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Solution-focused problem solving

This is a page to explore solution-oriented approaches to problem solving.

What is it?
An invitation (through a series of questions) to identify what it would be like if a situation was solved, what would need to be different to get to that solution, and actions to take to get there.

By following an action-reflection cycle, one can then iterate towards a solution.

The problem with problem-focused approaches to problem-solving is that often the frameworks used to describe and define problems can interfere with envisaging and implementing solutions.

When to use it?
Problem conditions which Swenson and Anstett (n.d.) suggest are particularly appropriate for a solution focused approach include:
 * 1) High complexity (elements, interconnections, or rate of change in elements or relationships)
 * 2) Recurrent problems (often indicates that first-order attempts at change have been unsuccessful)
 * 3) Problem escalation. (often indicates that first-order attempts at change have been unsuccessful)
 * 4) Inaction. (Stalemate due to lack of agreed definition, conclusion, or  action to take. Can indicate being too tied to past frameworks or investments).

Suggested readings

 * Solution-focused brief therapy
 * Solution-focused problem solving: Finding exceptions that work (Swenson & Anstett)

title: Solution-focused problem solving


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Talk:Solution-focused problem solving

Practical Example
James, as you know, one of your students (Laura Hale) prefers demonstration over theory.

Can the concepts here be demonstrated by applying them to any number of festering problems which have plagued Wikiversity these past 2 1/2 years?

In particular, how about applying the concepts to this long-festering problem.

Moulton 15:54, 25 February 2011 (UTC)
 * Do any of the question sequences suggested by Swenson and Anstett appeal to you in relation to your problem? -- Jtneill - Talk - c 16:01, 25 February 2011 (UTC)

 * Miracle question sequence.This question sequence is often one of the most productive. It seems to establish a more relaxed and creative attitude and frees the problem solver from the constraints of the problem focus.
 * 1. If a miracle occurred tonight, and when you woke up tomorrow the problem was solved, what would be the first just noticeable indication that things were different?
 * 2. What will have to be different for that to begin happening?
 * 3. When does that already happen, even if only a little? Who will have to do what to make that happen more?
 * 4. What will be an indication to you and others that the problem is really solved?
 * The first thing would be that the error-ridden BLPs would be systematically corrected, and letters of apology would be in the mail to all those who were unjustly defamed in the pages of Wikipedia.
 * WMF will need to establish normative standards for accuracy, excellence, and ethics in online media, especially with respect to the biographies of living persons.
 * It happens when someone who has some semblance of ethics tries to fix the problem. Charles Ainsworth and Scott MacDonald tried.  They (and others like them) will have to prevail in the general case, rather than in one or two hard-fought illustrative cases.
 * When the WMF Board of Trustees establishes a code of ethics for editors who are responsible for crafting biographies of living persons (be it in the main article space or in discussion pages outside of the main article space).
 * Moulton 17:50, 25 February 2011 (UTC)

Thanks, Moulton. Your responses at least help me to envisage and understand what you'd like to see happen. It may then perhaps be possible to describe steps that could be taken towards the goal, based on your experience of what works. To that end, some follow-up questions you might consider are (feel free to find other questions): -- Jtneill - Talk - c 22:11, 25 February 2011 (UTC)
 * 1) Is #4 your overall goal (preferred future)? (If not, clarify your overall goal).
 * 2) How progressed are you currently towards achieving this goal? (e.g., out of 10)
 * 3) What successful steps have you already taken towards achieving this goal?
 * 4) What were you doing when these successful steps towards this goal occurred?
 * 5) What action sequence (steps) do you think you could pursue which would make your your preferred future most likely to occur?

 * Outside of WMF, my overarching life goals are expressed here:

 * Inside of these WMF-sponsored projects, I have a narrower set of goals. Permit me to repost, verbatim, my Objectives, which have been prominently posted on all my user pages since December 2007:

 * After 3 1/2 years, my state of progress on the above objectives, stands at about 3/10.

 * With respect to just one person's BLP, my most successful maneuver was to post this excerpt on the talk page of her BLP. At the time, I was reading a newly published book by my colleague, Sherry Turkle, a sociologist at the MIT Media Lab.

 * I'm unclear what, if anything I can do to achieve Professor's Picard's goal of improving science education, as stated in that excerpt from Professor Turkle's book. Many of us have been working toward that goal for about 30 years now, and I am chagrined to say it appears we are losing ground (especially in the pages of Wikipedia and Wikiversity).

 * Moulton 23:13, 25 February 2011 (UTC)

-- Jtneill - Talk - c 01:30, 26 February 2011 (UTC)
 * 1) Why do you think your w:Talk:Rosalind Picard contribution has been (so far) your most successful maneuver for that particular BLP? What did you do differently on that occasion that made it so successful?
 * 2) Can you think of other examples of progress made by yourself (or others) towards your broader objectives? What was done by those people that helped to facilitate progress?
 * It succeeded because it supplied incontrovertible evidence of Professor Picard's beliefs, in the form of an autobiographical memoir on the very subject of dispute in her BLP. Prior to that, the allied editors of IDCab were absolutely confident that their haphazard theory of mind regarding Picard's beliefs was an unassailable true account of her frame of mind.  I knew they were dead wrong, but I had no way to convince them them of their error. The allied editors of IDCab fought me every step of the way, going so far as to recruit Jimbo Wales to site ban me and threaten to shut down Wikiversity if I didn't get out of their way.
 * Charles Ainsworth has made the most progress. He launched a protracted ArbCom case against FeloniousMonk, then the dominant leader of IDCab.  ArbCom unanimously adjudged FeloniousMonk guilty poor judgment, abuse of power, and leveling meritless accusations against others.  Ainsworth succeeded in getting ArbCom to act where I had failed.  When that case was in progress, NewYorkBrad wrote down his list of principles upon which he proposed to judge the case.  I copied NYBrad's principles here to Wikiversity so we could study them.  But at Jimbo's direction, Darklama baleted that review, thus crippling our effort to study NYBrad's concept of best principles for deciding a case.
 * Moulton 05:36, 26 February 2011 (UTC)

