理科畢業(yè)論文外文翻譯

1、<p>　　Assume that you have a guess U(n) of the solution. If U(n) is close enough to the</p><p>　　exact solution, an improved approximation U(n + 1) is obtained by solving the linearized problem</p>

2、;<p>　　where?is a positive number. (It is not necessary that ? have a solution.even if ? has. In this case, the Gauss-Newton iteration tends to be the minimizer of the residual, i.e., the solution of minU </p

3、><p>　　It is well known that for sufficiently small</p><p><b>　　And</b></p><p>　　is called a descent direction for , where | is the l2-norm. The</p><p>　　iter

4、ation is</p><p>　　where is chosen as large as possible such that the step has a reasonable descent.The Gauss-Newton method is local, and convergence is assured only when U(0)is close enough to the solution.

5、In general, the first guess may be outside thergion of convergence. To improve convergence from bad initial guesses, adamping strategy is implemented for choosing , the Armijo-Goldstein line search. It chooses the larges

6、t damping coefficient ??out of the sequence 1, 1/2,1/4, . . . such that the following inequ</p><p><b>　　|</b></p><p>　　which guarantees a reduction of the residual norm by at leastNo

7、te that each step of the line-search algorithm requires an evaluation of the residual</p><p><b>　　?</b></p><p>　　An important point of this strategy is that when U(n) approaches the

8、solution, then and thus the convergence rate increases. If there is a solution to the scheme ultimately recovers the quadratic convergence rate of the standard Newton iteration. Closely related to the above problem is th

9、e choice of the initial guess U(0). By default, the solver sets U(0) and then assembles the FEM matrices K and F and computes</p><p>　　The damped Gauss-Newton iteration is then started with U(1), which shoul

10、d be a better guess than U(0). If the boundary conditions do not depend on the solution u, then U(1) satisfies them even if U(0) does not. Furthermore, if the equation is linear, then U(1) is the exact FEM solution and t

11、he solver does not enter the Gauss-Newton loop.</p><p>　　There are situations where U(0) = 0 makes no sense or convergence is impossible.</p><p>　　In some situations you may already have a good

12、approximation and the nonlinear solver can be started with it, avoiding the slow convergence regime.This idea is used in the adaptive mesh generator. It computes a solution on a mesh, evaluates the error, and may refine

13、certain triangles. The interpolant of is a very good starting guess for the solution on the refined mesh.</p><p>　　In general the exact Jacobian</p><p>　　is not available. Approximation of Jn by

14、 finite differences in the following way is expensive but feasible. The ith column of Jn can be approximated by</p><p>　　which implies the assembling of the FEM matrices for the triangles containing grid poi

15、nt i. A very simple approximation to Jn, which gives a fixed point iteration, is also possible as follows. Essentially, for a given U(n), compute the FEM matrices K and F and set</p><p>　　Nonlinear Equations

16、</p><p>　　This is equivalent to approximating the Jacobian with the stiffness matrix. Indeed, sinceputting Jn = K yields In many cases the convergence rate is slow, but the cost of each iteration is cheap.&l

17、t;/p><p>　　The nonlinear solver implemented in the PDE Toolbox also provides for a compromise between the two extremes. To compute the derivative of the mapping , proceed as follows. The a term has been omitted

18、 for clarity, but appears again in the final result below.The first integral term is nothing more than Ki,j.The second term is “l(fā)umped,” i.e., replaced by a diagonal matrix that contains the row sums. Since ?j?j = 1, the

19、 second term is approximated bywhich is the ithcomponent of K(c')U, where K(c') i</p><p>　　which is the mass matrix associated with the coefficient . Thus the Jacobian of the residual ?(U) is approxi

20、mated by</p><p>　　where the differentiation is with respect to u. K and M designate stiffness and mass matrices and their indices designate the coefficients with respect to which they are assembled. At each

21、Gauss-Newton iteration, the nonlinear solver assembles the matrices corresponding to the equations</p><p>　　and then produces the approximate Jacobian. The differentiations of the coefficients are done nume

22、rically.</p><p>　　In the general setting of elliptic systems, the boundary conditions are appended to the stiffness matrix to form the full linear system: where the coefficients of and may depend on the solu

23、tion . The “l(fā)umped” approach approximates the derivative mapping of the residual by The nonlinearities of the boundary conditions and the dependencies of the coefficients on the derivatives of are not properly linearized

24、 by this scheme. When such nonlinearities are strong, the scheme reduces to the fix-pointit</p><p>　　Adaptive Mesh Refinement</p><p>　　The toolbox has a function for global, uniform mesh refinem

