、初期の応答曲面近似式（基板弾性率２〜６ＧＰａ）を、本発明に従い、基板弾性率２〜１２ＧＰａを対象にした応答曲面近似式に更新することを試みる。ここで観測点は、新たに得られた数値実験点６個と、部分領域（基板弾性率２〜６ＧＰａ）において作成した初期の高精度な応答曲面近似式をもとに得た１８データを逐次、各ステップにおいて算出することにより与えた。なお、図５において、５０は応答曲面近似範囲の拡張前を示し、５１は応答曲面近似範囲の拡張領域を示し、５２は各領域を包含した応答曲面近似範囲を示している。
【００６７】
以下のように、考慮する相互作用項の異なる７つの応答曲面モデルを対象に検証した。ここで、応答曲面モデル２〜７の相互作用項の係数は初期値０に設定した。
応答曲面モデル１：相互作用項なし（初期モデル）
応答曲面モデル２：Ｘ₁Ｘ₂を考慮した場合
応答曲面モデル３：Ｘ₁Ｘ₃を考慮した場合
応答曲面モデル４：Ｘ₁Ｘ₄を考慮した場合
応答曲面モデル５：Ｘ₂Ｘ₃を考慮した場合
応答曲面モデル６：Ｘ₂Ｘ₄を考慮した場合
応答曲面モデル７：Ｘ₃Ｘ₄を考慮した場合
ベイズの定理を用いた多重仮設検定法を上記７つの応答曲面モデルの適合判定に適用する。応答曲面モデルがアップデートされるごとに多重仮説検定を行い、各応答曲面モデルの有意性を算定する（図６参照）。
【００６８】
更新後の応答曲面近似式を以下に示す。
【００６９】

本実施例においては、実施形態にて説明した情報量基準（ＭＤＬ）の算出をサンプリング点に対する更新前後について行った。その結果を表１に示す。なお、表１における「ＲＳＭ」は「応答曲面モデル」の略記である。
【００７０】
【表１】

【００７１】
いずれの応答曲面モデルも初期に比べてＭＤＬ値は小さくなっている。多重仮説検定に基づく適合確率の変化を図７に示す。この結果から、応答曲面モデル２が適合度が良いことがわかる。近似式作成に用いた数値実験点以外の解析データ（一様乱数により数値実験点１０データ取得）により分散推定値を評価した結果を表２に示す。多重仮説検定結果と同じく、応答曲面モデル２の分散推定値が小さいことがわかる。本対象の場合、相互作用項Ｘ₁Ｘ₂を考慮すると汎化誤差が小さくなることを、本発明により算定できることがわかる。
【００７２】
【表２】

【００７３】
［実施例２］別領域の応答曲面近似式のサンプリング点による更新例
図８に示す要に、実施例２では、基板弾性率が８〜１２ＧＰａの範囲において新たに得られた応答曲面近似式をベースに算出したサンプリング点を観測データとして、基板弾性率２〜１２ＧＰａの全領域に対する応答曲面近似式を、初期の応答曲面近似式の更新によって構築することを試みる。
【００７４】
ここでは、基板弾性率が８〜１２ＧＰａの範囲（別領域）について、直交表と直交多項式により以下の応答曲面近似式が得られた場合について考える。
【００７５】
Y= -59.5+0.862 X₁ -0.000144 X₁ ² -2.38 X₂ + 0.103X₂ ²
+39.1 X₃-8.31 X₃ ²+0.500 X₄-0.00207X₄ ²
なお、図６において６０は初期応答曲面近似範囲を示し、６１は上記別領域の応答曲面近似範囲を示し、６２は各領域を包含した応答曲面近似範囲を示している。
【００７６】
サンプリング点については、上記別領域にて得られた応答曲面近似式が対象とする設計空間において一様乱数を発生させることにより得た１０データと、部分領域（基板弾性率２〜６ＧＰａ）において作成した初期の高精度な応答曲面近似式をもとに得た１０データを合わせた合計２０データを逐次、各ステップにおいて算出することにより得た。また、対象とする応答曲面モデルは、実施例１と同じく、考慮する相互作用項の異なる７モデルとした。該サンプリング点を観測データとして、カルマンフィルタによりアップデートを行った結果を以下に示す。
【００７７】

実施例２においても、情報量基準（ＭＤＬ）の算出をサンプリング点に対する更新前後について行った。その結果を表３に示す。
【００７８】
【表３】

【００７９】
いずれの応答曲面モデルも初期に比べてＭＤＬ値は小さくなっているが、各モデルの適合度の差が有意かどうかの判断が困難である。多重仮説検定に基づく適合確率の変化を図９に示す。この結果から、実施例１と同じく応答曲面モデル２の適合度が良いことがわかる。近似式作成に用いた数値実験点以外の解析データ（一様乱数により数値実験点１０データ取得）により分散推定値を評価した結果を表４に示す。多重仮説検定結果と同じく、応答曲面モデル２の分散推定値が小さいことがわかる。実施例１と同様に、相互作用項Ｘ₁Ｘ₂を考慮すると汎化誤差が小さくなることを、本発明により算定できることがわかる。
【００８０】
【表４】

