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「Γsu non-empty」は「Γsu which is non-empty」ってことでしょうね。 カラでない＝内部にソースが必ず含まれる境界、という意味だと思います。 >有限要素の匂いがプンプンしますか？その通りです！ >境界要素法の発展型のLBIE法（local boundary integral element method） >ってのをやってます． ほほぅ！ って、いや…、私は専門ではないので良くわからないのですが…。 境界要素法は学生のころ論文紹介しなくてはいけなくて、ほんの触りだけ勉強したことがあります。 補足をいただいたので、再度訳を試みたいと思います。 直訳。 「点xQにおける試験関数のMLS近似の定義域内にあるそれらの節点に対し、 形状関数φi(xQ) とその導関数φij(xQ)を空でないΓsuを持つ節点に対して計算せよ。」 「の」が多くて美しくないので、アレンジしてみる。 「点xQにおいてMLS近似した試験関数の定義域内にある、空でないΓsuを持つ節点に対して 形状関数φi(xQ) とその導関数φij(xQ)を計算せよ。」
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- nikorin
- ベストアンサー率24% (47/191)
工学系なシミュレーション関係（有限要素法関係？）っぽいがしますね。 "of"が多いなぁ…ぶつぶつ… どういった分野で、どのような流れでこの文が出てきたのか？ φi(xQ)，φij(xQ)，Γsuとはなにものなのか？ わからないことが多いので自信がないですけども。 「点xQでの試験関数のMLS近似の、定義域内における空でないΓsuを持つ節点に対して φi(xQ) と導関数φij(xQ)を計算せよ。」 （「for the nodes with Γsu non-empty」は前のほうに持ってきてしまいました。）
お礼
おおぉ！そのとうりです． 有限要素の匂いがプンプンしますか？その通りです！ 境界要素法の発展型のLBIE法（local boundary integral element method） ってのをやってます． φi(xQ)，は移動最小自乗法なるものを用いてだした形状関数． φij(xQ)，はその形状関数の微分（導関数に）したもの． Γsu は考えてる領域の境界でソース点ｙが含まれる境界． ですね．
- fushigichan
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こんにちは。 ＞for those nodes in the domain of definition of the MLS approximation of trial function at point xQ; 点xQにおいて、（点xQにおける）試験関数trial functionの 移動最小自乗近似による領域のなかにこれらの接点があるとき（あるため）、 ＞calculate φi(xQ) and the derivatives φij(xQ) for the nodes with Γsu non-empty; φi(xQ)を計算せよ。 また、Γsu non-emptyによってそれらの接点でのφij(xQ)を微分せよ。 ↑ 微分して求めよ、ということだと思います。 ということではないでしょうか。 一応数学科出身ですが、こころもとないです。
お礼
どうもです． nom-empty ってなんなんだろうって思うんですよね． Γsuってのはある領域の境界を指してるんですけれど． 回答ありがとうです．
Mathematics, Logic and System Theoryの事でしょうか。 数学のカテゴリの方が良いのでは？ これ程専門的な知識を持っている回答者はこの英語カテゴリにはいないかも。 文章そのものは難しくないのですが・・・
補足
nodes 節点 domain 領域 MLS approximation 移動最小自乗法近似 trial function 試験関数 derivatives 微分 で訳してもらえないでしょうか？
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for those nodes in the domain of definition of the MLS approximation of trial function at point xQ; calculate φi(xQ) and the derivatives φij(xQ) for the nodes with Γsu non-empty; xQ，φi(xQ)，φij(xQ)，Γsu はそのままでお願いします.
