I have an assignment to do in which I have attached the question. Only parts (b) and (c) need to be completed. I have also attached notes which should help on the projected SOR algorithm (pages 6 and 7 of the notes).
P-4 1-D Elliptic Inequality Consider now the 1D elliptic inequality problem for u : (0, 1) ! R subject to the Dirich- let BCs: �u00 f u (�u00 � f)(u� ) = 0 9 >= >; for x 2 (0, 1) (15) u(0) = 0 (16) u(1) = 0 (17) with f : (0, 1) ! R and : (0, 1) ! R (obstacle function). (a) PDE Discretisation [See Q-0(a)] Obtain the discretisation for a second-order differ- ence scheme as explained in the Lecture Notes on Inequality Problems. [5 marks] (b) Algorithm Implementation [See Q-0(b)] Implement the Projected SOR method with relaxation parameter !. [5 marks] (c) Code Verification Instead of the usual code verification with a complete exact solu- tion, consider the following data f(x) (x) ! Case 4: 50/3 1 1.8 • Take a mesh-width h = 1/16. Plot the first 8 iterations of the Projected SOR method (! = 1.8), assuming an initial guess ¯ u(0) = ¯ 0. • It may take many iterations for the Projected SOR method to converge. Come up with a criterion to terminate the iterations, and implement this. • For h = 1/16 compare the “converged” approximation to the unconstrained ap- proximation of the standard elliptic model (i.e., take ↵ = � = 0 and f(x) = 50/3 in P-1). Explain why you think the results are correct. • It is known that for the above data, the exact free boundary is at x = 1 5 p 3 and 1� 1 5 p 3 . Given this information on the exact solution, come up with a verification of your implementation. [10 marks] G14CAM — Coursework 2 Page 6 of 8 Lecture ⇤ Numerical Inequality Problems Contents ⇤.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 ⇤.2 Elementary example: Mass–spring–obstacle system . . . . . . . 2 ⇤.2.1 Without bottom wall . . . . . . . . . . . . . . . . . . . . . . . . . . 2 ⇤.2.2 With bottom wall . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 ⇤.3 Signorini’s problem . . . . . . . . . . . . . . . . . . . . . . . . . . 3 ⇤.3.1 Signorini’s problem in 1D . . . . . . . . . . . . . . . . . . . . . . . 4 ⇤.3.2 Discretization and solution algorithms . . . . . . . . . . . . . . . . 6 ⇤.3.3 Signorini’s problem in 2D . . . . . . . . . . . . . . . . . . . . . . . 8 ⇤.4 Bibliographical remarks . . . . . . . . . . . . . . . . . . . . . . . 9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 ⇤.1 Introduction Inequality problems arise in problems with inequality constraints. Such problems can be found in all areas of applied science and engineering. A classical applica- tion is contact problems in elastic mechanics, in which a deforming elastic body is constrained by the contacting object. Inequality problems are free-boundary problems. Indeed, if the solution to a in- equality problem satisfies the inequality (e.g., ) with a strict inequality (<) in="" one="" part="" of="" the="" domain,="" and="" with="" an="" equality="" (=")" in="" the="" remainder="" of="" the="" domain,="" then="" the="" interface="" separating="" these="" subdomains="" is="" a="" so-called="" free="" boundary="" whose="" position="" is="" part="" of="" the="" solution.="" a="" rigorous="" mathematical="" treatment="" (of,="" e.g.,="" well-posedness)="" of="" the="" subject,="" which="" k.g.="" van="" der="" zee,="" computational="" applied="" mathematics,="" lecture="" notes="" for="" module="" g14cam,="" 2017–2018,="" spring="" semester,="" university="" of="" nottingham="" version="" march="" 19,="" 2018="" numerical="" inequality="" problems="" 25.="" variational="" inequalities="" 2="" k="" m="" figure="" 5.1:="" exercise="" :="" a="" uniform="" four-element="" mesh="" and="" four="" basis="" functions.="" fig:fourelementbasis="" 5.2="" elementary="" example:="" mass–spring="" system="" with="" contact="" figure="" ⇤.1:="" a="" mass–spring="" system.="" the="" distance="" to="" the="" bottom="" wall="" is="" .="" is="" beyond="" the="" scope="" of="" this="" lecture,="" requires="" the="" study="" of="" so-called="" variational="" in-="" equalities.="" (these="" are="" weak="" (or="" variational)="" formulations="" for="" which="" the="" trial="" and="" test="" functions="" are="" members="" of="" a="" convex="" subset="" of="" some="" function="" space="" v="" ,="" instead="" of="" the="" whole="" of="" v="" as="" in="" ordinary="" weak="" formulations.)="" the="" contents="" of="" this="" lecture="" are="" as="" follows.="" we="" start="" with="" an="" elementary="" contact="" example="" for="" a="" mass–spring="" system,="" after="" which="" we="" investigate="" signorini’s="" problem="" involving="" an="" elliptic="" pde="" with="" an="" obstacle.="" ⇤.2="" elementary="" example:="" mass–spring–obstacle="" system="" consider="" the="" mass–spring="" system="" in="" figure="" ⇤.1="" with="" mass="" m="" and="" spring="" constant="" k.="" if="" no="" forces="" are="" acting="" on="" the="" mass,="" the="" distance="" to="" the="" bottom="" boundary="" is="" .="" ⇤.2.1="" without="" bottom="" wall="" let="" us="" first="" disregard="" the="" bottom="" wall.