外文翻譯--設(shè)計變量的分類
1 原文： Optimal Designs 2.2 Design variables The most important classification of design variables is into: SIZE design variables SHAPE design variables TOPOLOGY design variables stated here in order of difficulty to solve but also in order of increasing importance for the obtained objective value. It is therefore not surprising that recent research to some extent concentrates on topology design variables. The notion of size design variable, relates to the thickness of a beam, a plate or a shell (although this is often termed the shape of a beam, a plate or a shell). The area of a bar in a truss is also a size design variable, and the definition of size variable is related to the fact that the modelling domain is not changed. So, the line of the beam, rod or bar is unchanged, just like the reference surface of a plate or a shell is assumed unchanged when the concept of size design variable is used. In 3D-problems the mass density or a relative volume density is size. The orientations of non-isotropic material we also treat as size design variables. The notion of shape design variable, relates to the reference domain of the actual model. For beams, rods and bars we may treat the length as a design variable, which is then a shape design variable. Also the curvature of the reference line for these one-dimensional models is a shape design variable. For 2D-models likewise the boundary curve or the curvature of the reference surface are shape design variables. For 3D-models the boundary surface (including internal boundaries like holes) is a shape design variable. Stress concentration problems are often related to shapes of boundaries. Finally, the notion of topology design variable, relates to presence or absence of a certain design aspect. Should two joints in a truss be connected with a bar, - yes or no ?. Should a continuum like a plate have a hole, - yes or no ?. The complications in treating topology design variables are due to the fact that a change in topology results in a discontinuous change in the design response, while a 2 continuous change in size or shape design variables normally results in continuous change in the design response. Let us exemplify the difference between size, shape and topology design variables. In a truss (2D as well as 3D), the bar areas (uniform or non-uniform) are the sizes, the positions of the joints determine the shape, and the chosen bars (among many possibilities) give the topology. In a shell the thickness and material density distributions are the sizes, the boundaries of the reference surface and its curvatures are the shapes, and the number of holes in the reference surface is the topology. 2.2.1 Alternative classifications Many alternative names to classify design variables can be found in the literature, like cross-sectional, geometrical, configuration, layout etc. We try to avoid these names in order to avoid unnecessary confusion. The design variables may also be classified from other points of view. Let us first discuss the distinction between continuous and discrete design variables. If only a number of specific values for the design variable is acceptable, say when catalog values must be used, then the notion of discrete design variables is used, and procedures related to what is called Integer programming come into focus. This is not covered in the present book that concentrates on continuous descriptions, but we include absolute limitations for the values of the design variables. Another meaning of the ” continuous” and ” discrete” relates to the modelling of the design domain. A complete continuous