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ちょっと長いですが，和訳お願いします．分かりにくい単語の意味は 下に書いておきました．特に第２文目のIFがダブルで来ているところが 意味わかりません．よろしくおねがいします． The MLS approximation is well defined only when the matrix A in Eq. (20) is non-singular. It can be seen that this is the case if and only if the rank of P equals m. A necessary condition for a well-defined MLS approximation is that at least m weight functions are non-zero (i.e. n ＞m) for each sample point X∈Ω and that the nodes in Ωx will not be arranged in a special pattern such as on a straight line. Here a sample point may be a nodal point under consideration or a quadrature point. MLS approximation 移動最小自乗法近似 matrix A in Eq. (20) 方程式（20）でのA行列 singular 単位行列 the rank of P equals m Pのランクがmに等しい weight functions 重み関数 node 節点 n ＞m，X∈Ω，Ωx などの記号はそのままでお願いします．
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The numbers of collision orbits found in the present calculations are shown in Table 4 for the representative sets of (e,i). From these numbers we can expect the magnitude of statistical error in the evaluation of <P(e,i)> to be a few percent for small e, i and within 10% for large e, i for r_p=0.005 are shown in Table 5, together with those of the two-dimensional case. Interpolating these values, we have obtained the contour of <P(e,i)> and R(e,i) on the e-I plane. They are shown in Figs. 14 and 15. From Fig. 15 we can read out the general properties of the collisional rate in the three-dimensional case: (i) <P(e,i)> is enhanced over <P(e,i)>_2B except for small e and i, (ii) <P(e,i)> reduces to <P(e,i)>_2B for (e^2+i^2)^(1/2)≧4, and (iii) there are two peaks in R(e,i) near regions where e≒1 (i<1) and where i≒3 (e<0.1): the peak value is at most as large as 5. In the vicinity of small v(=(e^2+i^2)^(1/2)) and i, R(e,i) rapidly reduces to zero. This is due to a singularity of <P(e,i)>_2B at v=0 and i=0 in the ordinary expression given by Eq. (29) and hence unphysical; the behavior of collisional rate in the vicinity of small v and i will be discussed in detail later. Thus, we are able to assert, more strongly, the property (i) mentioned in the last paragraph: that is, solar gravity always enhances the collisional rate over that of the two-body approximation. One of the remarkable features of R(e,i) found in Fig. 15 is the property (ii). That is, the collisional rate between Keplerian particles is well described by the two-body approximation, for (e^2+i^2)^(1/2)≧4. This is corresponding to the two-dimensional result that R(e,0)≒1 for e≧4. よろしくお願いします。
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In Fig.13, we compare our results with those of Nishiida (1983) and Wetherill and Cox (1985). Nishida studied the collision probability in the two-dimensional problem for the two cases: e=0 and 4. For the case of e=0, his result (renormalized so as to coincide with our present definition) agrees accurately with ours. But for e=4, his collisional rate is about 1.5 times as large as ours; it seems that the discrepancy comes from the fact that he did not try to compute a sufficient number of orbits for e=4, thus introducing a relatively large statistical error. The results of Wetherill and Cox are summarized in terms of v/v_e where v is the relative velocity at infinity and v_e the escape velocity from the protoplanet, while our results are in terms of e and i. Therefore we cannot compare our results exactly with theirs. If we adopt Eq. (2) as the relative velocity, we have (of course, i=0 in this case) (e^2+i^2)^(1/2)≒34(ρ/3gcm^-3)^(1/6)(a_0*/1AU)^(1/2)(v/v_e). (34) According to Eq. (34), their results are rediscribed in Fig.13. From this figure it follows that their results almost coincide with ours within a statistical uncertainty of their evaluation. 7. The collisional rate for the three-dimensional case Now, we take up a general case where i≠0. In this case, we selected 67 sets of (e,i), covering regions of 0.01≦i≦4 and 0≦e≦4 in the e-i diagram, and calculated a number of orbits with various b, τ,and ω for each set of (e,i). We evaluated R(e,i) for r_p=0.001 and 0.005 (for r_p=0.0002 we have not obtained a sufficient number of collision orbits), and found again its weak dependence on r_p (except for singular points, e.g., (e,i)=(0,3.0)) for such values of r_p. Hence almost all results of calculations will be presented for r_p=0.005 (i.e., at the Earth orbit) here. Fig.13. Comparison of the two-dimensional enhancement factor R(e,0) with those of Nishida (1983) and those of Wetherill and Cox (1985).Their results are renormalized so as to coincide with our definition of R(e,0). 長文ですが、よろしくお願いします。