【００８１】
なお、本発明は上述した実施形態に限定されず種々変形して実施可能である。
【００８２】
［付記］
下記は参考文献の一覧である。
【００８３】
記
（１） 柏村孝義,白鳥正樹,于強，機論，63-607，A(1997)，624．
（２） 柏村孝義,白鳥正樹,于強，実験計画法による非線形問題の最適化，朝倉書店，(1998)．
（３） 福田隆文・清水久二，安全工学，28-4，(1989)，211-216．
（４） 清水久二，設備安全工学−新しい検査・監視技術−，裳華房，(1989)．
（５） K. Watanabe, R.Patton, et al. (ed)，Pretice Hall，(1989)，411．
（６） 福田隆文・ほか2名，機論，59-560，C，(1993)，982-988．
（７） 福田隆文，設備の異常検出手法に関する研究，横浜国立大学 学位論文，(1995)．
（８） 星谷勝・斉藤悦郎，データ解析と応用，鹿島出版会，(1991)．
（９） 片山徹，応用カルマンフィルタ，朝倉書店，(1983)．
（１０） 坂元慶行・石黒真木夫・北川源四郎，情報量統計学，共立出版，(1982)．
（１１） 韓太舜・小林欣吾，情報と符号化の数理，岩波書店，(1994)．
【００８４】
【発明の効果】
以上説明したように、本発明によれば、設計空間データが複数の領域において応答曲面近似式として蓄積されるにつれ、設計空間情報の未知の部分を、積極的に新たな相互作用項を含んだ形の応答曲面モデルで表現し、能動的に相互作用項等に関する情報抽出を行うことができる設計支援方法及びプログラムを提供できる。
【図面の簡単な説明】
【図１】 本発明の一実施形態に係る設計支援システムの概略構成を示すブロック図
【図２】 設計空間における応答曲面近似を示すグラフ
【図３】 実施形態に係る設計支援システムの処理手順を示すフローチャート
【図４】 本発明の実施例に係る圧着工法によるフリップチップを示す図
【図５】 実施例１に係る応答曲面近似式の更新を説明するための図
【図６】 多重仮説検定による応答曲面近似モデルの算定の流れを示す図
【図７】 実施例１に係る多重仮説検定に基づく適合確率の変化を示すグラフ
【図８】 実施例２に係る応答曲面近似式の更新を説明するための図
【図９】 実施例２に係る多重仮説検定に基づく適合確率の変化を示すグラフ
【符号の説明】
１１…入力装置
１２…ＣＰＵ
１３…ＲＯＭ
１４…ＲＡＭ
１５…出力装置
２０…設計変数データベース
３０…応答曲面モデルデータベース[0001]
BACKGROUND OF THE INVENTION
The present invention relates to a design support method and program using response surface methodology.
[0002]
[Prior art]
With the rapid progress of computers, stress / deformation analysis, thermal analysis, and vibration / noise analysis can be performed relatively easily even for large-scale nonlinear problems. Against this background, in recent years, the response surface method has been widely used as an approximate solution in the field of design optimization and quality engineering because of its simplicity in theory and convenience as an approximation method. The response surface methodology is a methodology for constructing and utilizing a response surface approximation formula for a target phenomenon or system. The response surface approximation formula is an approximation of the relational expression for the response y predicted from n (n> 1) variables xi.
[0003]
Design optimization and reliability prediction using CAE (Computer Aided Engineering) is often used as an approximate solution space, and when one CAE calculation cost can be ignored, each numerical experiment is performed on the response surface approximation formula. By substituting with calculation, we try to perform optimization or reliability prediction at high speed. Here, a method of creating an approximate expression of a response surface with high accuracy while reducing the number of numerical experimental points by a method based on an experimental design method is used. Then, using a technique such as stress simulation, characteristic values such as stress values and deformation amounts are extracted, and a response surface approximation formula is created.
[0004]
[Problems to be solved by the invention]
However, in the analysis of complex and large-scale design space data with non-linearity, the phenomenon or system can be accurately described by a relatively simple approximation expression (for example, an approximation expression for only the low-order main effect term). Often difficult. In addition, when constructing the response surface approximation formula, there are few effective guidelines for the introduction of the interaction term of each variable unless information regarding the interaction is confirmed in advance. In this case, if all interaction terms are taken into account, not only the numerical experimental points increase exponentially as the number of variables increases, but also an approximation model with strong redundancy (large generalization error) is obtained. Cost.
[0005]
Conventionally, a highly accurate response surface has been created for each region by dividing the design space into regions. However, when expanding the response surface space, or creating a more accurate response surface approximation formula, it is often necessary to re-execute experiment planning and analysis, and new design space data Therefore, there is a need for a practical method that can easily update the response surface approximation equation every time. In addition, as design space data is accumulated as response surface approximation equations in multiple regions, the unknown part of the design space information is a response surface model that actively includes new interaction terms (response surface approximation equations). : It is important to construct a methodology to express information actively and extract information by Response Surface Model.
[0006]
The present invention has been made paying attention to such problems, and the object of the present invention is to obtain an effective guideline for introducing an interaction term between design variables in the construction of a response surface model. An object of the present invention is to provide a design support method and program that can expand the response surface space or facilitate the creation of a response surface approximation formula with higher accuracy.
[0007]
[Means for Solving the Problems]
In order to achieve the above object, a design support method according to the present invention is a design support method for supporting creation of a response surface approximation formula in a design space, wherein a first response surface approximation formula is used for a partial region of the design space. A step of calculating sampling point data based on: a step of calculating a linear first-order response surface model with respect to a coefficient of a second response surface approximation formula for at least an inclusion region including the partial region; Each of the coefficients of the response surface approximation formula (2) is used as a state quantity, and a residual or variance estimated value is obtained by estimating the state quantity based on a state estimation algorithm using the observation data. Based on the residual or variance estimated value And a step of calculating the adaptability of the response surface model.
[0008]