description in space means design variables related to a point (like a design function) and not to a domain. Often this is termed distributed parameter description, in contrast to say a truss description where each bar is described as a unit. In a finite element modelling of a continuum, the element domains may be related to a number of design values, so in reality this is a discrete description. However, with the extensive number of elements and the fact that everything in a computer is discrete, the distinction between continuous and discrete related to the modelling of the design domain is of no practical importance. For a successful optimization the choice of design parametrization is of vital importance, perhaps the most important decision to take. In the experience of the author it is wise to start with as few design variables as possible. A hierarchical description is suggested, and also it is important to make sure that the design variables serve different purposes. It is asking for practical problems, 3 if the design variables are chosen such that different combinations of design variables can give the same design. The parametrization is also related to the chosen optimization procedure, so with an optimality criterion method large quantities, say 50.000 design variables, can be handled without problems. 2.3 Design objective The design objective is a function or a functional that returns a single value from which different designs can be compared. The optimal design is then the design with a minimum (or maximum) value of the objective. In this book we often use the notation to denote the objective. We shall not treat multi-objective formulations, which in most cases are reformulated into a single objective anyhow. Alternative names for the objective include criterion, cost, merit, goal as well as many others. The name ” criterion” is in this book used extensively in relation to optimality criterion formulations (see chapter 14), so we try only to use the name objective, although a name like cost may be more appealing. In fact, the objective value is often a measure of the cost of the design. A minimum and maximum formulation may be interchanged by simply changing the sign of the objective. However, it is important to notice that many methods just locate a stationary value of the objective, which means that the convergence of the procedure must be followed and the final design justified.A much more severe problem is related to the existence of local stationary solutions,and in reality very few (and often non-practical) methods are able to find a global optimal solution. Starting an optimal design procedure from different initial designs and always ending up in the same optimal design may be the most practical procedure for improving the probability of an obtained solution being the global optimal solution. It should be noticed that a number of optimal design formulations for idealized problems may include a proof of global optimal solution. However, for problems where a large number of practical constraints need to be taken into account, it is more safe to state that we have optimized the design as an alternative to obtaining the optimal design. Furthermore, it is not always easy to see from the formulation whether an optimal design exists. If an optimal design does not exist we talk about a not well formulated problem. 