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Finally, we will add a comment on comparison of our result with those of Wetherill and Cox (1985). Wetherill and Cox examined three-dimensional calculation for a swarm of planetesimals with a special distribution, i.e., e_2 has one value and i_2 is distributed randomly between 0.3e_2 and 0.7e_2 (<i_2>=e_2/2) while e_1=i_1=0, which corresponds, in our notation of Eq. (9), to <n_2>={n_sδ(e_2-e)δ(i_2-i)/0.4π^2e(2/2) for 0.3e_2<i_2<0.7e_2, {0 otherwise. (38) Integrating <P(e,i)> with above <n_2> according to Eq. (9), we compare our results with theirs. Figure 18 shows that their results almost agree with ours (the slight quantitative difference may come from the difference in definition of the enhancement factor); but their results contain a large statistical uncertainty because they calculated only 10~35 collision orbits for each set of e and i while 100~6000 collision orbits were found in our calculation (see Table 4). Furthermore, our results are more general than theirs in the sense that their calculations are restricted to the special distribution of planetesimals as mentioned above, while the collisional rate for an arbitrary planetesimal distribution can be deduced from our results. 8. Concluding remarks Based on the efficient numerical procedures to find collision orbits developed in Sect. 2 to 4, we have evaluated numerically the collisional rate defined by Eq. (10). The results are summarized as follows: (i) the collisional rate <P(e,i)> is like that in the two-dimensional case for i≦0.1 (when e≦0.2) and i≦0.02/e (when e≧0.2), (ii) except for such two-dimensional region, <P(e,i)> is always enhanced by the solar gravity, (iii) <P(e,i)> reduces to <P(e,i)>_2B for (e^2+i^2)^(1/2)≧4, where <P(e,i)>_2B is the collisional rate in the two-body approximation, and (iv) there are two notable peaks in <P(e,i)>/<P(e,i)>_2B at e≒1 (i<1) and i≒3 (e<0.1); but the peak value is at most 4 to 5. From the present numerical evaluation of <P(e,i)>, we have also found an approximate formula for <P(e,i)>, which can reproduce <P(e,i)> within a factor 5 but cannot express the peaks found at e≒1 (i<1) and i≒3 (e<0.1). These peaks are characteristic to the three-body problem. They are very important for the study of planetary growth, since they are closely related to the runaway growth of the protoplanet, as discussed by Wetherill and Cox (1985). This will be considered in the next paper (Ohtsuki and Ida, 1989), based on the results obtained in the present paper. Acknowledgements. Numerical calculations were made by HITAC M-680 of the Computer Center of the University of Tokyo. This work was supported by the Grant-in-Aid for Scientific Research on Priority Area (Nos. 62611006 and 63611006) of the Ministry of Education, Science and Culture of Japan. Fig. 18. Comparison of the enhancement factors with those of Wetherill and Cox (1985). The error bars in their results arise from a small number (10~35) of collision orbits which they found for each e. Our results are averaged by the distribution function which they used (see text). Fig. 18.↓ http://www.fastpic.jp/images.php?file=0990654048.jpg かなりの長文になりますが、どうかよろしくお願いします。
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To avoid this difficulty, we consider the scale height to be (i+αr_G) rather than i, where α is a numerical factor; α must have a value of the order of 10 to be consistent with Eq. (35). For the requirements that in the limit of i=0, <P(e,i)>_2B has to naturally tend to <P(e,0)>_2B given by Eq. (28), we put the modified collisional rate in the two-body approximation to be <P(e,i)>_2B=Cπr_p^2{1+6/(r_p(e^2+i^2))}(e^2+i^2)^(1/2)/(2(i+ατ_G)) (36) with C=((2/π)^2){E(k)(1-x)+2αE(√(3/4))x}, (37) where x is a variable which reduces to zero for i>>αr_G and to unity for i<<αr_G. The above equation reduces to Eq. (29) when i>>αr_G while it tends to the expression of the two-dimensional case (28) for i<<αr_G. Taking α to be 10 and x to be exp(-i/(αr_G)), <P(e,i)> scaled by Eq. (36) is shown in Fig. 17. Indeed, the modified <P(e,i)>_2B approximates <P(e,i)> within a factor of 5 in whole regions of the e-I plane, especially it is exact in the high energy limit (v→∞). However, two peaks remain at e≒1 and i≒3, which are closely related to the peculiar features of the three-body problem and hence cannot be reproduced by Eq. (36). Fig. 16a and b. Behaviors of r_min(i,b): a i=0, b i=2, 2.5, and 3.0. The level of the planetary radius (r_p=0.005) is denoted by a dashed line. Fig. 17. Contours of <P(e,i)> normalized by the modified <P(e,i)>_2B given by Eq. (36). Fig. 16a and b.↓ http://www.fastpic.jp/images.php?file=4940423993.jpg Fig. 17.↓ http://www.fastpic.jp/images.php?file=5825412982.jpg よろしくお願いします。