The program according to the present invention is a design support program for supporting the creation of a response surface approximation formula in a design space, wherein a sampling point based on the first response surface approximation formula is obtained for a partial region of the design space on a computer. Calculating a linear first-order response surface model with respect to the coefficient of the second response surface approximation formula for at least the inclusion region including the partial region, and the second response surface approximation formula Each state coefficient is used as a state quantity, and a residual or variance estimated value is obtained by estimating the state quantity based on a state estimation algorithm using the observation data, and the response surface model is calculated based on the residual or variance estimated value. This is a design support program for executing the process of calculating the degree of conformity.
[0009]
DETAILED DESCRIPTION OF THE INVENTION
Hereinafter, embodiments of the present invention will be described with reference to the drawings.
[0010]
FIG. 1 is a block diagram showing a schematic configuration of a design support system according to an embodiment of the present invention. This design support system can be realized by using a general-purpose or dedicated computer (for example, PC or EWS) and a program operating on the computer, and as shown in FIG. (Processing device) 12, ROM 13, and RAM 14, and at least a design database 20 and a response surface database 30.
[0011]
These databases and data such as temporary calculation results generated during the data processing according to the present invention are stored and held in an auxiliary storage device (not shown). Such an auxiliary storage device is composed of, for example, a floppy disk device, a magnetic disk device, a CD-ROM device, a DVD device, or a combination thereof. The program for realizing the data processing according to the present invention is supplied through, for example, the auxiliary storage device, read out to the RAM 14, and executed by the CPU 12. The input device 11 includes, for example, a keyboard and a mouse, and is used by the user to give various instructions such as start and end of execution of the program to the system. The output device 13 is for outputting the result of the data processing according to the present invention, and is composed of a CRT, a printer or the like.
[0012]
Such a design support system according to the present embodiment executes the data processing described below with reference to the design variable database 20 and the response surface database 30 to generate a wide range of response surface models (see FIG. 2) in the design space. It helps to easily perform work such as constructing or building a more accurate response surface model from the already obtained response surface model when new data is obtained in the design space. .
[0013]
FIG. 3 is a flowchart showing a schematic procedure of data processing for realizing the design support of the present invention.
[0014]
The general procedure shown in FIG. 3 is to increase the accuracy of the response surface approximation formula based on sampling points calculated based on a plurality of initial response surface approximation formulas or newly obtained numerical experimental points. Preparation of initial response surface approximation formula (STEP 0), assumption of various response surface models (STEP 1), update of response surface approximation formula by Kalman filter algorithm (STEP 2), response surface model multiplexing based on Bayes' theorem It consists of four steps of hypothesis test (STEP4). Hereinafter, each step of STEP0 to STEP4 will be described.
[0015]
(STEP 0: Creation of initial response surface approximation formula)
First, a state where the design space is appropriately divided into regions and a response surface approximation formula is accurately created in the subspace is assumed as an initial state. This region division may be divided into regions for each design plan, a region may be divided according to a difference in structure response mode such as a deformation mode, or a design space may be divided at equal intervals.
[0016]
Conventionally, as a response surface approximation formula, a polynomial that is easy to handle and to which a statistical method can be applied has been used. However, many nonlinear functions that can be linearized by performing variable transformation are also used. For example, exponential functions, power functions, rational functions, logarithmic functions, logistic functions, and the like. Kashimura et al. Introduced an experimental design method and a response surface method to structural analysis, and constructed an orthogonal polynomial-based response surface method applicable to nonlinear problems. It shows that the structural reliability evaluation based on the optimized and extended first-order approximate second-moment method (AFOSM) can be performed efficiently and universally. In the present embodiment, for example, the design space is defined by using the response surface method using the orthogonal table and the orthogonal polynomial proposed by Kashimura et al. Referring to the design variable database 20 storing various design variables, an approximate expression is created as a function of variables inherent in the design, and this is used as an initial response surface approximate expression.
[0017]
In this embodiment, Chebyshev's orthogonal polynomial (reference document (1)) is used as the initial response surface approximation formula. Numerical experimental points are determined by an experimental design using an orthogonal table. Such a response surface approximation formula corresponds to a regression formula created using the analysis result of each numerical experimental point (that is, a column of the orthogonal table). The orthogonal polynomial has a feature that the low-order term is prioritized and the respective order terms are independent of each other, and therefore shows the best regression equation in the remaining order even if the approximate expression is cut off at an arbitrary order.
[0018]
The response is y, and the variable inherent in the design is x_iAnd the average value of each variable is μ_iThen, such a response is expressed by equation (1). Where x₁, X₂, ... indicate design variables and μ₁, Μ₂,... Indicate average values, a indicates a coefficient, and k and h indicate the number of levels and the level interval, respectively.
[0019]
[Expression 1]