25、ent. It divides each</p><p>　　triangle into four similar triangles by creating new corners at the midsides, adjusting for curved boundaries. You can assess the accuracy of the numerical solution by comparing

26、 results from a sequence of successively refined meshes.</p><p>　　If the solution is smooth enough, more accurate results may be obtained by extra polation. The solutions of the toolbox equation often have g

27、eometric features like localized strong gradients. An example of engineering importance in elasticity is the stress concentration occurring at reentrant corners such as the MATLAB favorite, the L-shaped membrane. Then it

28、 is more economical to refine the mesh selectively, i.e., only where it is needed. When the selection is based ones timates of errors in th</p><p>　　The adaptive refinement generates a sequence of solutions

29、on successively finer meshes, at each stage selecting and refining those elements that are judged to contribute most to the error. The process is terminated when the maximum number of elements is exceeded or when each tr

30、iangle contributes less than a preset tolerance. You need to provide an initial mesh, and choose selection and termination criteria parameters. The initial mesh can be produced by the init mesh function. The three compon

31、en</p><p>　　The Error Indicator Function</p><p>　　The adaption is a feedback process. As such, it is easily applied to a lar gerrange of problems than those for which its design was tailored. Yo

32、u wantes timates, selection criteria, etc., to be optimal in the sense of giving the mostaccurate solution at fixed cost or lowest computational effort for a given accuracy. Such results have been proved only for model p

33、roblems, butgenerally, the equid is tribution heuristic has been found near optimal. Element sizes should be chosen such that each eleme</p><p>　　solutions in terms of the source function f. For none lli pti

34、c problems such abound may not exist, while the refinement scheme is still well defined and has been found to work well.</p><p>　　The error indicator function used in the toolbox is an element-wise estimate

35、of the contribution, based on the work of C. Johnson et al. For Poisson'sequation –?u = f on ?, the following error estimate for the FEM-solution uhholds in the L2-norm </p><p>　　where h = h(x) is the lo

36、cal mesh size, and</p><p>　　The braced quantity is the jump in normal derivative of v across edge ?, hr is the length of edge ?, and the sum runs over Ei, the set of all interior edges of the triangulation.

37、The coefficients ??and ??are independent of the train gulation. This bound is turned into an element-wise error indicator function E(K) for element K by summing the contributions from its edges. The final form for the to

38、olbox equation</p><p><b>　　Becomes</b></p><p>　　where n is the unit normal of edge and the braced term is the jump in flux</p><p>　　across the element edge. The L2 norm

39、is computed over the element K. This error indicator is computed by the pdejmps function.</p><p>　　The Mesh Refiner</p><p>　　The PDE Toolbox is geared to elliptic problems. For reasons of accura

40、cy and ill-conditioning, they require the elements not to deviate too much from beingequilateral. Thus, even at essentially one-dimensional solution features, such as boundary layers, the refinement technique must guaran

41、tee reasonably shaped triangles.</p><p>　　When an element is refined, new nodes appear on its mid sides, and if the neighbor triangle is not refined in a similar way, it is said to have hanging nodes. The fi

42、nal triangulation must have no hanging nodes, and they are removed by splitting neighbor triangles. To avoid further deterioration of</p><p>　　triangle quality in successive generations, the “l(fā)ongest edge bi

43、section” scheme</p><p>　　Rosenberg-Stenger [8] is used, in which the longest side of a triangle is always split, whenever any of the sides have hanging nodes. This guarantees that no angle is ever smaller th

44、an half the smallest angle of the original triangulation. Two selection criteria can be used. One, pdead worst, refines all elements with value of the error indicator larger than half the worst of any element. The other,

45、 pdeadgsc, refines all elements with an indicator value exceeding a user-defined dimensionless tol</p><p>　　The Termination Criteria</p><p>　　For smooth solutions, error equi distribution can be

46、 achieved by the pde adgsc selection if the maximum number of elements is large enough. The pdead worst adaption only terminates when the maximum number of elements has been exceeded. This mode is natural when the soluti

47、on exhibits singularities. The error indicator of the elements next to the singularity may never vanish, regardless of element size.</p><p><b>　　外文翻譯</b></p><p>　　假定估計值，如果是最接近的準確的求解,

48、通過解決線性問題得到更精確的值</p><p>　　當(dāng)為正數(shù)時,( 有一個解，即使也有一個解都是不需要的。在這種情況下，高斯-牛頓疊代傾向于殘余量，即，minU的解答minimizer</p><p><b>　　也就是?足夠小</b></p><p>　　沿著下降方向?其中||??||是L2準則。</p><p>　　