4 Even so a procedure may return an optimized design, and the convergence often reveals the missing aspect(s) in the formulation. An important part of an optimization procedure is to decide when to stop. We talk about convergence tests. Two different aspects of convergence must be clarified, convergence of the design objective and convergence of the design variables. Often the rates of these two convergences are very different. Also the formulation of the specific stop condition can be mathematically formulated more or less complicated. The favourite formulation of the present author is as follows: When the design changes are somewhat smaller than the actual accuracy in the design production, then the design procedure should be stopped. At that instant the design objective is often converged at a much earlier step. 5 翻譯譯文 2.2 個設(shè)計變量 設(shè)計變量的分類是最重要的： 尺寸設(shè)計變量 形狀設(shè)計變量 拓?fù)湓O(shè)計變量 這里所說的在解決難 題 也越來越重視，以便獲得客觀的價值。因此毫不奇怪，在某種程度上，最近的研究集中在拓?fù)湓O(shè)計變量。 尺寸設(shè)計變量的概念，涉及一種梁的厚度，板或殼（雖然這是通常被稱為一束，形狀的板或殼）。在桁架桿地區(qū)也是一個尺寸設(shè)計變量，和尺寸變量的定義是這樣的事實，建模領(lǐng)域是沒有改變的關(guān)系。因此，梁的線，桿或棒是不變的，就像一個板或殼參考表面被假定不變時的尺寸設(shè)計變量的概念的使用。在三維問題的質(zhì)量密度或相對密度的大小。取向的非各向同性材料我們也把尺寸設(shè)計變量。形狀設(shè)計變量的概念， 涉及到實際的模型參考域。梁，棒可以將長度為設(shè)計變量，然后一個形狀設(shè)計變量。也為這些一維模型的參考線的曲率是一個形狀設(shè)計變量。對于二維模型同樣 有 邊界曲線或基準(zhǔn)表面的曲率形狀設(shè)計變量。 三維模型的邊界表面（包括內(nèi)部邊界像孔）是一個形狀設(shè)計變量。應(yīng)力集中問題往往是相關(guān)的邊界的形狀。 最后，拓?fù)湓O(shè)計變量的概念，涉及到一個特定的設(shè)計方面存在或不存在。應(yīng)在桁架節(jié)點連接桿， -是或不是？一個連續(xù)的。應(yīng)該像一個盤子上有一個洞，是還是不是？。在處理拓?fù)湓O(shè)計變量的并發(fā)癥是由于拓?fù)渲械淖兓谠O(shè)計中的響應(yīng)不連續(xù)變化的結(jié) 果，而在大小或形狀的設(shè)計響應(yīng)不斷變化的設(shè)計變量連續(xù)變化的一般結(jié)果。 讓我們舉例說明之間的差的大小，形狀和拓?fù)湓O(shè)計變量。 6 在桁架（ 2D 和 3D）， 邊界區(qū)域 （均勻或不均勻）的大小，關(guān)節(jié)的位置確定的形狀，和所選擇的 邊界 （其中許多可能性）給拓?fù)?。在殼的厚度和材料密度分布的大小，參考面及其曲率的邊界的形狀，并在參考表面孔的?shù)量是拓?fù)洹?2.2.1 另一種分類 許多其他的名字將設(shè)計變量可以在文獻(xiàn)中找到，如橫截面，幾何，結(jié)構(gòu)，布局等，我們盡量避免以避免不必要的混淆這些名字。 設(shè)計變量也可以從其 他角度分類。讓我們先討論連續(xù)和離散設(shè)計變量之間的區(qū)別。如果只為設(shè)計變量的特定值的數(shù)量是可以接受的，說的時候必須使用目錄的值，然后使用離散的設(shè)計變量的概念，并稱之為整數(shù)規(guī)劃成為關(guān)注的焦點，有關(guān)的程序。這是不包括在本書致力于不斷的描述，但我們將設(shè)計變量的值的絕對限制。 另一種意義上的 “ 連續(xù) ” 和 “ 離散 ” 涉及到設(shè)計領(lǐng)域的建模。連續(xù)空間中的一個完整的描述手段一點相關(guān)的設(shè)計變量（如設(shè)計功能）和不到域。這通常被稱為分布參數(shù)描述，相反，說一個桁架的描述 ， 描述為一個單元，每一桿。在一個連續(xù)的有限元模型，單元域可能要數(shù)設(shè)計值相關(guān)，因此在現(xiàn)實中這是一個離散的描述。然而，隨著元素的大量事實和計算機(jī)中的一切都是離散的，連續(xù)的和離散的設(shè)計領(lǐng)域建模的相關(guān)之間的區(qū)別是沒有實際意義的。 一個成功的優(yōu)化設(shè)計參數(shù)的選擇是至關(guān)重要的，也許是最重要的決定。在本文開始的一些設(shè)計變量可能是明智的經(jīng)驗。提出了一種分層描述，并確保設(shè)計變量為不同目的的重要。它要求的實際問題，如果設(shè)計變量的選擇，不同的組合設(shè)計變量可以提供相同的設(shè)計。參數(shù)化也是選擇的優(yōu)化過程相關(guān)，所以一個優(yōu)化準(zhǔn)則法大量，說 50 個設(shè)計變量，可以在未經(jīng)處理的問題。 2.3 設(shè)計的目的 設(shè)計的目的是一個函數(shù)或函數(shù)返回單個值，不同的設(shè)計，可以進(jìn)行比較。優(yōu)化設(shè)計是一個設(shè)計最小（或最大）的客觀價值。在這本書中我們經(jīng)常使用的 符號 表示目的。我們不應(yīng)當(dāng)把多目標(biāo)的 規(guī)劃 ，這在大多數(shù)情況下，轉(zhuǎn)化為單目標(biāo)總之。為目的的替代名稱包括標(biāo)準(zhǔn)，成本，價值，目標(biāo)，以及其他許多人。 “ 在這本書中被廣泛使用，在關(guān)系到最優(yōu)準(zhǔn)則的配方標(biāo)準(zhǔn) ” （見 14 章），所以我們只使用名稱的目的，雖然這樣的名字，成本可能更具吸引力。事實上，客觀的價值往往是衡量成本的設(shè)計。 7 最小和最大的 規(guī)劃 可以互換，通過簡單地改變目 標(biāo)的 符號 。然而，這是要注意，很多方法只找到一個固定的目標(biāo)值的重要，這意味著收斂的過程中必須遵循和最終的設(shè)計合理。 一個更嚴(yán)重的問題是局部平穩(wěn)解的存在性，并在現(xiàn)實中很少（通常是非現(xiàn)實）的方法是能夠找到全局最優(yōu)解。從不同的初始設(shè)計的優(yōu)化設(shè)計程序和總是結(jié)束在相同的優(yōu)化設(shè)計可以提高所得到的解是全局最優(yōu)解的概率最實用的程序。值得注意的是，理想化的問題的一些優(yōu)化設(shè)計的配方可以包含全局最優(yōu)解的一個證明。 然而，在大量的實際約束，需要考慮的問題，它是更安全的國家，我們已經(jīng)優(yōu)化設(shè)計作為一種替代獲得最優(yōu)設(shè)計。此 外，它并不總是很容易看到，從是否存在的配方優(yōu)化設(shè)計。如果一個優(yōu)化設(shè)計不存在我們談的不是制定問題。即便如此，一個程序可能會返回一個優(yōu)化設(shè)計，和收斂性往往揭示了失蹤的方面（ S）的制定。 在 優(yōu)化過程中的一個重要組成部分，是決定何時停止。我們談?wù)摿耸諗啃詸z驗。收斂的兩個不同方面必須澄清，對設(shè)計變量的設(shè)計目的和收斂的收斂性。通常這兩種收斂率有很大的不同。還制定了具體的停止條件可以更多或更少的復(fù)雜的數(shù)學(xué)公式。作者最喜歡的配方如下：當(dāng)設(shè)計的變化比在設(shè)計生產(chǎn)的實際精度稍小，然后設(shè)計程序應(yīng)該停止。在那一瞬間，其設(shè)計 目的是經(jīng)常會聚在更早的階段。