[0020]
In the process of creating the response surface approximation formula, analysis of variance can be performed by using a method in which the influence of each variable on the response is evaluated by orthogonal decomposition into polynomial components. Here, the degree of influence represents the degree of change in response accompanying the variation of each variable. By this method, the degree of influence can be obtained for characteristics decomposed such as first-order and second-order in the process of creating a response surface approximation formula.
[0021]
In this way, in STEP 0, the target phenomenon or the design space in the system is divided into regions, and a (partial) initial response surface approximation formula for each divided region is created. The sampling point is calculated as observation data (value) from the initial response surface approximation formula, and can be used in later processing (response surface model update: STEP 2). Note that the initial response surface approximation formula created in STEP 0 can also be used for the assumption of a new response surface model described below.
[0022]
(STEP1: Assumption of various response surface models)
Even complex and large-scale phenomena and systems with nonlinearity can be approximated as response surface models of linear linear equations by redefining multivariable functions created by variable transformation as new variables. It can be expressed in many cases. However, since there are many ways of redefining such variables, there can be many response surface models having a multivariable function that becomes a new variable even if it is limited to a certain degree or less. Therefore, in constructing a more accurate response surface approximation formula or a response surface approximation formula in a wider area, a method based on practical use is required that can efficiently assume a response surface model.
[0023]
Each design variable is stored in the design variable database 20, and each design variable is linearized in advance with respect to a nonlinear function that can be linearized, for example, an exponential function, a power function, a rational function, a logarithmic function, a logistic function, or the like. It may be given. The response surface model database 30 stores a set of multivariable functions of each design variable that can be assumed.
[0024]
In STEP 1, the response surface model database 30 is referred to, and a plurality of different response surface models are assumed. That is, it is assumed that a multivariable function is selected from the response surface model database 30 and redefined as a new variable, and these are combined so as to be a linear linear expression with respect to the coefficient of each redefined variable.
[0025]
In the assumption of the response surface model, the selection of multivariable functions may be narrowed down by dimensional analysis in consideration of the characteristic value unit system. Alternatively, assume a response surface model by adding various first-order variable terms to the initial response surface approximation formula (considering only the main effects of design variables) and redefining the multivariable function. Also good.
[0026]
In this embodiment, a sampling point calculated from a response surface approximation formula constructed in another region or a newly obtained numerical experimental point is used as observation data by a Kalman filter based on Bayesian theory (references (3) to (7)). While updating (STEP 2), each response surface model is sequentially calculated based on the index of the probability of matching by the multiple hypothesis test (STEP 3).
[0027]
Multivariable functions created by variable transformation are replaced with new variable X_iThe response surface model of the linear linear equation that is redefined as follows is generally expressed as follows.
[0028]
[Expression 2]