Relevant next questions might be: -- Jtneill - Talk - c 02:00, 2 March 2011 (UTC)
 * 1) How can that be made to occur more often? (Q3 from Exception question sequence)
 * 2) Who will have to do what to make that happen more? (Part of Q3 from Miracle question sequence)
 * 3) How are those things helpful? What else would be helpful? (Q3 from Coping question sequence)
 * I suppose if more scientists wrote and published more autobiographical memoirs that got included in Sherry Turkle's books on the Sociology of Science, the erratic OR PoV pushers in IDCab would find it harder to sustain their haphazard theories of mind regarding the libelous and defamatory BLPs they are crafting.
 * Scientists and academics will have to spend more time writing and publishing personal memoirs detailing their philosophical beliefs, but writing them in sufficiently simple and concise language that people like Tracy Walker, Paul Mitchell, name redacted as (i) that person has not given (implicit or explicit) permission to use it & (ii) it contains nothing even-vaguely-resembling "representations of [their] views", Ian Ramjohn, Mark Pellegrini, and Bob Stevens can understand them.
 * Miracles are helpful, but not terribly reliable. What else would be helpful is to have well-educated editors who are required to take and pass a course in ethics before they are allowed anywhere near a BLP. And then they must take personal responsibility for accuracy and fairness.

 * Moulton 03:37, 2 March 2011 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Solution-focused problem solving


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User talk:Moulton/Agnosimnesia

A great deal of what takes place in Wikiversity and in the pages of its sister WMF sites is Gnosimnemic Recovery of long lost cultural knowledge and wisdom.

My preferred method to revert back to a mindful state of Gnosimnesia is called The Method of Mimnetic Mirroring (or Mimnetics for short). It's basically a meditative and research technique to reach back into the cultural mists to decode old stories, songs and myths, and to decipher their long lost meanings. It's another application of Semiotics, using Wild Mind Associations, also known as Nostalgic Free Association or (in literary circles) Poetry.

Above personal opinion and comment by Moulton 11:39, 8 February 2011, moved to Talk by Abd 17:35, 25 February 2011 (UTC)

title: User talk:Moulton/Agnosimnesia


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Engineering Experience 4: Design a Small Solar Vehicle/Nl/Team AM4

Introduction
Naar aanleiding van het EE4-project op Groept Leuven werd de opdracht gegeven aan verschillende groepjes studenten een zonnewagen te maken. Het doel van dit project is het ontwerpen, bouwen en optimaliseren van een Small Solar Vehicle (SSV). Op 29 april dient dit wagentje afgewerkt te zijn en neemt het team er mee deel aan een race. In deze race legt de SSV een recht parcours van 14 meter af waarvan 6 meter vlak en de laatste 8 meter onder een helling van 3 procent. Het meest innovatieve en mooiste wagentje zullen tevens extra beloond worden. Voor de realisatie van project krijgt elk team een zonnepaneeltje met bijhorende motor ter beschikking. Daarnaast is er een budget voorzien van maximum 200 euro en de mogelijkheid tot het gebruiken van Fablab.

Teamleden
Bruno Crouzé

Denise Jeurissen

Lars Mannaerts

Bart Mentens

Raf Schoenaers

Aurélie Smeekens

Wim Van Mulken

Christof Van Poucke

Blog
Week 1

Wat de EE4 betreft in de eerste week was er op dinsdag het eerste, inleidende, seminarie. We kregen een uitleg over het verloop van het vierde EE project en ons team werd willekeurig door de coach ingedeeld. Meteen werd ook de eerste deadline vastgelegd. We kregen tot vrijdag om de Work Breakdown Structure (WBS), Gantt Chart, Plan Van Aanpak en het Samenwerkingscontract samen te stellen. De eerste deadline werd op tijd behaald.

Week 2

In de tweede week werden we op dinsdag verder ingelicht over concrete berekeningen, hulpmiddelen, problemen etc. waar we gedurende ons project zeker mee te maken zullen hebben. Na dit 2 uur durende seminarie kregen we feedback op onze ingeleverde documenten van deadline 1. Op enkele kleine verbeteringen na was de eerste deadline tot een goed einde gebracht. Zo restte ons ‘enkel’ nog de opdracht om deze verbeteringen tegen vrijdag van dezelfde week in orde te brengen. Het vermelden waard is dat we op dinsdag een nieuwe teamgenoot erbij kregen: Bruno Crouzé. Om 16:00 hielden we onze eerste officiële vergadering. Zo kreeg Bruno ook meteen de kans om kennis te maken met de rest van het team. Tijdens deze vergadering werd de belangrijke basis gelegd van het verdere verloop van het project. Aangezien het hele project nu als een blok voor ons stond was het noodzakelijk om iedereen een duidelijke taak te geven. Tevens werd er vastgelegd dat de vergaderingen steeds op een vast tijdstip zullen plaatsvinden: dinsdag om 16:00 uur. De volgende deadline die voor ons ligt is in week 6. Dit lijkt nog een hele tijd, al wordt er veel van ons verwacht. Dit was de voornaamste reden dat we een persoonlijke deadline oplegden op de donderdag van week 3. Op die manier kunnen we op de vergadering van week 3 eens checken hoever iedereen staat met zijn of haar taak en waar de problemen zitten.

Week 3

Aangezien we deze week geen seminaries hoeven bij te wonen konden we dinsdag voormiddag nuttig besteden aan de wekelijkse vergadering. Na een korte feedback van onze coach over onze gecorrigeerde documenten trokken we naar een computerlokaal. Dit was een uitgelezen kans om samen met het team te brainstormen over het uiteindelijke concept. Iedereen had zijn inbreng waardoor enerzijds onhaalbare fantasieën terug werden opgeborgen, maar anderzijds ook interessante vondsten werden uitgewerkt. Het is op deze bijeenkomst dat onze SSV een eerste concrete vorm kreeg. Het vooropgestelde doel was om het gewicht zo laag mogelijk te houden. Na een snelle berekening van het gewicht van de bodemplaat concludeerde we dat dit geen eenvoudige opdracht zal worden. Het uitkiezen van het geschikte materiaal zal uiteraard cruciaal zijn voor het uiteindelijk gewicht, waardoor we dit ook eens grondig zullen moeten uitzoeken. Ook stootten we op het probleem van hoe we de wielen op de as zullen bevestigen en de motor op de bodemplaat. Verder werd er besproken hoever iedereen stond met zijn persoonlijke opdracht die voor donderdag moest worden verwezenlijkt. De berekeningen vergen wel enige tijd en speciale aandacht aangezien deze cruciaal zijn om een mooie tijd te kunnen neerzetten met de race. Hier zullen we dus ook de nodige tijd voor uittrekken.