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1 原文： Optimal Designs 2.2 Design variables The most important classification of design variables is into: SIZE design variables SHAPE design variables TOPOLOGY design variables stated here in order of difficulty to solve but also in order of increasing importance for the obtained objective value. It is therefore not surprising that recent research to some extent concentrates on topology design variables. The notion of size design variable, relates to the thickness of a beam, a plate or a shell (although this is often termed the shape of a beam, a plate or a shell). The area of a bar in a truss is also a size design variable, and the definition of size variable is related to the fact that the modelling domain is not changed. So, the line of the beam, rod or bar is unchanged, just like the reference surface of a plate or a shell is assumed unchanged when the concept of size design variable is used. In 3D-problems the mass density or a relative volume density is size. The orientations of non-isotropic material we also treat as size design variables. The notion of shape design variable, relates to the reference domain of the actual model. For beams, rods and bars we may treat the length as a design variable, which is then a shape design variable. Also the curvature of the reference line for these one-dimensional models is a shape design variable. For 2D-models likewise the boundary curve or the curvature of the reference surface are shape design variables. For 3D-models the boundary surface (including internal boundaries like holes) is a shape design variable. Stress concentration problems are often related to shapes of boundaries. Finally, the notion of topology design variable, relates to presence or absence of a certain design aspect. Should two joints in a truss be connected with a bar, - yes or no ?. Should a continuum like a plate have a hole, - yes or no ?. The complications in treating topology design variables are due to the fact that a change in topology results in a discontinuous change in the design response, while a 2 continuous change in size or shape design variables normally results in continuous change in the design response. Let us exemplify the difference between size, shape and topology design variables. In a truss (2D as well as 3D), the bar areas (uniform or non-uniform) are the sizes, the positions of the joints determine the shape, and the chosen bars (among many possibilities) give the topology. In a shell the thickness and material density distributions are the sizes, the boundaries of the reference surface and its curvatures are the shapes, and the number of holes in the reference surface is the topology. 2.2.1 Alternative classifications Many alternative names to classify design variables can be found in the literature, like cross-sectional, geometrical, configuration, layout etc. We try to avoid these names in order to avoid unnecessary confusion. The design variables may also be classified from other points of view. Let us first discuss the distinction between continuous and discrete design variables. If only a number of specific values for the design variable is acceptable, say when catalog values must be used, then the notion of discrete design variables is used, and procedures related to what is called Integer programming come into focus. This is not covered in the present book that concentrates on continuous descriptions, but we include absolute limitations for the values of the design variables. Another meaning of the ” continuous” and ” discrete” relates to the modelling of the design domain. A complete continuous description