49、當(dāng)選擇盡可能大導(dǎo)致這步驟有合理的降低。高斯-牛頓法是具有局限性。并且集合到一起，只有當(dāng)U（0）足夠接近準確結(jié)果。通常，第一猜測可以是集合范圍之外，從不合理的猜測來改進這個集合，通過選擇來實施被約束的策略。線選擇最大的阻止的系數(shù)在序列?這樣以下不等式</p><p>　　殘余標(biāo)準的降低至少為注意每一線一線搜索算法要求估計殘差</p><p>　　針對這一措施重要一點是非常接近結(jié)果并且收斂速度

50、增加?如果組合最終恢復(fù)二次收斂速度標(biāo)準牛頓法，相對于選擇假設(shè)通過誤差,求解設(shè)定并收斂于FEM矩陣K和F</p><p>　　使高斯-牛頓疊代減弱開始于遠遠好于如果分界條件不在于U，滿足要求, 不必要滿足.更進一步,如果方程式是線性的, 是FEM求解器進行求解不能使用高斯-牛頓環(huán)。在某些情況下,你可以找到一個標(biāo)準近似值,并且非線性求解器進行求解了，避免低于集合范圍,應(yīng)采用適應(yīng)性網(wǎng)格。在一個網(wǎng)格中計算U通常采用矩陣是

51、不能得到的。</p><p>　　不同的特定近似的很難得到但很有特征性. 是近似的的次冪</p><p>　　三角形網(wǎng)格點i是FEM矩陣集合點, 近似的提供一個固定點,實際上,給定,計算FEM矩陣K和F 設(shè)定</p><p>　　近似的矩陣和剛度矩陣等效.更深入的,因為, 帶入得到</p><p>　　在很多情況下，收斂度是降低的,但迭代是很

52、容易得到的。非線性求解器進行求解在PDE工具箱中，提供兩個極限折中.計算,如下計算, 項被刪除,但在以下的求解中會出現(xiàn)。</p><p>　　第一完整項不及于第二項是組合體,例如,被對角矩陣代替包括排列總數(shù),因為, 第二項近似為</p><p>　　的指數(shù)是剛度矩陣與系數(shù)組合并不是，同樣使,最終注意到，準確的是</p><p>　　大眾矩陣與系數(shù)組合一起,殘差近似為

53、：</p><p>　　相關(guān)的和M清楚的標(biāo)出剛度矩陣和大眾矩陣的區(qū)別,并且它們的指數(shù)可以指定參數(shù).在高斯-牛頓法中,非線性求解器進行求解矩陣對應(yīng)方程式為：</p><p>　　并且產(chǎn)生近似的矩陣,通常設(shè)定橢圓系統(tǒng),分界條件是附加剛度矩陣來完成線性系統(tǒng).</p><p>　　非線性分界條件和U的系數(shù)不可能是線性的，當(dāng)這個非線性性能很強時,組合降低固定點的迭代，收斂度不

54、可能降低.當(dāng)分界條件是線性的，不能影響迭代組合的收斂度.在類型中是不可見的(H是空矩陣)并且在類型中,它們的狀態(tài)殘余量是零在對應(yīng)分界點。</p><p><b>　　適應(yīng)性網(wǎng)格加密</b></p><p>　　工具箱有一個總函數(shù)，網(wǎng)格加密，把三角形分成四個相似三角形，創(chuàng)建一個新文件在文本框中，調(diào)整分界。通過網(wǎng)格加密次序可以得到準確的求解的解,如果解足夠平緩，通過推算可

55、以獲得精確的解.工具欄方程的求解經(jīng)常有幾何圖形特征，如斜率強度，一個工程設(shè)計重要的是強壓收斂會產(chǎn)生返回文件,例如MATLAE軟件，L型薄膜,它更好的進行網(wǎng)格加密,只有需要時，計算結(jié)果要進行誤差估計和后置估計，我們稱適應(yīng)性網(wǎng)格加密,參見適應(yīng)性網(wǎng)格求解例子,通常改進網(wǎng)格需要6000多個參數(shù)，而適應(yīng)性網(wǎng)格加密值需要500個參數(shù)。適應(yīng)性改進會更好的得到網(wǎng)格求解的次序，每次選定和改進這些參數(shù)都可以進行誤差判斷.約束工程的產(chǎn)生是在最大參數(shù)的準確性和

56、每一三角形少于它本身的調(diào)整容限。網(wǎng)格初值通過函數(shù)獲得.算法的三個組成部分是通過誤差指示函數(shù)，計算參數(shù)誤差估計,選擇網(wǎng)格加密和進一步分析得到最終標(biāo)準。</p><p><b>　　誤差指示函數(shù)</b></p><p>　　這是一個反饋過程，例如，提出問題范圍要比它設(shè)定的大,估計選擇標(biāo)準.正確求解可以在最低誤差估計給定.模型問題可以提高求解結(jié)果,通常使用均勻分配法.改進組