Week 4

Dinsdag was er opnieuw een seminarie gepland. Er werd ons uitgelegd hoe we Simulink moesten gebruiken. Het duurde echter maar een halfuurtje waardoor het seminarie slechts beperkt bleef tot een inleiding. Het leek niet dat we er echt veel wijzer uit zijn geworden. Een nadeel waarmee we onmiddellijk werden geconfronteerd was dat we Simulink enkel in GroepT ter beschikking hebben en we er bijgevolg dus niet thuis aan kunnen werken. Hier zullen we uiteraard rekening mee moeten houden. Na de seminarie hadden we nog drie en half uur die ons restte. Net zoals vorige week konden we deze tijd weer nuttig besteden. Zo konden we onder andere eindelijk de gearratio bepalen waar we al een hele tijd ons hoofd over hebben gebroken. Verder zijn deze week al concrete ontwerpen in Solid Edge getekend, waardoor het hele team het ‘einddoel’ eens heeft kunnen aanschouwen. Het was een stimulans om verder te gaan zoals we bezig zijn. Verder hebben we een mail gekregen met de materialen die ter beschikking zijn in Fablab Leuven. Onze voorkeur in verband met het materiaal ligt daardoor al zo goed als vast. Vrijdag 11/03/2011 zal er echter voor het eerst iemand ook effectief naar Fablab Leuven gaan. Bruno zal op die manier verdere informatie verzamelen en horen of ons ontwerp in alle opzichten ook echt wel te verwezenlijken zijn.

Week 5

Week 5 is de week waarbij we voor het eerst een glimp van de bodemplaten van andere teams hebben kunnen opvangen. De ene bleek gelijkaardig aan die van ons, de andere was dan weer wat groter. In dat opzicht leken we qua gewicht wel op de goede weg. Verder hebben we een belangrijke aanpassing kunnen bedenken waar we in de race misschien wel eens voordeel zouden kunnen uithalen. Om evidente redenen kan ik die hier niet verder behandelen. De stemming en werkijver binnen het team zitten beide goed. Iedereen heeft zijn taak en meestal worden de persoonlijk opgestelde taken tijdig uitgevoerd. Dit draagt bij aan de goede stemming, alsook het feit dat we op de vergadering alles eens goed op een rijtje hebben gezet om een duidelijk overzicht te krijgen van waar we precies staan. Dit leerde ons dat we goed op schema zitten en met een paar puntjes zelfs al wat voor zitten op de planning wat natuurlijk nooit kwaad kan. Deze week is het de bedoeling dat onze SSV ook in de praktijk wordt omgezet: de onderdelen worden deze week gehaald en ook de bodemplaat zal in Fablab Leuven gelaserd worden. We beseffen wel dat de kans klein is dat hetgeen dat we nu in elkaar zullen puzzelen meteen goed zal zijn. We zijn benieuwd hoe ver we zullen staan als we een eerste keer echt aan onze SSV zullen geknutseld hebben. Nog even wachten.

Week 6

Deze week is de week van de eerste serieuze vooropgestelde deadline. Vrijdag moeten de berekeningen in verband met 1m afrollen van de helling, de gearratio, het Sankey diagram en - last but not least! – Multisim afgegeven worden. De vergadering van dinsdag is vrij rustig verlopen. De reden hiervan is dat we op een paar aanpassingen na al volledig klaar waren voor de deadline van vrijdag. Het enige dat nog diende te gebeuren was Multisim. Bart en Christof hebben zich daarover ontfermd. Na wat samenwerken met andere teams, na wat hulp van de coach, na heel wat gevloek en vooral na heel wat uren hard doorwerken hebben we dankzij hen een degelijke simulatie in Simulink. Ere wie ere toekomt. Enkele onderdelen van het wagentje werden ook reeds gefabriceerd in Fablab: een prototype van het wiel, de bodemplaat, en een blokje waarin de kogellagers en de as zal komen. Het gehele team heeft de opdracht mee naar huis gekregen om een degelijke oplossing te vinden voor het ondersteunen en regelen van het zonnepaneel. Wanneer iemand het afgesproken onderdeel vindt, zou het wel eens erg snel kunnen gaan. Een volgende hindernis zal het vinden van de juiste tandwielen zijn. Geen simpele taak, maar uiteraard niet onoverkomelijk.

Week 7

We zijn nu al 7 weken druk bezig met ons project. De eerste deadline is achter de rug. Dinsdag hebben we feedback gekregen op onze ingeleverde documenten. Over het algemeen mogen we tot nu toe tevreden zijn over het geleverde werk. Merendeel van de kritiek was te wijten aan enkele details die we nog moeten aanpassen zoals symbolen beter uitleggen, assen benoemen, de berekeningen die in Maple zijn gebeurd moeten duidelijk uitgelegd worden en de Maple-layout moet nog beter worden weggewerkt. Een steroperator mag niet gebruikt worden, en in het tekenen van het Sankeydiagram moet de volgorde van de energie-pijlen verandert worden. Dit laatste is op zich een simpele aanpassing al moet daarvoor wel het diagram helemaal opnieuw worden getekend. Multisim was zeer goed uitgevoerd, enkel de assen van de grafieken moesten benoemd worden en de ruis moest verklaard worden. Over het algemeen moet er ter verduidelijking wat meer uitleg bijgevoegd worden. In tegenstelling tot andere teams was Multisim bij ons goed genoeg. We hebben immers gehoord dat er andere teams zijn die met een verbeterde versie opnieuw moeten terugkomen. De berekeningen van de krachten op de as zijn ook het vermelden waard: niet alleen waren deze berekeningen goed uitgewerkt, maar tevens lopen we hiermee voor op de planning aangezien dit onderdeel slechts voor de tweede deadline moest afgewerkt worden. Hierdoor hebben we niet alleen wat ademruimte gecreëerd, maar hebben we ook extra feedback gekregen op dit onderdeel. Aan het deel marketing wordt ook vlijtig gewerkt en alsook aan het procesverslag. Verder komt de deadline van het ontwerp van het wagentje dichterbij. Normaal komen eind van deze week de bestelling van tandwielen aan en hebben we daarmee zo goed als alle onderdelen voor ons wagentje.