in space means design variables related to a point (like a design function) and not to a domain. Often this is termed distributed parameter description, in contrast to say a truss description where each bar is described as a unit. In a finite element modelling of a continuum, the element domains may be related to a number of design values, so in reality this is a discrete description. However, with the extensive number of elements and the fact that everything in a computer is discrete, the distinction between continuous and discrete related to the modelling of the design domain is of no practical importance. For a successful optimization the choice of design parametrization is of vital importance, perhaps the most important decision to take. In the experience of the author it is wise to start with as few design variables as possible. A hierarchical description is suggested, and also it is important to make sure that the design variables serve different purposes. It is asking for practical problems, 3 if the design variables are chosen such that different combinations of design variables can give the same design. The parametrization is also related to the chosen optimization procedure, so with an optimality criterion method large quantities, say 50.000 design variables, can be handled without problems. 2.3 Design objective The design objective is a function or a functional that returns a single value from which different designs can be compared. The optimal design is then the design with a minimum (or maximum) value of the objective. In this book we often use the notation to denote the objective. We shall not treat multi-objective formulations, which in most cases are reformulated into a single objective anyhow. Alternative names for the objective include criterion, cost, merit, goal as well as many others. The name ” criterion” is in this book used extensively in relation to optimality criterion formulations (see chapter 14), so we try only to use the name objective, although a name like cost may be more appealing. In fact, the objective value is often a measure of the cost of the design. A minimum and maximum formulation may be interchanged by simply changing the sign of the objective. However, it is important to notice that many methods just locate a stationary value of the objective, which means that the convergence of the procedure must be followed and the final design justified.A much more severe problem is related to the existence of local stationary solutions,and in reality very few (and often non-practical) methods are able to find a global optimal solution. Starting an optimal design procedure from different initial designs and always ending up in the same optimal design may be the most practical procedure for improving the probability of an obtained solution being the global optimal solution. It should be noticed that a number of optimal design formulations for idealized problems may include a proof of global optimal solution. However, for problems where a large number of practical constraints need to be taken into account, it is more safe to state that we have optimized the design as an alternative to obtaining the optimal design. Furthermore, it is not always easy to see from the formulation whether an optimal design exists. If an optimal design does not exist we talk about a not well formulated problem. 