57、合原理是利用分解求解原函數(shù)f,對于非線性問題分界條件不存在,同時改進組合被確定。</p><p>　　誤差指示函數(shù)在工具箱中應(yīng)用,參數(shù)估計,基于矩陣方程在幾何中,FEM求解誤差估計在L2文本框中</p><p>　　h = h(x)是本網(wǎng)格函數(shù), </p><p>　　穩(wěn)定性能是V跳轉(zhuǎn)通過邊緣線段,是邊緣線段的高度,總的數(shù)量超過,三角形的內(nèi)邊緣線段給定, 參數(shù)和取決

58、于三角形,輸入分界參數(shù)誤差指示函數(shù)E(K)，參數(shù) K最終形成.方程式：</p><p><b>　　因此</b></p><p>　　是邊緣線段的標(biāo)準數(shù)，每項跳轉(zhuǎn)通過邊緣線L2文本框，估計參數(shù)K。誤差指示是通過函數(shù)。</p><p><b>　　網(wǎng)格加密</b></p><p>　　PDE工具箱調(diào)整

59、容限問題,因為準確性,它們需要參數(shù)不能偏離等邊，甚至基本一維求解特征，求解器改進技術(shù)必須保證合理形狀的三角形。</p><p>　　當(dāng)參數(shù)改進時，新的標(biāo)志在它們的中線上，如果鄰近三角形不能用類似的方法改進，三角形不需要標(biāo)志，避免三角性能改變“最長邊二等分”組合在應(yīng)用三角形最長邊是不連續(xù)的，此角要小于三角形最小角的一。兩中選擇的標(biāo)準應(yīng)用：第一改進所有參數(shù)具有誤差指示值大于最大參數(shù)的一半 改進所有參數(shù)指示值的確定。&

60、lt;/p><p><b>　　約束標(biāo)準</b></p><p>　　得到平緩的求解，誤差平均分配通過選擇，如果最大參數(shù)足夠大。參數(shù)被確定參數(shù)誤差指示是可見的。無論參數(shù)大小，平均分配都是理想化。</p><p><b>　　橢圓系統(tǒng)</b></p><p><b>　　拋物線系統(tǒng)</b&

61、gt;</p><p><b>　　雙曲線系統(tǒng)</b></p><p><b>　　和本征值系統(tǒng)</b></p><p>　　PDE工具箱不檢查問題的橢率，并且它盡可能在數(shù)學(xué)上不定義橢圓形系統(tǒng)。標(biāo)量的計算步驟在計算中是通用的，當(dāng)需要特殊微分計算時就產(chǎn)生了一個對稱、正的方程。</p><p>　　一般

62、來說,邊界條件在每點是混和的，即， Dirichlet和廣義Neumann的邊界條件組合：</p><p>　　ODE系統(tǒng)是有限制條件的.利用時間積分器在每個時間階段解決一個橢圓問題，應(yīng)用間接算法器可能也很費時。ODE系統(tǒng)數(shù)值綜合化由解這類問題是高效的MATLAB ODE Suite函數(shù)體現(xiàn)。 時間步進值控制著誤差允許范圍,并在必要時執(zhí)行矩陣系數(shù)的分解。當(dāng)系數(shù)是時變的，盡管拋物線復(fù)評僅歲時間變化，矩陣的復(fù)評暖和

63、復(fù)參數(shù)在沒一時間步進的解法仍然很費時。在特定條件下，時變Dirichlet矩陣h (t)也許造成誤差錯誤，甚而造成問題的合理性和u (t)是平滑的。ODE積分器也能減少算式v， u = Bv + ud的難度。 因為h改變，恒定值在軸上旋轉(zhuǎn)的改變從一步到另一步的消除次序。這就以為著B、v和ud全部不連續(xù)地改變，盡管u本身不變。</p><p><b>　　雙曲線方程</b></p>

64、<p>　　對于拋物線的方程使用相同的方法,雙曲線的數(shù)學(xué)解法：</p><p>　　如下例,方程的解是波一速度移動的。在給定的區(qū)域的三角形范圍內(nèi)，在我們除去通過Dirichlet邊界條件約束的未知領(lǐng)域之后，線性方法產(chǎn)生的二次ODE系統(tǒng)：</p><p><b>　　初始條件：</b></p><p>　　之前，剛度矩陣K和大眾矩陣