[0029]
However, N in the equation means the number of variables, and a means an unknown parameter.
[0030]
In the update of the response surface approximation formula in STEP 2, sampling points calculated based on the response surface approximation formula of another region (other partial regions) or newly obtained numerical experimental points are used as observation values. In step k, the observed value Y₁, ..., Y_NIs obtained, the following observation equation (3) can be constructed.
[0031]
[Equation 3]

[0032]
Where ν_kIndicates Gaussian white noise.
[0033]
Where the parameter (a₁, a₂..., a₉) Is a state quantity, the state equation of Expression (4) is obtained.
[0034]
[Expression 4]

[0035]
The prior distribution of the coefficient vector a is f (a). The obtained sampling point (Y_k, C_k), F (a | Y expressed by equation (6)_k) Is estimated using Bayesian theory.
[0036]
[Equation 5]

[0037]
At this time, the expected value of ak has the property of effective estimation (minimum variance estimation value). In this embodiment, the algorithm of the Kalman filter (reference documents (8) and (9)) is used as an algorithm for efficiently obtaining the expected value and variance of the state quantity.
[0038]
(STEP 2: Update of response surface approximation formula by Kalman filter)
The Kalman filter is one of the least square estimation methods. Consider a case in which a target system can be expressed by a state equation (8) and an observation equation (9) including uncertainties in the system and uncertainties in observation. At this time, the state quantity estimation in step k is calculated using equation (10), the Kalman gain is calculated using equation (11), and the error covariance matrix is calculated using equation (13).
[0039]
[Formula 6]

[0040]
However, a (k) is a state vector, y (k) is an observation vector, A and C are matrices, w (k) is system noise, v (k) is observation noise, and k is a discrete time system. If the time interval is Δt, kΔt is the observation time, and the matrices A and C are determined by modeling. Each time an observed value y (k) is obtained, an estimated value of a (k) is obtained. By assuming the error covariance matrix of the initial state estimation vector, the parameter (a₁, A₂... a₉) And the response surface approximation formula can be updated.
[0041]
In the process of calculating sampling points from a plurality of response surface approximation formulas, the embodiment may be configured to control the breakdown of sampling points calculated from each approximation formula, that is, the breakdown of the elements of the observation vector, in step k. . For example, it is preferable to increase the sampling point breakdown ratio in the vicinity of a region where it is desired to improve the accuracy of the response surface approximation formula. Alternatively, it is also preferable to control the breakdown ratio of the number of samplings calculated from each approximate expression in all steps. In this case, it is preferable to use a resampling method such as a cross-validation method (Cross-Validation method) or a Boosttrap method.
[0042]
(STEP3: Multiple temporary tests using Bayes' theorem)
The multiple temporary test method using Bayes' theorem has been proposed and applied by Fukuda and Shimizu et al. As a method for judging abnormal signs in plants and the like (references (3) (4) (6) (7 )). In the present embodiment, the response surface model is determined to be compatible by a multiple hypothesis testing method based on Bayes' theorem. Specifically, the multiple hypothesis test is performed each time the response surface approximation formula is updated (STEP 2), and the significance of each response surface model is calculated.
[0043]
Bayes' theorem is formulated as shown in Equation (16).
[0044]
[Expression 7]

[0045]
However, P (E_i) Is event E_iPrior probability of P (E_i| A) is event E_iA posteriori probability (specific event E under event A)_iProbability of occurrence), P (A | E_i) Is event E_iThe probability that event A will occur under. From this equation, a specific E under the condition of event A_iEvent E occurs_iPosterior probability P (E_iWhen | A) is updated, it gradually approaches the correct probability. In other words, even if it is necessary to estimate the state of the system based on judgment based on past experience or a small amount of observation data, the posterior probability is corrected each time new data is obtained. You can approach the probability. By applying Bayes' theorem to multiple hypothesis testing, the state of the observation system can be estimated sequentially from the residual (difference between the observed value and the estimated value) of the response surface approximation equation updated by the Kalman filter algorithm.
[0046]
The residual calculated from the Kalman filter follows a normal distribution with an average of 0 when the model described from the state equation and the observation equation matches the target phenomenon or system. The residual γ (k) output from the Kalman filter for the response surface model i has an average of 0 and a covariance V_iNormal distribution N (0, V) of (k)_iFollow (k)). The probability that the residual γ (k) appears between γ + dγ can be expressed as Equation (17).
[0047]
[Equation 8]

[0048]
This probability is expressed as P (A | E in equation (16)._i).
[0049]
Where m is the dimension of y and V_i(K) = E (γ_i ^T(K) γ_i(K)).
[0050]
Therefore, the posterior probability P_i(K) (= P (E_i| A)) can be expressed as equation (18).
[0051]
[Equation 9]