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As mentioned above, there are two peaks in R(e,i) in the e-i diagram: one is at e≒1 (i<1) and the other at i≒3 (e<0.1). The former corresponds directly to the peak in R(e,0) at e≒1 found in the two-dimensional case. The latter is due to the peculiar nature to the three-dimensional case, as understood in the following way. Let us introduce r_min (i, b,ω) in the case of e=0, which is the minimum distance during encounter between the protoplanet and a planetesimal with orbital elements i, b, and ω. In Fig. 16, r_min(i,b)=min_ω{r_min(i,b,ω)} is plotted as a function of b for various i, where r_min<r_p (=0.005) means “collision”; there are two main collision bands at b≒2.1 and 2.4 for i=0. For i≦2, these bands still exist, shifting slightly to small b. This shift is because a planetesimal feels less gravitational attracting force of the protoplanet as i increases. As i increases further, the bands approach each other, and finally coalesce into one large collision band at i≒3.0; this large band vanish when i≧4. In this way, the peculiar orbital behavior of three-body problem makes the peak at i≒3 (e<0.1). Though there are the peaks in R(e,i), the peak values are not so large: at most it is as large as 5. This shows that the collisional rate is well described by that of the two-body approximation <P(e,i)>_2B except for in the vicinity of v, i→0 if we neglect a difference of a factor of 5. Now we propose a modified form of <P(e,i)>_2B which well approximates the calculated collisional rate even in the limit of v, i→0. We find in Fig.14 that <P(e,i)> is almost independent of i, i.e., it behaves two-dimensionally for i≦{0.1 (when e≦0.2), {0.02/e (when e≧0.2). (35) This transition from three-dimensional behavior to two-dimensional behavior comes from the fact that the isotropy of direction of incident particles breaks down for the case of very small i (the expression <P(e,i)>_2B given by Eq. (29) assumes the isotropy). In other words, as an order of magnitude, the scale height of planetesimals becomes smaller than the gravitational radius r_G=σ_2D/2 (σ_2D given by Eq. (3)) and the number density of planetesimals cannot be uniform within a slab with a thickness σ_2D for small i. Table 4. Numbers of three-dimensional collision events found by orbital calculations for the representative sets of e and i. In the table r_p is the radius of the protoplanet. Table 5. The three-dimensional collision rate <P(e,i)> for the case of r_p=0.005 (r_p being the protoplanetary radius), together with two-dimensional <P(e,i=0)> Fig. 14. Contours of the evaluated <P(e,i)>, drawn in terms of log_10<P(e,i)> Fig. 15. Contours of the enhancement factor R(e,i) Table 4.↓ http://www.fastpic.jp/images.php?file=1484661557.jpg Table 5.↓ http://www.fastpic.jp/images.php?file=6760884829.jpg Fig. 14. ＆Fig. 15.↓ http://www.fastpic.jp/images.php?file=8798441290.jpg 長文になりますが、よろしくお願いします。
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We evaluated <P(e, 0)> for 12 cases of e between 0 and 6: e=0.0, 0.01, 0.1, 0.5, 0.75, 0.9, 1.0, 1.2, 1.5, 2.0, 4.0, and 6.0. As for r_p, we considered three cases: r_p=0.005, 0.001, and 0.0002. These are representative values of radii of protoplanets at the Earth, Jupiter, and Neptune orbits regions, respectively. The numbers of collision orbits found by our orbital calculation are shown in Table 3 for representative values of e. From Table 3 we can expect the statistical errors in the evaluated collisional rate to be within 5% for the cases of e≦1.5 and within 8% for e=4 and 6; they are smaller than that of the previous studies by Nishida (1983) and by Wetherill and Cox (1985). The calculated collisional rate is summarized in terms of the enhancement factor defined by Eq. (27) and shown in Fig.11, as a function of e and r_p. From Fig.11 one can see that the collisional rate is always enhanced by the effect of solar gravity, compared with that of the two-body approximation <P(e,0)>_2B. In particular, in regions where e≦1, R(e,0) is almost independent of e, having a value as large as 3. At e≦1, R(e,0) has a notable peak beyond which the enhancement factor decreases gradually with increasing e. For large values of e, i.e., e≧4, <P(e,0)> tends rapidly to <P(e,0)>_2B. As seen in the next section, we will find a similar dependence on e even in the three-dimensional case (i≠0) as long as we are concerned with cases where i≦2. お手数ですが、よろしくお願いします。