Week 8

Het belangrijkste dat nog in orde moeten maken is zonder twijfel de overbrenging. Het is jammer dat we hier nog niet verder mee zijn geraakt. Zoals eerder geschreven staat hebben we tandwielen op Conrad besteld. Deze zijn weldegelijk aangekomen alleen voldeden ze absoluut niet aan onze verwachtingen en kunnen we er eigenlijk totaal niets mee doen. Het tweede idee is om een kleine tandwielenset te kopen op Conrad waarmee je de gewenste overbrenging kan maken. Het nadeel hiervan is dat de boring in de tandwielen voor de as slechts 3mm groot is. Dit is een probleem aangezien de lagers en de wielen al voorzien zijn voor een as van 4mm. Het gat uitboren is geen oplossing aangezien er tanden vlak rond het gat zitten om de gewenste overbrenging te kunnen maken. Door het gat uit te boren naar 4mm zouden we deze tanden wegboren. Een andere oplossing is de volgende. We zouden een as kunnen kopen van 3mm waar de tandwielen op zouden passen. We kopen daarna een holle buis met een binnendiameter van 3mm en een buitendiameter van 4mm. Door deze dan op de plaats van de lagers en de wielen over de as van 3mm te schuiven lossen we op die manier het probleem op. Of het in praktijk even goed zal lukken als het in de theorie klinkt valt nog af te wachten. Voor de rest hebben we op de rubberen oorringen na alle onderdelen. Deze oorring komt rond de mdf wielen. Op de rand is vakkundig een sleuf getrokken om deze ring mooi te geleiden en laten passen. In de paas-’vakantie’ zal Christof zich dan bezighouden met het assembleren van het zonnewagentje zodat we in week 9 al weer een heel eind verder zijn.

Week 9

De paasvakantie zit erop en Christof heeft de nodige tijd aan de SSV gespendeerd zodat het wagentje volledig is afgewerkt en -al zeggen we het zelf- het resultaat mag er zeker wezen. Uiteindelijk hebben we rond de as een holle as met een iets grotere buitendiameter gebruikt, de tandwielen uit een afstandsbediend autootje, de bodemplaat vergroot en de centrale opening in de bodemplaat weggelaten. Week 9 begint al zeer interessant aangezien we vandaag voor de eerste en ook meteen de laatste keer de SSV kunnen testen op de ‘echte’ racetrack. Nu ja, ‘echte’ is niet het juiste woord aangezien de rail vrij krom lag, maar erger nog, ze lag verkeerd. De doorsnede van de rail kan beschouwd worden als een ‘L’ waarvan het horizontale beentje wordt vastgemaakt op de baan. Vandaag lag de ‘L’ met het beentje naar rechts als je met de rijrichting meekijkt, terwijl het weldegelijk andersom moet zijn. Voor ons is dit vrij belangrijk aangezien ons systeem om de rail te volgen specifiek gemaakt is om rechts van de rail te rijden en niet op het horizontale beentje zoals vandaag dus wel het geval was. Dit hebben we bij de coach gemeld en voor de echte race zou dit in orde moeten zijn. Voor de rest was de test met de SSV zeer positief. Op iets meer dan 6 seconden legt ons wagentje de 14 meter af. Het enige dat we nog kunnen verbeteren is de gleuf in het voorwiel breder maken zodat het wiel beter de rail kan volgen. Een tweede verbeterpunt is dat de achteras iets vlotter mag rollen in de lagers. Aangezien het mogelijk is dat volgende week de race plaatsvindt, moet het eind dit weekend sowieso volledig klaar zijn voor de race. Het vlotte verloop van het testen schept vertrouwen, maar we mogen het vel van de beer niet verkopen voor het geschoten is. We hebben een week om onze wagen echt op punt te stellen dus zullen we de tijd die ons rest verstandig benutten om het maximale uit het wagentje te halen voor de race.

Week 10

Vandaag is het een bijzondere dag geworden! Maandag 2 mei in de namiddag hebben we vernomen dat we ’s anderendaags de Solar Race zullen hebben. Aangezien de lagers bij het testen vorige week een beetje te fel schuurden moesten deze vervangen worden. Omdat we niet tijdig aan de lagers geraakten hebben we die op maandag nog snel moeten vinden en heeft Christof het wagentje thuis helemaal race-klaar gemaakt. Vandaag hebben we vanaf half 11 nog even de tijd gehad om het wagentje te testen. Maar goed ook: bij het testen schuurden de tandwielen namelijk over elkaar! Door het motortje vast te zetten was dit gelukkig verholpen en waren we helemaal klaar voor de race! De race is schitterend verlopen. We werden uiteindelijk 2e van de 19 deelnemende teams! Door een foute handeling bij het opvangen van het wagentje is bij de allereerste race een achterwiel gesneuveld maar door de reserveonderdelen hebben we gelukkig een zeer geslaagde pitstop kunnen afwerken tussen 2 races in. Met deze herstelling hebben we het helemaal tot in de grote finale geschopt. De beste tijd die vandaag gereden is bedroeg iets meer dan 4,9 seconden terwijl wij onze snelste tijd klokte in de finale met nauwelijks 5 seconden! Ook dit was goed voor een tweede plaats! Je merkt dat het zelfs in de finale niet meer veel scheelde of we hadden helemaal bovenaan gestaan! We zijn uiteraard ontzettend tevreden met onze tweede plaats, dit overtreft onze mooiste dromen. We zijn erin geslaagd om van nul af aan een solarwagentje te ontwerpen dat later de op 1 na snelste auto van het tweede jaar zal blijken! Onze welgemeende proficiat aan het snelste team. Volgende week volgt nog een laatste onderdeel, dan zal het origineelste/mooiste autootje gekozen worden. Ons vooropgestelde doel ging om de snelheid en niet om het esthetische. Volgende week zullen we naar onze verwachtingen geen potten breken, iets dat we vandaag duidelijk wél hebben gedaan! Wat een prachtige dag.

