21MAT31 Lecture : Numerical Solutions of Partial Differential Equations

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amankg4251

24 Mar 2023

School

Visvesvaraya Technological University - VTU

Department

Engineering

Course

21MAT31

Professor

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Document Summary

The partial differetial eaualion o said: be () ellpic i. I which pant h e (,a) plane euatioy elliplt elliptic. the fol" rust ng. B eu a l fon elliphc if e-ac<0. Con sd e r plane a nef work o. Ax= h cm a eo r in h e. Xrawing the e iney intevseekiim k j o"2,,. thexe awe caled mexh poins. 9rid pointo g i (,9), a+h, ) , +ah, 3) Finile deme opp-imat:on for padial de vai ves i,r h. Conside he laplace equ a t i on u = Keplacing theiv. theiv. deremce appoti mabims we hane u. 9ui,j+ 4f,j, mij-1-24j+sh-0 h for for aauare meh h =k. 4-,j 2ut,j +j+lj , uij-1-2uij tj+l =o thenw h2 h2 by h. 4-,j 2ut,j +j+lj , uij-1-2uij tj+l =o mulhiply. Thi s at a aveyae ou to the elt, oight. Amtevi ov mesh point the neigh bouvi ng points above amd belous.

Related Questions

Norwegian Binding

Norwegian is an SVO language that is syntactically similar to English. Unlike English, Norwegian has two possessive determiners that can be used with male antecedents: hans and sin. (English only has one: his.) In this task you’ll ﬁrst have to ﬁgure out whether hans is an anaphor or a pronoun. Then you’ll have to ﬁgure out whether sin is an anaphor or pronoun. In order to show that something is a pronoun or an anaphor, you’ll have to consider whether it follows Principle A or Principle B of the Binding Theory. Remember: your answers should discuss the Norwegian examples, NOT their English translations!

The Possessor Hans

Let’s start with Hans. Here are the two hypotheses under consideration:

Hypothesis A1: Possessive hans is an anaphor.

Hypothesis B1: Possessive hans is a pronoun.

The data below illustrate how hans can - and cannot - be used. NOTE: I have glossed hans as POSS. HANS. In the English translations, hans is translated as ‘his’. Do NOT base your conclusions about what the sentences mean on the English translations! Base your conclusions on the co-indexation in the examples. Pay attention to which co-indexations are NOT acceptable!

(15) Dave(i) sa at Frank k spiste hans (i) mat.

Dave said that Frank ate POSS . HANS food

‘Dave said that Frank ate his food.’

(16) Dave(i) sa at hans(i) mat var kald.

Dave said that POSS . HANS food was cold

‘Dave said that his food was cold.’

(17) Daves(i) far k spiste hans(i) mat.

Dave’s father ate POSS . HANS food

‘Dave’s dad ate his food.’

(18) Dave(i) kom inn. Hans(i) mat var kald.

Dave came in POSS . HANS food was cold

‘Dave came in. His food was cold.’

Now answer these questions. Again, make your answers complete, precise, and as concise as possible.

A. According to Hypothesis A1 does the antecedent of hans have to be local to its antecedent (i.e. within the same clause)? Why or why not?

B. According to Hypothesis A1 does the antecedent of hans have to c-command hans?

C. According to Hypothesis B1 does the antecedent of hans have to be local to its antecedent (i.e. within the same clause)? Why or why not?

D. According to Hypothesis B1 does the antecedent of hans have to c-command hans?

E. Can hans corefer with a non-local antecedent? Yes or no?

- Identify which sentence(s) support(s) your conclusion. (Give the sentence number(s))

- Explain how the sentence(s) support(s) your conclusion.

F. Can hans corefer with a DP does not c-command it? Yes or no?

- Identify which sentence(s) support(s) your conclusion. (Give the sentence number(s))

- Explain how the sentence(s) support(s) your conclusion.

G. Based on your answers above, state which hypothesis you think is correct.

Here are the two new hypotheses under consideration:

Hypothesis A2: Possessive sin is a pronoun.

Hypothesis B2: Possessive sin is an anaphor.

The data below illustrate how sin can - and cannot - be used. Once again sin is translated as ‘his’ in the English translations, but this should not be taken to mean that sin and his are the same. Do NOT base your conclusions about what the sentences mean on the English translations! Base your conclusions on the coindexation in the examples. Pay attention to which co-indexations are NOT acceptable!

(19) Dave(i) sa at Frank(k) spiste sin(k) mat.

Dave said that Frank ate POSS.SIN food

‘Dave said that Frank ate his food.’

