<jupyter_start><jupyter_text>Fast PDE/IE course, Skoltech, Spring 2015  Problem Set 1  Modeling a Cantilever Beam Consider a beam fixed at one end (called a [cantilever](http://en.wikipedia.org/wiki/Cantilever) beam) as shown here: Its deflection from the equilibrium position is described by the deflection $u = u(x)$ which satisfies the boundary-value problem for the [Euler-Bernoulli equation](http://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory)$$ (EIu'')'' = 0$$$$u(0)=0 ~~~ u'(0)=0$$$$u''(L)=0 ~~~ (EIu'')'(L)=P$$where $P$ is the force applied to the beam’s end, $E$ is the (constant) elastic modulus (a material’s property) and I is the second moment of the area of the cross-section. If the cross-section is a rectangle with height $~H = H(x)$ and the width is $W$ then  $~I = I(x) = WH(x)^3/12$ (see more details in Wikipedia under the link above).  Proposition 1.Assume that $~I = I(x)$ is constant.* **(a)** $$\frac{u(x − 2h) − 4u(x − h) + 6u(x) − 4u(x + h) + u(x + 2h)}{h^4}$$  approximates  $u''''(x)$  with the second order.* **(b)** Consider a grid $~h = L/N~$,  $~xi = h(i − 0.5)~$,  $~i = 0,..., N + 2~$. The last line of ([1](eq)) can be approximated by $$u(L+1.5h)−u(L+0.5h)−u(L−0.5h)+u(L−1.5h) = 0$$$$u(L + 1.5h) − 3u(L + 0.5h) + 3u(L − 0.5h) − u(L − 1.5h) = h^3 P/(EI)$$. Problem 1 (60pt)Assume that $~I = I(x)$ is constant and the mesh is chosen as in Proposition 1.* **(10pt)** Prove part (a) of Proposition 1.* **(10pt)** Formulate the finite difference problem approximating ([1](eq)) with the second order.* **(10pt)** Write a code that implements this finite difference problem by constructing an $(N + 3) × (N + 3)$ matrix $A_h$ and the right-hand side vector $b_h$.* **(10pt)** Given parameters $E = 2.5\, GPa$, $H = 7mm$, $W = 3mm$, $L = 25cm$, $P = 200gm$, $h = L/20$ use your code to compute the solution $u_h = (u_h(−0.5h), . . . , u_h(L + 1.5h))$. What is the computed deflection of the right end (let us denote it as $u_h(L)$ with a slight abuse of notation)? (Don’t know what the [abuse of notation](http://en.wikipedia.org/wiki/Abuse_of_notation) is?)*  (Optional.) Approximate the values $u(0), u(h), . . . , u(L)$. Upload your answer to Stellar as a text file named *PS1P1.txt* with numbers separated by a newline. The first one who does it correctly will get a non-material bonus.* **(10pt)** Find the solution $u$ in the analytic form. Compare $~u(L)~$ with $~u_h(L)~$ for $~h = 25,50,100~$. As you increase $~h~$ by a factor or 2, by what factor does the difference $~|u_h(L) − u(L)|~$ decrease?* **(10pt)** Compute $~λ_{\min}(A_h)~$ and $~λ_{\max}(A_h)~$ for $~h = 25, 50, 100$. As you increase h by a factor or 2, by what factor does $cond(A_h)$ increase? Can you find, approximately, for what $h$ your code computes the value $~|u_h(L)−u(L)|~$ most accurately? Explain why your code’s answers are worse if $~h~$ is less than this value and if $~h~$ is larger than this value. Problem 2 (40pt)Let all parameters be the same as in Problem 1 except for $H(x) = (3 − 2x/L)(2 + \cos(18πx/L)) · 6mm$.* **(10pt)** Formulate the corresponding finite difference problem.* **(10pt)** Write the corresponding code.* **(10pt)** Assuming that the beam fractures at a point where the modulus of the quantity $\sigma(x) = u''(x)H(x)$ is largest, find the point where the beam should fracture. Give details on how you compute it (e.g., what value of h you used).* **(10pt)** Give an example of a smooth function $~H = H(x)$, satisfying $~3mm ≤ H ≤ 15mm$ everywhere, such that the beam fractures at $~x ≈ L/3$.  Problem 3 (bonus problem)Suppose all parameters, except $H=H(x)$ are the same. You need to find $H(x)$ such that* The load that can be applied to the beam before it fractures is maximal subject to the constraints below:* The total material is fixed: $\int_{0}^L H(x) dx = 2000mm^2$* $3mm \leq H \leq 15mm$Take $n=100$. Upload 101 values $u(0), u(1/n), . . . , u(1)$ to Stellar as a text file named *PS1P3.txt* with numbers separated by a newline.<jupyter_code>from IPython.core.display import HTML
def css_styling():
    styles = open("./styles/custom.css", "r").read()
    return HTML(styles)
css_styling()<jupyter_output><empty_output>
{'fraction_non_alphanumeric': 0.16945107398568018, 'fraction_numerical': 0.03532219570405728, 'mean_word_length': 4.325285895806862, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'chain_length': '1', 'dir': 'jupyter-structured-clean-dedup', 'id': '634173', 'license': 'CC0-1.0', 'n_tokens_mistral': 1566, 'n_tokens_neox': 1444, 'n_words': 610, 'path': 'PS1.ipynb', 'repo_name': 'oseledets/fastpde', 'size': '4190'}