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The second feature seen from Fig.11 is that the profile of R(e,0) does not depend significantly on r_p (for r_p=0.005 to 0.0002). Only an exception is found near e≒1, but this is, in some sense, a singular point in R(e,0), which appears in a narrow region around e≒1 ( in fact, for e=0.9 and 1.2, there is no appreciable difference between r_p=0.005 and 0.0002). Thus, neglecting such fine structures in R(e,0), we can conclude that R(e,0) does depend very weakly on r_p. In other words, the dependence on r_p of <P(e,0)> is well approximated by that of <P(e,0)>_2B given by Eq. (28). Now, we will phenomenalogically show what physical quantity is related to the peak at e≒1. We introduce the collisional flux F(e,E) for orbits with e and E, where E is the Jacobi energy given by (see Eq. (15)) E=e^2/2-(3b^2)/8+9/2. (31) The collisional flux F(e,E) is defined by F(e,E)=(2/π)∫【‐π→π】p_col(e,i=0, b(E), τ)dτ. (32) From Eqs. (11) and (31), we obtain <P(e,0)>=∫F(e,E)dE. (33) In Fig.12, F(e,E) is plotted as a function of E for the cases of e=0, 0.5, 1.0, and 2.0. We can see from this figure that in the case of e=1 a large fraction of low energy planetesimals contributes to the collisional rate compared to other cases (even to the cases with e<1). In general, in the case of high energy a solution for the three-body problem can be well described by the two-body approximation: in other words, in the case of low energy a large difference would exist between a solution for the three-body problem and that in the two-body approximation. As shown before, this difference appears as an enhancement of the collisional rate. Thereby an enhancement factor peak is formed at e≒1 where a large fraction of low-energy planetesimals contributes to the collisional rate. よろしくお願いいたします。
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Before a detailed description of our numerical procedures, we comment on above simplifications. As seen in an example of e=1.0 and i=0.5, n-recurrent (n≧3) collision orbits contribute by only 1% or less to the collision rate. On the one hand, from calculations with other e and i, the degree of contribution by n-recurrent (n≧3) collision orbits is found to be largest in the case e≒1. Hence, the error in <P(e, i)> introduced by simplification (i) is of the order of 1% or less. As for the applicability of the two-body approximation, we have confirmed in PaperII that the orbit are well described by the two-body formula inside the two-body sphere, whose radius is given by Eq. (13). No appreciable error in <P(e, i)> comes from simplification (ii). Simplification (iii) follows the discussion in the last section. Using above simplifications, we have developed numerical procedures for obtaining <P(e, i)> efficiently; their flow chart is illustrated in Fig. 10. Choosing initial values of orbital elements (e, i, b, τ_s, ω_s), we start to compute numerically Hill’s equations (6) by an ordinary fourth-order Runge-Kutta method from a starting point given by Eqs. (21) and (22). The distance r is checked at every time step of the numerical integration. If the particle flies off to a sufficient distance from the protoplanet after approaching it, i.e., if |y|>y_0+2e, (26) then, the orbital computation is stopped. If a particle approaches the protoplanet and crosses the two-body sphere surface, i.e., if r≦r_cr, the two-body formula is employed to predict whether or not a collision occurs. When no collision occurs at the first encounter, the numerical integration of Hill’s equations is continued. Since a particle which enters the two-body sphere inevitably escapes from the sphere (see Paper II), the particle follows alternatives: one is that it departs to such a distance that Eq. (26) is satisfied, and the other is that it crosses the two-body sphere surface again. In the former case, we stop the computation, considering that the orbit is non-collisional. In the latter case, the occurrence of collision is checked in the same way as earlier by means of the two-body formula, and the orbital calculation is terminated. Using the numerical procedures developed in this way, we obtain <P(e, i)> in many sets of (e, i); the results are presented in the preceeding sections. Fig. 10. Flow chart of orbital calculation for finding collision orbits. Fig. 10. 拡大画像↓ http://www.fastpic.jp/images.php?file=2113240192.jpg 長文になりますが、よろしくお願いします。
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お礼
ありがとうございました． 分かりやすい訳で助かりました． 境界要素法を説明したってどんなことをやってたんだろう？ でも，ほんとありがとうございました．