Week 11

De weken lopen op hun einde, al is deze week nog een belangrijke week aangezien onze laatste deadline vrijdag is. Dit is eigenlijk dé deadline aangezien dan alles af moet zijn: het rapport, procesverslag etc. Voor de rest hebben we volgende weken enkel nog feedbacksessies opstaan, aanstaande dinsdag hebben we de voorstelling van het autootje in het Atrium. Dan zal het origineelste en mooiste, maar ook het innovatiefste wagentje gekozen worden. We verwachten niet hier nog eens in de prijzen te vallen, aangezien we ervoor gekozen hebben om het snelste wagentje te bouwen. Vandaar dat het wagentje licht moest blijven waardoor het esthetische heeft moeten onder doen voor het functionele. We zijn echter zeer blij met onze tweede plaats in de race, het voelt bijna net zo goed aan alsof we eerste zijn geworden.

Ontwerp
Het ontwerp van onze SSV is vrij simpel. We kozen voor 1 wiel vooraan en 2 achteraan. De motor bevindt zich achteraan en drijft via tandwielen en een as de 2 wielen aan. De basis van de carrosserie is gemaakt uit MDF (Medium Density Fibreboard). Dit kozen we omdat het een stevig materiaal is, dat tevens licht en lijmvast is. Ook de wielen en de lagervoetjes zijn van MDF. Al deze onderdelen werden getekend in Solid Edge en geproduceerd bij Fablab Leuven. Het zonnepaneel zal op de grondplaat bevestigd worden via een staander met daarop een kogelgewricht. Zo is het paneel beweegbaar en kan het gedraaid worden richting de zon voor optimale prestaties. In het voorste wiel is een systeem voorzien om de wagen recht te houden op het parcours. Het wiel bestaat uit 2 delen waardoor de rail op het parcours gevolgd wordt.


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User:Eml5526.s11.team2/topsecret topsecret4

=Problem 4.1: Solve the PDE using a half Fourier Basis with the Weighted Residual Method =

Given: PDE, Boundary Conditions, and Basis Function
PDE:
 * {| style="width:100%" border="0"

$$  \displaystyle 2\frac{d^2u}{dx^2}+3=0 $$     (4.1.1)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

Boundary Conditions:
 * {| style="width:100%" border="0"

$$  \displaystyle u(x=1)=0 $$     (4.1.2)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle -\frac{du}{dx}(x=0)=4 $$     (4.1.3)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

Half Fourier Basis:
 * {| style="width:100%" border="0"

$$  \displaystyle b_j(x)={1, sin(x), sin(2x),...,sin(jx)} $$     (4.1.4)
 * style="width:95%" |
 * style="width:95%" |
 *
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Find: Unknown Coefficients in $$ u^h(x) $$ expansion to approximate $$ u(x) $$ of the PDE.
We want to approximate the solution of Eq.4.1.1 by a series expansion.
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$$  \displaystyle u^h(x)=\sum_{i=0}^n{d_ib_i(x)} $$     (4.1.5) To solve for the unknown coefficients $$ d_i, $$ we must develop the same number of linearly independent equations as we have unknowns. The number of unknowns we generate with the expansion of $$ u^h(x) $$ is entirely up to us, i.e. is it dependent on the number of degrees of freedom we decide to use for the problem formulation. The following solutions are examples of using 3, 5 and 7 degrees of freedom.
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Develop 2 equations which satisfy the given Boundary Conditions.
First let's expand Eq.4.1.1 out to n=2 terms.
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$$  \displaystyle u^h(x)=d_0b_0+d_1b_1+d_2b_2 $$     (4.1.6) Now consider Eq.4.1.4, and substitute the appropriate terms into Eq.4.1.6.
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$$  \displaystyle u^h(x)=d_0(1)+d_1\Big(sin(x)\Big)+d_2\Big(sin(2x)\Big) $$     (4.1.7) Let's rewrite this equation in terms of a product between 2 vectors.
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$$  \displaystyle u^h(x)=\left[1\quad sin(x) \quad sin(2x)\right ]^T\begin{bmatrix} d_0\\d_1\\d_2\end{bmatrix} $$     (4.1.8)
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In order for us to solve for the unknown $$d_i's$$ in Eq.4.1.8, we must have 3 linearly independent equations describing linear combinations of the $$d_i's.$$Two of the equations are easily obtained directly from the given boundary conditions in Eq.4.1.3 and Eq.4.1.4. We want our approximation to closely resemble our exact solution, therefore we must require identical behavior between Eq.4.1.6 and Eq.4.1.1 at the boundaries.
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$$  \displaystyle \begin{align} u^h(x=1)=d_0(1)+d_1sin(1)+d_2sin(2)&=0\\ \end{align} $$     (4.1.9)
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$$  \displaystyle \frac{du}{dx}(x=0)=d_1cos(0)+2d_2cos(0)=-4 $$     (4.1.11)
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Use the Weighted Residual Method to develop a $$ 3^{rd} $$ linearly independent equation.
Now we just need one more linearly independent equation and we can then readily solve for the $$d_i's.$$ To develop another set of equations which are linearly independent of Eq.4.1.10 and Eq.4.1.12, we take the inner product between, Eq.4.1.15 and our basis set of functions Eq.4.1.4. Before getting to the inner product first, let's define our linear differential operator as the following
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$$  \displaystyle P(\cdot)=\frac{d^2}{dx^2}+\frac{3}{2}=0 $$     (4.1.13) Now, using our operator in Eq.4.1.13, we can operate on our approximation in Eq.4.1.8.
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$$  \displaystyle \begin{align} P(u^h)=\frac{d^2}{dx^2}\left[1\quad sin(x)\quad sin(2x)\right ]^T\begin{bmatrix} d_0\\d_1\\d_2\end{bmatrix}+\frac{3}{2}&=0\\ \left[0\quad -sin(x)\quad -4sin(2x)\right ]^T\begin{bmatrix} d_0\\d_1\\d_2\end{bmatrix}+\frac{3}{2}&=0 \end{align} $$     (4.1.14) Which we can then rewrite Eq.4.1.14 as
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$$  \displaystyle \left[0\quad -sin(x)\quad -4sin(2x)\right ]^T\begin{bmatrix} d_0\\d_1\\d_2\end{bmatrix}=-\frac{3}{2} $$     (4.1.15) Now, we are in a position to take the inner product of Eq.4.1.15(our approximation) with the basis set in Eq.4.1.4 with $$ n=2.$$
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$$  \displaystyle \left \langle b_j, P(u^h) \right \rangle =\int_0^1b_jP(u^h)=0 $$     (4.1.16)
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$$  \displaystyle \int_0^1\begin{bmatrix} 1\\sin(x)\\sin(2x)\end{bmatrix} \left[0\quad -sin(x)\quad -4sin(2x)\right ]^T\,dx\begin{bmatrix} d_0\\d_1\\d_2\end{bmatrix}=-\frac{3}{2}\int_0^1\begin{bmatrix} 1\\sin(x)\\sin(2x)\end{bmatrix}\,dx $$     (4.1.17)
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$$  \displaystyle \int_0^1\begin{bmatrix} 0&-sin(x)&-4sin(2x)\\0&-sin^2(x)&-4sin(x)sin(2x)\\0&-sin(x)sin(2x)&-4sin^2(2x)\end{bmatrix}\,dx\begin{bmatrix} d_0\\d_1\\d_2\end{bmatrix}=-\frac{3}{2}\int_0^1\begin{bmatrix} 1\\sin(x)\\sin(2x)\end{bmatrix}\,dx $$     (4.1.18)
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$$  \displaystyle \begin{bmatrix} 0&-1&2cos(2)-2\\0&sin(2)/4-1/2&2sin(3)/3-2sin(1)\\0&sin(3)/6-sin(1)/2&sin(4)/2-2\end{bmatrix}\begin{bmatrix} d_0\\d_1\\d_2\end{bmatrix}=\begin{bmatrix} -3/2\\-3/2\\-3sin^2(1)/2\end{bmatrix} $$     (4.1.19) We can use any one of the above equations as our $$ 3^{rd} $$ equation. For simplicity, let's choose the $$ 1^{st} $$ equation in the above matrix expression. This would be
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(4.1.20)
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Now we are set to expression our formulated problem in Matrix Form.