(20) *Dave(i) sa at Frank(k) spiste sin(i) mat.

Dave said that Frank ate POSS . SIN food

‘Dave said that Frank ate his food.’

(21) *Dave(i) sa at sin(i) mat var kald.

Dave said that POSS . SIN food was cold.

‘Dave said that his food was cold.’

(22) Daves(i) far(k) spiste sin( k/∗i) mat.

Dave’s father ate POSS . SIN food

‘Dave’s dad ate his food.’

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Now answer these questions. Again, make your answers complete, precise, and as concise as possible.

A. According to Hypothesis A2 does the antecedent of sin have to be local (i.e. within the same clause)? Why or why not?

B. According to Hypothesis A2 does the antecedent of sin have to c-command sin? Why or why not?

C. According to Hypothesis B2 does the antecedent of sin have to be local (i.e. within the same clause)? Why or why not?

D. According to Hypothesis B2 does the antecedent of sin have to c-command sin? Why or why not?

E. Can sin corefer with a non-local antecedent? Yes or no?

- Identify which sentence(s) support(s) your conclusion. (Give the sentence number(s))

- Explain how the sentence(s) support(s) your conclusion.

F. Can sin corefer with a DP does not c-command it? Yes or no?

- Identify which sentence(s) support(s) your conclusion. (Give the sentence number(s))

- Explain how the sentence(s) support(s) your conclusion.

G. Based on your answers above, state which hypothesis you think is correct.

astray

Maasai

Consider the data below from Maasai (Eastern Nilotic; Kenya and Tanzania), which consists of verbs marked for first person singular (I), second person singular (you), and third person singular (she/he/it) agreement. Maasai vowels show a [ ±ATR ] contrast, which is similar (but not identical to) the tense/lax distinction in a language like English. In Maasai, [i u e o] are [ +ATR ] , while [ɪ ʊ ɛ ɔ a] are [ −ATR ] .

1st sing. 2nd sing. 3rd sing. Gloss

a-rɪk ɪ-rɪk ɛ-rɪk ‘cause nausea’

a-ɪlɛp ɪ-ɪlɛp. ɛ-ɪlɛp ‘ascend’

a-bɔ ɪ-bɔl ɛ-bɔl ‘hold its mouth’

a-bʊl ɪ-bʊl ɛ-bʊl ‘prosper’

a-raɲ ɪ-raɲ ɛ-raɲ ‘sing’

a-pɛt ɪ-pɛt ɛ-pɛt ‘lean on’

a-rik i-rik e-rik ‘lead it’

a-lep i-lep e-lep ‘milk it’

a-bol i-bol e-bol ‘open it’

a-bul i-bul e-bul ‘pierce it’

1. List all the alternants of the following prefixes.

• 1st singular:

• 2nd singular:

• 3rd singular:

2. What is the natural class of vowels that show an alternation in this data? Using features, characterize the prefix vowels that alternate, making sure that you distinguish those vowels from the prefix vowels that do not alternate.

3. Some of the alternants are a vowel that’s [ +ATR ] and some are a vowel that’s [ −ATR ]. For the alternating prefixes, list the first vowel that comes after it in the root. For instance, [ɪ] is the first vowel in the root after the prefixes in the words with the meaning “cause nausea.”

• After [ +ATR ] prefixes:

• After [ −ATR ] prefixes:

4. Characterize the phonological environments from the previous question using distinctive features.

• After [ +ATR ] prefixes:

• After [ −ATR ] prefixes:

5. First, state in plain language how the vowel in the prefix gets its specification for [ ±ATR ]. When is the vowel [ +ATR ], and when is it [−ATR]? Just a sentence or two is fine, but be sure to use technical vocabulary when appropriate.

6. Now, write a rule in formal notation describing the alternation using the information you’ve learned above. You will probably need to write an α-rule.

7. The UR for the first person prefix is clearly /a/, since it does not alternate. What about the second and third person prefixes? From this data, is it possible to decide which alternant is the UR (e.g. is it /e/ or /ɛ/ for the third person prefix)? Explain your reasoning.

astray

Saturn V First-Stage Structural Analysis The first stage of the Saturn V (also called the S-IC) provided thrust to lift the second stage, third stage, and payload to an altitude of about 100km within the first 150 seconds of flight, at which point it experienced a maximum acceleration of 38𝑚/𝑠2 . The first stage structure (the shell around the propellant tanks) transmitted the force produced by the engines to the upper stages through its outer skin. This skin was composed of 2219-T87 aluminum, which has a Young's modulus of 73.1GPa, a yield strength of 390MPa, a density of 2840 𝑘𝑔/𝑚3 , and a thermal expansion coefficient of 22 ∗ 10-6𝐾-1 . We will perform an analysis of the S-IC structure and the loads it carried through a series of scenarios. The exception is for Part 5, which must be done by hand as it requires significant drawing and only uses symbolic variables, no numerical calculations. I also recommend setting up the equations for Part 4 by hand and then using the .m file to perform the actual computations.