="" suppose="" that="" a="" force,="" for="" example="" mg="" (due="" to="" gravity),="" acts="" on="" the="" mass.="" the="" (equilibrium)="" vertical="" displacement="" u="" (positive="" if="" downwards)="" can="" then="" be="" computed="" using="" force="" balance:="" k="" u="mg" .="" obviously,="" this="" yields="" the="" solution:="" u="mg" k="" .="" numerical="" inequality="" problems="" 3="" ⇤.2.2="" with="" bottom="" wall="" the="" situation="" is="" more="" complicated="" if="" the="" bottom="" wall="" is="" included="" in="" our="" considerations.="" indeed,="" there="" are="" two="" situations="" that="" may="" occur:="" i)="" the="" force="" is="" small="" enough="" so="" that="" no="" contact="" occurs="" with="" the="" wall.="" the="" displace-="" ment="" is="" then="" governed="" by="" force="" balance.="" in="" other="" words:="" no="" contact:="" u="">< and="" k="" u="mg" .="" (⇤.1a)="" ii)="" the="" force="" is="" large="" enough="" so="" that="" contact="" occurs="" with="" the="" wall.="" the="" displacement="" then="" equals="" the="" gap="" distance,="" and="" mg="" is="" at="" least="" as="" large="" as="" the="" spring="" force.="" in="" other="" words:="" contact:="" u="and" k="" u="" ="" mg="" .="" (⇤.1b)="" notice="" that="" the="" above="" two="" sets="" of="" conditions="" (⇤.1a)–(⇤.1b)="" only="" apply="" to="" one="" particular="" situation.="" let="" us="" now="" give="" a="" formulation="" which="" encapsulates="" both="" situations="" simultaneously.="" the="" formulation="" is="" known="" as="" a="" linear="" complementarity="" system="" (or="" linear="" complemen-="" tarity="" problem):="" (k="" u�mg)="" (u�="" )="0" ,="" u�="" ="" 0="" ,="" k="" u�mg="" ="" 0="" .="" (⇤.2a)="" (⇤.2b)="" (⇤.2c)="" it="" is="" worth="" verifying="" its="" equivalence="" with="" (⇤.1a)="" and="" (⇤.1b).="" firstly,="" note="" that="" (⇤.1a)="" and="" (⇤.1b)="" both="" separately="" imply="" (⇤.2).="" indeed,="" u="">< implies="" (⇤.2b)="" while="" ku="mg" implies="" (⇤.2a)="" and="" (⇤.2c).="" similarly="" for="" (⇤.1b).="" conversely,="" the="" linear="" complementarity="" system="" (⇤.2)="" implies="" either="" of="" the="" two="" situations:="" no="" contact="" (⇤.1a)="" or="" contact="" (⇤.1b).="" indeed,="" suppose="" that="" u="" �="">< 0="" (no="" contact),="" then="" (⇤.2a)="" implies="" that="" ku="mg." however,="" if="" ku="">< mg,="" then="" (⇤.2a)="" implies="" that="" u="(contact)." the="" trick="" that="" is="" used,="" is="" that="" a="" strict="" inequality=""><) in="" (⇤.2b)="" or="" (⇤.2c)="" implies="" (via="" (⇤.2a))="" an="" equality="" (=")" in="" the="" other.="" ⇤.3="" signorini’s="" problem="" in="" this="" section="" we="" consider="" signorini’s="" problem.="" this="" problem="" is="" a="" standard="" model="" inequality="" problem="" that="" involves="" an="" elliptic="" pde="" for="" which="" the="" solution="" is="" constrained="" numerical="" inequality="" problems="" 4="" f(x)="" x="" u(x)="" �="" (x)="" +="" figure="" ⇤.2:="" a="" distributed="" load="" f="" is="" applied="" to="" an="" elastic="" string="" [left="" ].="" an="" obstacle="" acts="" as="" a="" constraint="" on="" the="" displacement="" u="" [right="" ].="" by="" an="" obstacle.1="" ⇤.3.1="" signorini’s="" problem="" in="" 1d="" signorini’s="" problem="" in="" 1d="" corresponds="" to="" the="" following="" physical="" situation.="" consider="" a="" perfectly-elastic="" string="" which="" is="" held="" fixed="" at="" its="" end-points.="" the="" string="" is="" deformed="" transversely="" because="" of="" some="" distributed="" load,="" and="" thereby="" partly="" comes="" into="" contact="" with="" an="" obstacle;="" see="" figure="" ⇤.2.="" to="" describe="" the="" mathematical="" model,="" let="" the="" domain="" ⌦="" :="(0," `)="" with="" `="" denoting="" the="" length="" of="" the="" undeformed="" string,="" and="" let="" f="" :="" ⌦="" !="" r="" be="" the="" distributed="" load.="" the="" vertical="" displacement="" field="" of="" the="" string="" is="" denoted="" by="" u="" :="" ⌦="" !="" r="" and="" taken="" as="" positive="" when="" downwards2.="" the="" boundary="" conditions="" imply="" that="" u(0)="0" and="" u(`)="0." what="" is="" the="" problem="" for="" solving="" u?="" without="" the="" obstacle,="" the="" solution="" u="" satisfies="" the="" pde:="" �e0="" d2u="" dx2="f" in="" ⌦="" .="" where="" e0=""> 0 denotes a material constant (elastic modulus), With the obstacle, the problem may be written as the linear complementarity 1The model problem we study is generally referred to as an obstacle problem and actually not as a Signorini problem. Only obstacle problems which induce an inequality on the boundary of the domain are generally referred to as Signorini problems. We shall nevertheless not make this distinction. 2This is just a sign convention that makes sense for the specific physical situation. Numerical inequality problems 5 system: ⇣ � E0 d2u dx2 � f ⌘⇣ u� ⌘ = 0 u� 0 �E0 d2u dx2 � f 0 9 > > > > > = > > > > > ; for a.e. x 2 ⌦ . (⇤.3a) (⇤.3b) (⇤.3c) where : ⌦ ! R denotes the position of the obstacle (positive when downwards). In particular, note that Eq.