Display in Matrix Form

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$$  \displaystyle \begin{bmatrix} 1&sin(1)&sin(2)\\0&1&2\\0&-1&2\Big(cos(2)-1 \Big)\end{bmatrix}\begin{bmatrix} d_0\\d_1\\d_2\end{bmatrix}=\begin{bmatrix} 0\\-4\\-3/2\end{bmatrix} $$     (4.1.21)
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Solve for unknown d.
Our stiffness matrix is
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$$  \displaystyle \underline{\mathbf{K}}=\begin{bmatrix} 1&sin(1)&sin(2)\\0&1&2\\0&-1&2\Big(cos(2)-1 \Big)\end{bmatrix} $$     (4.1.22) Using MATLAB to aid in our solution we find the inverse of our stiffness matrix to be
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$$  \displaystyle \underline{\mathbf{K}}^{-1}=\begin{bmatrix} 1&0&0\\1327/1577&1&-1\\401/441&2&-2644/1443\end{bmatrix} $$     (4.1.23) Therefore, we find $$ {\mathbf{d}} $$ to be
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$$  \displaystyle {\mathbf{d}} = {\mathbf{K}}^{-1}{\mathbf{f}} $$     (4.1.24)
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$$  \displaystyle \begin{bmatrix} d_0\\d_1\\d_2\end{bmatrix}=\begin{bmatrix} 1&0&0\\1327/1577&1&-1\\401/441&2&-2644/1443\end{bmatrix} \begin{bmatrix}0\\-4\\-3/2\end{bmatrix} $$     (4.1.25)
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Construct,Plot and find the error between the approximate and exact solutions for u(x).
A plot of both the approximate and exact solutions can be found below.

We can see that our approximate solution is though very close to exact, does not converge as quickly as we saw the polynomial basis function converge. This to be due to the slower converging half Fourier Basis when approximating a polynomial, versus using a polynomial basis. When using a polynomial basis of degree n, it will provide us with the exact solution of any polynomial of degree n or less. This can be easily proven through the use of the Integral Mean Value Theorem. Let's move on to using 5 degrees of freedom (n=4) and see our or approximate solution can improve.
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$$  \displaystyle \Delta \mathbf{E_{abs}}=u^h(x=0.5)-u(x=0.5) $$     (4.1.27)
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Or in other words approximately 3%. Not to bad but lets see if we can get better.

Construct,Plot and find the error between the approximate and exact solutions for u(x)..
Being that we have extensively explained the mathematics for case 2 above, we will just show the MATLAB code that I have written to solve this problem while focus more on the results of increasing the degrees of freedom, i.e. the number of coefficients that we expand to in our approximate solution. This not only makes it easier on the writer, but on the reader as well. The following is the Plot for the case n=4.

We see that we have a little further away with the 2 extra degrees of freedom. Let's calculate our error below.
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$$  \displaystyle \begin{align} \Delta \mathbf{E_{abs}}=u^h(x=0.5)-u(x=0.5) \end{align} $$     (4.1.29) Or in other words we have increased our error with an increase in expansion terms of our approximate solution.
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Computational Difficulties
This case proved to be too computational demanding for my computer. I let the code, which worked for both cases 2 and 4 above run for 45 min and still MATLAB said it was busy. Therefore, I force quit my program and can conclude our best result is obtained with the n=2 case.

=Problem 4.2: Heat Conduction Fish & Belytschko Page 73 Problem 3.5 =

Given: The Following Boundary Conditions
Where (4.2.1) is the essential boundary condition defined at $$ \displaystyle x = 0 $$ and (4.2.2) is the natural boundary condition defined at $$ \displaystyle x = 10 $$ on the domain $$ \Omega = ]0,10[ \quad $$

Find:The Weak Form for 1-D Heat Conduction From the Strong Form
Obtain the weak form for the equation of the heat conduction with the boundary conditions stated above. The condition on the natural boundary condition is a convection condition.

Solution: Obtaining the Weak Form
The strong form has the following equation for a 1-D heat conduction body.