Part 1:

The S-IC stage shell was required to support the load of the second stage, third stage, instrument unit, and Apollo spacecraft modules, both on the pad and during launch. The maximum mass it had to support was when the fully-fueled vehicle was sitting on the pad, and the maximum acceleration it experienced was immediately before stage burnout (when the S-IC was empty but all the other stages were full). Assume the S-IC tank structure is a smooth cylinder with an outer diameter of 10 meters, a thickness of 5.5 mm, and a height of 41 meters when nothing is mounted on top of it. The empty mass of everything above the stage is 9.7 ∗ 104 𝑘𝑔, and the mass of everything above the stage is 2.9 ∗ 106 𝑘𝑔 when fully fueled.

Find:

• How much the stage shrinks when the empty stages are placed atop the S-IC

• How much the stage shrinks when the vehicle is fully fueled

• How much the stage shrinks under full acceleration

• The average axial stress experienced in each of these three cases

• The factor of safety in each of these three cases Ignore the mass of the S-IC structure itself and propellant; all we care about is how the tank has to deal with supporting the masses above it.

Part 2:

In reality, the stage did not have a uniform cross section; it was composed of steps of decreasing thickness toward the top of the S-IC, to save structural mass. Suppose the tank wall thickness decreased from 6.5 mm at the bottom to 4.3 mm at the top in eight steps of equal height. Each of the eight "rings" would have a constant wall thickness, the same height as the other rings, and the total height of the stage would still be 41 meters. Repeat the first three bullet points from Part 1, to find how much the stage will contract with the stepped cross sections instead of the uniform. Additionally, find the mass savings by constructing the tank in this manner.

Part 3:

If the 41m height measurement given in Part 1 was taken at an ambient temperature of 25C, how much would the entire S-IC shrink due to thermal contraction (with no load atop the stage) at an operating temperature of -100C? You may choose either the uniform or stepped cross-section configuration from Part 1 or Part 2 to solve this problem, but there is one optimal solution within the scope of this project. Explain which option you chose and why.

Part 4: Consider the planar truss section shown. Suppose four of these held the S-IC stage upright on the mobile launch platform. If the weight each had to support was 6.9MN (labeled F in the diagram), find the reaction forces at joints B and C (a total of three quantities) and the internal forces in the rods AB, AC, and BC. You may assume that the load is balanced and there is no moment about point A. ABC is an equilateral triangle. Be sure to include your sanity check of the static equilibrium of the whole structure. Briefly describe why the analysis would be complicated by making point C a fixed reaction joint like point B instead of a roller joint.

Part 5: Draw a SMD for the transverse axis of the S-IC using symbolic values (no numbers). The diagram should reflect: 1. A constant, even transverse pressure the vehicle experiences due to wind at the pad (note that this must include the wind pressure experienced by the upper stages, collectively referred to as 𝑃𝑊). 2. The fixed base of the vehicle on the pad. 3. A fueling arm located at 𝑙/2 from the base of the rocket that exerts a load 𝑃𝑓 in a direction counter to the wind. 𝑃𝑓 is not large enough to counteract all the shear experienced by the stage at station 𝑙/2. 07 Apr 2022 Hint: for this problem, continue drawing the SMD along the X axis, with the fixed base at the bottom of the rocket.

Part 6:

The LOX tank bottom is hemispherical, with a total tank height of 19.5 meters tip-to-tip. Suppose the tank were pressurized to 151 kPa and contained LOX with a density of 1141 𝑘𝑔/𝑚3 . Use the Bernoulli equation to find the pressure at the bottom of the tank (at surface gravity) and the thin wall pressure vessel stress experienced by the bottom of the hemispherical tank bottom. At burnout, the tank was empty of liquid oxygen but still under gaseous pressure of 151 kPa. Find the longitudinal and circumferential stresses about the central cylindrical section of the tank. For this part, ignore acceleration, axial forces, thermal deformation, ambient pressure, etc.

Part 7: Even if composite materials (like carbon fiber) had been widely available in the 1960s, give three reasons why they would be less desirable for an application like the S-IC stage than aluminum or titanium

m2gollc

21MAT31 Lecture : Numerical Solutions of Partial Differential Equations

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