[0052]
The multiple hypothesis test here is a method of probabilistically determining which model is being observed by assuming several models for the system to be tested. It consists of the following procedures.
[0053]
(1) Response surface model 1, response surface model 2,..., Response surface model n are set.
[0054]
(2) Hypothesis H_iThat is, “the system under observation matches the response surface model i (i = 1, 2,..., N)” is established. This event is called E_iAnd
[0055]
(3) Assuming that the hypothesis is correct T (True), the hypothesis is H₁, H₂, ..., H_nProbability P (E₁| T), P (E₂| T), ..., P (E_n| T) is calculated.
[0056]
Every time new data is given, the above calculations (1) to (3) are repeated, and a hypothesis that takes a maximum value and exceeds a preset threshold value is adopted. As a result, various response surface models can be sequentially calculated using the matching probability as an index.
[0057]
As a simple method for evaluating the generalization ability that works effectively even for unknown data in the target space in the comparison of the suitability and redundancy of the linear regression model, Rissanen's MDL (Minimum Description Length ) And Akaike's Information Criterion (Akaike's Information Criterion) are known (reference documents (10) and (11)). In the present embodiment, the MDL value can be calculated. MDL is derived as a description of the description length minimization (Minimal Description Length) in encoding by Rissanen, and is defined as shown in equation (19).
[0058]
MDL = − (maximum log likelihood) + (N / 2) · log P = − (P / 2) · log (SSR / P) + (N / 2) · log P (19)
Here, N is the degree of freedom of the model, SSR is the residual sum of squares, and P is the number of samples. The approximation error of the model can be evaluated from the first term, and the degree of freedom can be evaluated from the second term. These are effective as indices for calculating a model with a small approximation error and a small degree of freedom. Accordingly, various response surface models assumed in the above-described STEP 2 are narrowed down using the information amount reference value as an index, or the response surface model is calculated using both the information amount reference value index and multiple hypothesis testing. It is preferable to configure the embodiment as described above.
[0059]
The design support system of the present embodiment described above updates the response surface approximation formula such as expanding the response surface space or creating a more accurate response surface approximation formula every time new design space data is obtained. It can be easily carried out without requiring a great deal of labor to re-execute experiment planning and analysis, and is practical.
[0060]
Thereby, useful information regarding the interaction term of each variable in the target design space can be easily extracted. Specifically, as the analysis results using the CAE technique are stored in the design space every time the design is repeated, design space information (for example, interaction information of variables) can be positively extracted. This is extremely effective in the sense of accumulating know-how for efficient design.
[0061]
Hereinafter, more specific examples of the present invention will be described.
[0062]
[Example]
As electronic devices become lighter, thinner, and smaller, demands for miniaturization of mounted parts are increasing. Against this background, flip chip connection technology capable of high-density mounting is being actively developed.
[0063]
As shown in FIG. 4, the flip chip by the crimping method connects the stud bump 44 of the chip 46 and the TAB electrode 43 by thermocompression bonding via the connection resin 42. In order to secure connection reliability, the structure around the bump 44 is fixed and sealed with a resin 42, and resin selection is an important item for improving the reliability of flip chip mounting. In the case of a flip chip by the crimping method, the final bump repulsive force f2, that is, the connection margin, is determined by the bonding load (B'g load) f1 and the curing shrinkage, stress relaxation, and thermal shrinkage of the resin 42. Depending on the selection of the resin and the setting of the B′g load, connection open (increase in electric resistance) may occur due to resin swelling in the high temperature / moisture absorption test. In order to ensure long-term reliability, it is important to clarify the influence of the characteristics of the resin 42 and the material characteristics of the TAB (or substrate) 40, and to optimize the B'g load f1.
[0064]
In accordance with STEP 0 described in the above embodiment, the results of creating an initial response surface approximation formula for the characteristics of the resin 42, the material characteristics of the TAB 40, and the bump repulsive force f2 with the B'g load f1 as a design variable are shown below.
[0065]
(Initial response surface approximation)
Gold bump / pad interface compressive load after high temperature moisture absorption (gf) = -96.4 + 0.679 X₁+0.00132 X₁ ²+4.64 X₂-0.500 X₂ ²+22.9 X_Three-3.89 X_Three ²+1.16 X_Four-0.00593 X_Four ²
This response surface approximation formula is constructed by setting an approximation formula for TAB elastic modulus of 2 to 6 Gpa. Hereinafter, (Example 1) Update example of response surface approximation formula when analysis data is newly obtained when substrate elastic modulus is 12 GPa, (Example 2) New in range of substrate elastic modulus of 8 to 12 GPa An example of constructing a response surface approximation formula for the entire region of 2 to 12 GPa when sampling points calculated based on the obtained response surface approximation formula are used as observation data is shown.
[0066]
[Example 1] Example of renewal with new numerical experimental points
As shown in FIG. 5, in Example 1, the analysis data obtained when the substrate elastic modulus is 12 GPa is newly obtained about six times (the other variables are set with uniform random values). An attempt is made to update the response surface approximation formula (substrate elastic modulus 2 to 6 GPa) to a response surface approximation equation targeting the substrate elastic modulus 2 to 12 GPa according to the present invention. Here, the observation points were successively obtained from six newly obtained numerical experimental points and 18 data obtained based on the initial high-accuracy response surface approximation formula created in the partial region (substrate elastic modulus 2 to 6 GPa). , Given by calculating at each step. In FIG. 5, 50 indicates the response surface approximation range before expansion, 51 indicates the expansion region of the response surface approximation range, and 52 indicates the response surface approximation range including each region.
[0067]
As described below, seven response surface models with different interaction terms to be considered were examined. Here, the coefficient of the interaction term of the response surface models 2 to 7 was set to an initial value of 0.
Response surface model 1: no interaction term (initial model)
Response surface model 2: X₁X₂When considering
Response surface model 3: X₁X_ThreeWhen considering
Response surface model 4: X₁X_FourWhen considering
Response surface model 5: X₂X_ThreeWhen considering
Response surface model 6: X₂X_FourWhen considering
Response surface model 7: X_ThreeX_FourWhen considering
A multiple temporary test method using Bayes' theorem is applied to the above-mentioned seven response surface model conformity determinations. Each time the response surface model is updated, a multiple hypothesis test is performed to calculate the significance of each response surface model (see FIG. 6).
[0068]
The updated response surface approximation formula is shown below.
[0069]