Assuming a steady-state from for the heat conduction and a constant area ( 4.2.3) reduces to

Multiplying by an arbitrary weighting function $$ \displaystyle w(x) $$ and integrating with respect to x across the domain leads to the following form

Making use of Integration by Parts on the first term of (4.2.5) leads to the following equation.

Using the requirement of the essential boundary condition on the weighting function satisfying the essential boundary location implies that $$ \displaystyle w(0) = 0 $$ and also using the natural boundary condition (4.2.2) $$ k \frac{\partial T}{\partial x}\bigg|_{x=0} = hT $$ leads to the following

Therefore the weak form can be represented finding a $$ \displaystyle T(x) $$ such that for all smooth $$ \displaystyle T(x) $$ with $$ \displaystyle T(0) = 100 $$ (4.2.8) is satisfied.

= Problem 4.3: Construct weak form of heat problem and find solution =

Given: Strong Form and Boundary Conditions
Statement:- Given the strong form for the heat conduction problem in a circular plate:
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$$  \displaystyle k\frac{d}\left( {r\frac} \right) + rs = 0,0 < r \leqslant R $$ (4.3.1) natural boundary condition :$$ \frac\left( {r = 0} \right) = 0$$
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essential boundary condition :$$T\left( {r = R} \right) = 0$$

where R is the total radius of the plate, s is the heat source per unit length along the plate radius, T is the temperature and k is the conductivity. Assume that k, s and R are given:

a. Weak Form
Construct the weak form for the above strong form.

b. Solution Using Quadratic Polynomial Trial Function
Use quadratic trial (candidate) solutions of the form $$ \displaystyle T = {\alpha _0} + {\alpha _1}r + {\alpha _2}{r^2}$$ and weight functions of the same form to obtain a solution of the weak form.

c. Solve the Differential Equation
Solve the differential equation with the boundary conditions and showthat the temperature distribution along the radius is given by:

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$$  \displaystyle T = \frac{s}\left( {{R^2} - {r^2}} \right) $$     (4.3.2)
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a. Weak Form Derivation
Strong Form Can be written as:

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$$  \displaystyle p\left( T \right) = 0 $$
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$$  \displaystyle p\left( T \right) = k\frac{d}\left( {r\frac} \right) + rs $$
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$$w\left( r \right)$$ is weight function Weighted Residual Form can be written as

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$$  \displaystyle \int\limits_0^R {w\left( r \right)p\left( T \right)dr = 0} $$     (4.3.3)
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$$  \displaystyle \int\limits_0^R {w\left( r \right)\left[ {k\frac{d}\left( {r\frac} \right) + rs} \right]dr} = 0 $$
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$$  \displaystyle \int\limits_0^R {w\left( r \right)k\frac{d}\left( {r\frac} \right)dr + } \int\limits_0^R {w\left( r \right)rsdr} = 0 $$
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Write $$w\left( r \right) = w$$ Integrating the first term by parts.

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$$  \displaystyle \left[ {wkr\frac} \right]_0^R - \int\limits_0^R {\left( {\frackr\frac} \right)} dr + \int\limits_0^R {wrsdr} = 0 $$
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Assuming $$w\left( R \right) = 0$$ and using natural boundary condition $$\frac\left( {r = 0} \right) = 0$$

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$$  \displaystyle 0 - 0 - \int\limits_0^R {\left( {\frackr\frac} \right)} dr + \int\limits_0^R {wrsdr} = 0 $$
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$$  \displaystyle \int\limits_0^R {\left( {\frackr\frac} \right)} dr = \int\limits_0^R {wrsdr} $$     (4.3.4)
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Writing in standard form:

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b. Solution using quadratic polynomial trial function
let $$T$$ and $$w$$ are of the following
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$$  \displaystyle T = {\alpha _0} + {\alpha _1}r + {\alpha _2}{r^2} $$ and $$ \displaystyle w = {\beta _0} + {\beta _1}r + {\beta _2}{r^2} $$     (4.3.6)
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Now using essential Boundary Condition i.e $$T\left( R \right) = 0$$

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$$  \displaystyle {\alpha _0} + {\alpha _1}R + {\alpha _2}{R^2} = 0 $$     (4.3.7)
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$$  \displaystyle {\alpha _0} = - \left( {{\alpha _1}R + {\alpha _2}{R^2}} \right) $$     (4.3.8)
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Now using Natural boundary condition i.e. $$\frac\left( {r = 0} \right) = 0$$

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$$  \displaystyle {\alpha _1} + 2{\alpha _2}\left( 0 \right) = 0 $$     (4.3.9)
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$$  \displaystyle {\alpha _1} = 0 $$     (4.3.10)
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$$  \displaystyle {\alpha _0} = - {\alpha _2}{R^2} $$     (4.3.11)
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Now using homogeneous boundary condition $$w\left( R \right) = 0$$
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$$  \displaystyle {\beta _0} + {\beta _1}R + {\beta _2}{R^2} = 0 $$     (4.3.12)
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$$  \displaystyle {\beta _0} = - \left( {{\beta _1}R + {\beta _2}{R^2}} \right) $$     (4.3.13)
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Also

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$$  \displaystyle \frac = {\alpha _1} + 2{\alpha _2}r $$     (4.3.14)
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As $${\alpha _1} = 0$$
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$$  \displaystyle \frac = 2{\alpha _2}r $$     (4.3.15)
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Now differentiating $$w$$ with respect to $$r$$
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$$  \displaystyle \frac = {\beta _1} + 2{\beta _2}r $$     (4.3.16)
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Putting values of $$w,\frac,\frac$$ in the standard weak form.