4 Even so a procedure may return an optimized design, and the convergence often reveals the missing aspect(s) in the formulation. An important part of an optimization procedure is to decide when to stop. We talk about convergence tests. Two different aspects of convergence must be clarified, convergence of the design objective and convergence of the design variables. Often the rates of these two convergences are very different. Also the formulation of the specific stop condition can be mathematically formulated more or less complicated. The favourite formulation of the present author is as follows: When the design changes are somewhat smaller than the actual accuracy in the design production, then the design procedure should be stopped. At that instant the design objective is often converged at a much earlier step. 5 翻譯譯文 2.2 個設(shè)計變量 設(shè)計變量的分類是最重要的： 尺寸設(shè)計變量 形狀設(shè)計變量 拓?fù)湓O(shè)計變量 這里所說的在解決難 題 也越來越重視，以便獲得客觀的價值。因此毫不奇怪，在某種程度上，最近的研究集中在拓?fù)湓O(shè)計變量。 尺寸設(shè)計變量的概念，涉及一種梁的厚度，板或殼（雖然這是通常被稱為一束，形狀的板或殼）。在桁架桿地區(qū)也是一個尺寸設(shè)計變量，和尺寸變量的定義是這樣的事實，建模領(lǐng)域是沒有改變的關(guān)系。因此，梁的線，桿或棒是不變的，就像一個板或殼參考表面被假定不變時的尺寸設(shè)計變量的概念的使用。在三維問題的質(zhì)量密度或相對密度的大小。取向的非各向同性材料我們也把尺寸設(shè)計變量。形狀設(shè)計變量的概念， 涉及到實際的模型參考域。梁，棒可以將長度為設(shè)計變量，然后一個形狀設(shè)計變量。也為這些一維模型的參考線的曲率是一個形狀設(shè)計變量。對于二維模型同樣 有 邊界曲線或基準(zhǔn)表面的曲率形狀設(shè)計變量。 三維模型的邊界表面（包括內(nèi)部邊界像孔）是一個形狀設(shè)計變量。應(yīng)力集中問題往往是相關(guān)的邊界的形狀。 最后，拓?fù)湓O(shè)計變量的概念，涉及到一個特定的設(shè)計方面存在或不存在。應(yīng)在桁架節(jié)點連接桿， -是或不是？一個連續(xù)的。應(yīng)該像一個盤子上有一個洞，是還是不是？。在處理拓?fù)湓O(shè)計變量的并發(fā)癥是由于拓?fù)渲械淖兓谠O(shè)計中的響應(yīng)不連續(xù)變化的結(jié) 果，而在大小或形狀的設(shè)計響應(yīng)不斷變化的設(shè)計變量連續(xù)變化的一般結(jié)果。 讓我們舉例說明之間的差的大小，形狀和拓?fù)湓O(shè)計變量。 6 在桁架（ 2D 和 3D）， 邊界區(qū)域 （均勻或不均勻）的大小，關(guān)節(jié)的位置確定的形狀，和所選擇的 邊界 （其中許多可能性）給拓?fù)?。在殼的厚度和材料密度分布的大小，參考面及其曲率的邊界的形狀，并在參考表面孔的?shù)量是拓?fù)洹?2.2.1 另一種分類 許多其他的名字將設(shè)計變量可以在文獻(xiàn)中找到，如橫截面，幾何，結(jié)構(gòu)，布局等，我們盡量避免以避免不必要的混淆這些名字。 設(shè)計變量也可以從其 他角度分類。讓我們先討論連續(xù)和離散設(shè)計變量之間的區(qū)別。如果只為設(shè)計變量的特定值的數(shù)量是可以接受的，說的時候必須使用目錄的值，然后使用離散的設(shè)計變量的概念，并稱之為整數(shù)規(guī)劃成為關(guān)注的焦點，有關(guān)的程序。這是不包括在本書致力于不斷的描述，但我們將設(shè)計變量的值的絕對限制。 另一種意義上的 “ 連續(xù) ” 和 “ 離散 ” 涉及到設(shè)計領(lǐng)域的建模。連續(xù)空間中的一個完整的描述手段一點相關(guān)的設(shè)計變量（如設(shè)計功能）和不到域。這通常被稱為分布參數(shù)描述，相反，說一個桁架的描述 ， 描述為一個單元，每一桿。在一個連續(xù)的有限元模型，單元域可能要數(shù)設(shè)計值相關(guān)，因此在現(xiàn)實中這是一個離散的描述。然而，隨著元素的大量事實和計算機(jī)中的一切都是離散的，連續(xù)的和離散的設(shè)計領(lǐng)域建模的相關(guān)之間的區(qū)別是沒有實際意義的。 一個成功的優(yōu)化設(shè)計參數(shù)的選擇是至關(guān)重要的，也許是最重要的決定。在本文開始的一些設(shè)計變量可能是明智的經(jīng)驗。提出了一種分層描述，并確保設(shè)計變量為不同目的的重要。它要求的實際問題，如果設(shè)計變量的選擇，不同的組合設(shè)計變量可以提供相同的設(shè)計。參數(shù)化也是選擇的優(yōu)化過程相關(guān)，所以一個優(yōu)化準(zhǔn)則法大量，說 50 個設(shè)計變量，可以在未經(jīng)處理的問題。 2.3 設(shè)計的目的 設(shè)計的目的是一個函數(shù)或函數(shù)返回單個值，不同的設(shè)計，可以進(jìn)行比較。優(yōu)化設(shè)計是一個設(shè)計最小（或最大）的客觀價值。在這本書中我們經(jīng)常使用的 符號 表示目的。我們不應(yīng)當(dāng)把多目標(biāo)的 規(guī)劃 ，這在大多數(shù)情況下，轉(zhuǎn)化為單目標(biāo)總之。為目的的替代名稱包括標(biāo)準(zhǔn)，成本，價值，目標(biāo)，以及其他許多人。 “ 在這本書中被廣泛使用，在關(guān)系到最優(yōu)準(zhǔn)則的配方標(biāo)準(zhǔn) ” （見 14 章），所以我們只使用名稱的目的，雖然這樣的名字，成本可能更具吸引力。事實上，客觀的價值往往是衡量成本的設(shè)計。 7 最小和最大的 規(guī)劃 可以互換，通過簡單地改變目 標(biāo)的 符號 。然而，這是要注意，很多方法只找到一個固定的目標(biāo)值的重要，這意味著收斂的過程中必須遵循和最終的設(shè)計合理。 一個更嚴(yán)重的問題是局部平穩(wěn)解的存在性，并在現(xiàn)實中很少（通常是非現(xiàn)實）的方法是能夠找到全局最優(yōu)解。從不同的初始設(shè)計的優(yōu)化設(shè)計程序和總是結(jié)束在相同的優(yōu)化設(shè)計可以提高所得到的解是全局最優(yōu)解的概率最實用的程序。值得注意的是，理想化的問題的一些優(yōu)化設(shè)計的配方可以包含全局最優(yōu)解的一個證明。 然而，在大量的實際約束，需要考慮的問題，它是更安全的國家，我們已經(jīng)優(yōu)化設(shè)計作為一種替代獲得最優(yōu)設(shè)計。此 外，它并不總是很容易看到，從是否存在的配方優(yōu)化設(shè)計。如果一個優(yōu)化設(shè)計不存在我們談的不是制定問題。即便如此，一個程序可能會返回一個優(yōu)化設(shè)計，和收斂性往往揭示了失蹤的方面（ S）的制定。 在 優(yōu)化過程中的一個重要組成部分，是決定何時停止。我們談?wù)摿耸諗啃詸z驗。收斂的兩個不同方面必須澄清，對設(shè)計變量的設(shè)計目的和收斂的收斂性。通常這兩種收斂率有很大的不同。還制定了具體的停止條件可以更多或更少的復(fù)雜的數(shù)學(xué)公式。作者最喜歡的配方如下：當(dāng)設(shè)計的變化比在設(shè)計生產(chǎn)的實際精度稍小，然后設(shè)計程序應(yīng)該停止。在那一瞬間，其設(shè)計 目的是經(jīng)常會聚在更早的階段。
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