In this example, the calculation of the information criterion (MDL) described in the embodiment was performed before and after the update with respect to the sampling point. The results are shown in Table 1. Note that “RSM” in Table 1 is an abbreviation for “response surface model”.
[0070]
[Table 1]

[0071]
In any of the response surface models, the MDL value is smaller than the initial value. FIG. 7 shows changes in the probability of matching based on the multiple hypothesis test. From this result, it can be seen that the response surface model 2 has good adaptability. Table 2 shows the result of evaluating the variance estimation value by analysis data other than the numerical experimental points used for the approximate expression creation (10 numerical experimental point data obtained by uniform random numbers). As with the multiple hypothesis test result, it can be seen that the estimated variance of the response surface model 2 is small. In this case, the interaction term X₁X₂It can be understood that the present invention can calculate that the generalization error is reduced by considering the above.
[0072]
[Table 2]

[0073]
[Embodiment 2] An example of updating a response surface approximation formula in another region by sampling points
In short, in Example 2, the sampling point calculated based on the response surface approximation equation newly obtained in the range of the substrate elastic modulus of 8 to 12 GPa is used as observation data, and the substrate elastic modulus of 2 to 12 GPa is shown in FIG. An attempt is made to construct a response surface approximation formula for the entire region by updating the initial response surface approximation formula.
[0074]
Here, let us consider a case where the following response surface approximation formula is obtained from the orthogonal table and the orthogonal polynomial for a range (another region) where the substrate elastic modulus is 8 to 12 GPa.
[0075]
Y = -59.5 + 0.862 X₁ -0.000144 X₁ ² -2.38 X₂ + 0.103X₂ ²
+39.1 X_Three-8.31 X_Three ²+0.500 X_Four-0.00207X_Four ²
In FIG. 6, reference numeral 60 denotes an initial response surface approximation range, 61 denotes the response surface approximation range of the other region, and 62 denotes a response surface approximation range including each region.
[0076]
Sampling points are created in 10 data obtained by generating uniform random numbers in the design space targeted by the response surface approximation formula obtained in the above-mentioned separate area and a partial area (substrate elastic modulus 2 to 6 GPa). A total of 20 data obtained by combining the 10 data obtained based on the initial high-accuracy response surface approximation formula was sequentially calculated in each step. In addition, the response surface model to be processed was set to seven models having different interaction terms to be considered as in the first embodiment. The results of updating by the Kalman filter using the sampling points as observation data are shown below.
[0077]

Also in Example 2, the calculation of the information criterion (MDL) was performed before and after the update with respect to the sampling point. The results are shown in Table 3.
[0078]
[Table 3]

[0079]
In any response surface model, the MDL value is smaller than the initial value, but it is difficult to determine whether the difference in the fitness of each model is significant. FIG. 9 shows changes in the probability of matching based on the multiple hypothesis test. From this result, it can be seen that the matching degree of the response surface model 2 is good as in the first embodiment. Table 4 shows the result of evaluating the variance estimation value based on analysis data other than the numerical experimental points used to create the approximate expression (obtaining 10 numerical experimental points using uniform random numbers). As with the multiple hypothesis test result, it can be seen that the estimated variance of the response surface model 2 is small. Similar to Example 1, the interaction term X₁X₂It can be understood that the present invention can calculate that the generalization error is reduced by considering the above.
[0080]
[Table 4]

[0081]
The present invention is not limited to the above-described embodiment, and can be implemented with various modifications.
[0082]
[Appendix]
Below is a list of references.
[0083]
Record
(1) Takayoshi Tsujimura, Masaki Shiratori, Yu Qiang, Mechanism, 63-607, A (1997), 624.
(2) Takayoshi Tsujimura, Masaki Shiratori, Yu Qiang, Optimization of nonlinear problems by experimental design, Asakura Shoten, (1998).
(3) Takafumi Fukuda and Kuji Shimizu, Safety Engineering, 28-4, (1989), 211-216.
(4) K. Shimizu, Facility Safety Engineering-New Inspection / Monitoring Technology-, Suikabo, (1989).
(5) K. Watanabe, R. Patton, et al. (Ed), Pretice Hall, (1989), 411.
(6) Takafuku Fukuda, et al., 2nd, Theory, 59-560, C, (1993), 982-988.
(7) Takafuku Fukuda, Research on anomaly detection methods for facilities, Dissertation in Yokohama National University, (1995).
(8) Hoshiya Masaru and Saito Goro, Data Analysis and Application, Kashima Press, (1991).
(9) Toru Katayama, Applied Kalman Filter, Asakura Shoten, (1983).
(10) Sakamoto Yoshiyuki, Ishiguro Makio, Kitagawa Genshiro, Information Statistics, Kyoritsu Shuppan, (1982).
(11) Han Tao, Kobayashi, Mathematics of Information and Coding, Iwanami Shoten, (1994).
[0084]
【The invention's effect】
As described above, according to the present invention, as design space data is accumulated as a response surface approximation formula in a plurality of regions, an unknown part of design space information is positively included in a new interaction term. It is possible to provide a design support method and program that can be expressed by a response surface model of a shape and can actively extract information on interaction terms and the like.
[Brief description of the drawings]
FIG. 1 is a block diagram showing a schematic configuration of a design support system according to an embodiment of the present invention.
FIG. 2 is a graph showing response surface approximation in the design space.
FIG. 3 is a flowchart showing a processing procedure of the design support system according to the embodiment.
FIG. 4 is a view showing a flip chip by a crimping method according to an embodiment of the present invention.
FIG. 5 is a diagram for explaining the update of the response surface approximation formula according to the first embodiment.
FIG. 6 is a diagram showing a flow of calculating a response surface approximation model by multiple hypothesis testing.
FIG. 7 is a graph showing changes in the probability of matching based on the multiple hypothesis test according to the first embodiment.
FIG. 8 is a diagram for explaining the update of the response surface approximation formula according to the second embodiment.
FIG. 9 is a graph showing changes in the probability of matching based on multiple hypothesis testing according to Example 2;
[Explanation of symbols]
11 ... Input device
12 ... CPU
13 ... ROM
14 ... RAM
15 ... Output device
20 ... Design variable database
30 ... Response surface model database

Claims (16)

In a design support device that supports the creation of a response surface approximation formula in the design space,
Means for calculating sampling point data based on a first response surface approximation formula for a partial region of the design space;
Means for calculating a linear first-order response surface model for the coefficient of the second response surface approximation formula for at least the inclusion region including the partial region;
Each coefficient of the second response surface approximation formula is used as a state quantity, and a residual or variance estimation value is obtained by estimating the state quantity based on a state estimation algorithm using data of the sampling points, and the residual or variance design support apparatus characterized by comprising means for calculating the goodness of fit of the response surface model based on the estimated value. The means for calculating the response surface model is:
Means for redefining a multivariable function of a design variable as a new variable;
The design support apparatus according to claim 1, further comprising: means for introducing the redefined new variable into the second response surface approximation formula. The design support apparatus according to claim 1, wherein sampling point data used for the state estimation algorithm is a newly obtained experimental point. Means for calculating the fitness of the response surface model is:
Means for calculating the fitting probability of the response surface model by multiple hypothesis testing;
4. The design support apparatus according to claim 1, further comprising: means for comparing the matching probability with a predetermined threshold value. 5. The design support apparatus according to claim 2, further comprising means for selecting the multivariable function by linearization conversion and dimension analysis in consideration of a unit system of characteristic values of the design variable. 6. The apparatus according to claim 1, further comprising means for controlling the number or dimension of sampling points from the plurality of first response surface approximation equations for each partial region in the design space. The design support apparatus according to the item. The design support apparatus according to claim 1, further comprising a unit that calculates an information criterion in order to evaluate the response surface model. The design support according to claim 2, wherein the response surface model is calculated by adding or deleting a term relating to the new redefined variable to the first response surface approximation formula. Equipment . In a design support program that supports the creation of response surface approximation formulas in the design space,
On the computer,
Calculating a sampling point data based on a first response surface approximation formula for a partial region of the design space;
Calculating a linear first-order response surface model with respect to the coefficient of the second response surface approximation formula for an inclusion region including at least the partial region;
Each coefficient of the second response surface approximation formula is used as a state quantity, and a residual or variance estimation value is obtained by estimating the state quantity based on a state estimation algorithm using data of the sampling points, and the residual or variance A design support program for executing a step of calculating a fitness of the response surface model based on an estimated value. The step of calculating the response surface model includes
Redefining a multivariable function of design variables as a new variable;
The design support program according to claim 9, further comprising: introducing the redefined new variable into the second response surface approximation formula. Design assistance program according to claim 9 or 10, characterized in that the data of the sampling points used in the state estimation algorithm, the newly obtained experimental point. Calculating the fitness of the response surface model,
Calculating the probability of fit of the response surface model by multiple hypothesis testing;
The design support program according to claim 9, further comprising: comparing the matching probability with a predetermined threshold value.   The design support program according to claim 10, further comprising a step of selecting the multivariable function by linearization conversion and dimension analysis in consideration of a unit system of characteristic values of the design variable.   14. The method according to claim 9, further comprising a step of controlling the number or dimension of sampling points from the plurality of first response surface approximation equations for each partial region in the design space. The design support program described in the section.   15. The design support program according to claim 9, further comprising a step of calculating an information criterion in order to evaluate the response surface model. The design support according to claim 10, wherein the response surface model is calculated by adding or deleting a term relating to the redefined new variable with respect to the first response surface approximation formula. program.

Images