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$$  \displaystyle \int\limits_0^R {kr\left( {{\beta _1} + 2{\beta _2}r} \right)} \left( {2{\alpha _2}r} \right)dr = \int\limits_0^R {sr\left( {{\beta _0} + {\beta _1}r + {\beta _2}{r^2}} \right)} dr $$
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$$  \displaystyle k\int\limits_0^R {\left( {2{\beta _1}{\alpha _2}{r^2} + 4{\beta _2}{\alpha _2}{r^3}} \right)dr} = s\int\limits_0^R {\left( {{\beta _0}r + {\beta _1}{r^2} + {\beta _2}{r^3}} \right)dr} $$
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$$  \displaystyle \left[ {\frac{3} + k{\beta _2}{\alpha _2}{r^4}} \right]_0^R = \left[ {\frac{2} + \frac{3} + \frac{4}} \right]_0^R $$
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Putting value of $$$$ from equation

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$$  \displaystyle \frac{3} + k{\beta _2}{\alpha _2}{R^4} = \frac{2} + \frac{3} + \frac{4} $$
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Rearranging terms

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$$  \displaystyle {\beta _1}\left[ {\frac{3} + \frac{2} - \frac{3}} \right] + {\beta _2}\left[ {k{\alpha _2}{R^4} + \frac{2} - \frac{4}} \right] = 0 $$
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As $${\beta _1},{\beta _2}$$ are arbitrary they can't be zero, putting bracketed terms equal to zero both equation give same value of $${\alpha _2}$$.

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$$  \displaystyle {\alpha _2} = \frac $$     (4.3.17)
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Using equation (4.3.10) $${\alpha _0}$$
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$$  \displaystyle {\alpha _0} = - {\alpha _2}{R^2} = \frac $$     (4.3.18)
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Putting these values in $$T = {\alpha _0} + {\alpha _1}r + {\alpha _2}{r^2}$$

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$$  \displaystyle T = \frac - \frac $$
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Now checking this solution by putting in given strong form given in equation.....

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$$  \displaystyle k\frac{d}\left( {r\frac{d}\left( {\frac{s}\left( {{R^2} - {r^2}} \right)} \right)} \right) + rs = 0 $$
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$$  \displaystyle k\frac{d}\left( { - \frac} \right) + rs = 0 $$
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$$  \displaystyle k\left( { - \frac{k}} \right) + rs = 0 $$
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$$  \displaystyle - \cancel{sr} + \cancel{sr} = 0 $$     (4.3.20)
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Comments
As we are not getting any particular interval of 'r' by putting the weak form solution in the given differential equation, so this solution is valid on given domain of 'r'.

c. Exact solution for given differential equation
Starting from 4.3.1 and rearranging we get

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$$  \displaystyle d\left( {r\frac} \right) = - \frac{s}{k}rdr $$     (4.3.21) Integrating this we get
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$$  \displaystyle r\frac = - \frac{s}{r^2} + {c_1} $$     (4.3.22) Using the boundary condition and solving
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$$  \displaystyle \frac(r = 0) = 0 $$     (4.3.23)
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$$  \displaystyle \Rightarrow {c_1} = 0 $$     (4.3.24) Next we rearrange again and integrate
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$$  \displaystyle dT = - \frac{s}rdr $$     (4.3.25)
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$$  \displaystyle T = - \frac{s}{r^2} + {c_2} $$     (4.3.26) Again using the boundary condition to solve for the constant of integration
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$$  \displaystyle T(R) = 0 $$     (4.3.27)
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$$  \displaystyle \Rightarrow {c_2} = \frac{s}{R^2} $$     (4.3.28) We arrive at the exact solution to the strong form which was found from the weak form
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$$  \displaystyle T(r) = \frac{s}\left( {{R^2} - {r^2}} \right) $$     (4.3.29)
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Solution from weak form and exact solution from the differential equation are same for given heat conduction problem.

=Problem 4.4: Deriving the Weak Form from the Strong Form for the Torsion of a Bar Fish & Belytschko Page 73 Problem 3.7 =

Given: The Strong Form
Given the strong form for the circular bar in torsion (Fig. 1):

Where the natural boundary condition is

Where the essential boundary condition is

Where the essential boundary condition $$ \displaystyle m(x) $$ is a distributed moment per unit length, M is the torsion moment, $$ \displaystyle \phi $$ is the angle of rotation, $$ \displaystyle G $$ is the shear modulus and $$ \displaystyle J $$ is the polar moment of intertia given by $$ \displaystyle J = \frac{\pi C^4}{2} $$, where $$ \displaystyle C $$ is the radius of the circular shaft.

b. Determine Constants of Integration
Assume that $$ \displaystyle m(x) = 0 $$ and integrate the differential equation given above. Find the integration constants using the boundary conditions.

(a) Obtaining the Weak Form
Multiplying (4.4.1) by an arbitrary weighting function $$ \displaystyle w(x) $$ and integrating with respect to x across the domain leads to the following form

Making use of Integration by Parts on the first term of (4.4.4) leads to the following equation.

Using the requirement of the essential boundary condition on the weighting function satisfying the essential boundary location implies that $$ \displaystyle w(0) = 0 $$ and also using the natural boundary condition (4.4.2) $$ JG \frac{\partial \phi}{\partial x}_{x=l} = M $$ leads to the following

Therefore the weak form can be represented finding a $$ \displaystyle \phi(x) $$ such that for all smooth $$ \displaystyle \phi(x) $$ with $$ \displaystyle \phi(0) = 0 $$ (4.4.7) is satisfied.

(b)Obtaining the Constants of Integration
Applying equation (4.4.1) and with $$ \displaystyle m(x) = 0 $$ leads to the following form.

Integrating (4.4.8) with respect to x generates a constant of integration and provides the following form.

Applying the natural boundary condition from (4.4.2) leads to the following

Solving (4.4.9) for $$ \displaystyle \frac{\partial \phi}{\partial x} $$ and substituting in $$ \displaystyle C_1 = -M $$ leads to the following

Integrating (4.4.11) with respect to x generates a constant of integration and provides the following form, which is the well known equation for angle of twist of a bar.

Applying the essential boundary condition from (4.4.3) to (4.4.12) leads to the following

Therefore the constants and the angle of twist are as follows.

=Problem 4.5: Constraint Breaking Solution (CBS) for Multiple Families of Basis Functions=

Given: Multiple Families of Basis Functions on a Domain
For

The possible basis function families:

e) Exponential Family:
Where the above families satisfy the the CBS with a domain of

1) Show the Following and Plot $$ \displaystyle b_j $$ and $$ \displaystyle \bar{b_j} $$ for $$ \displaystyle j=1,2,3$$
i) $$ \displaystyle \overline  (x) = const $$ ii) And plot $$\displaystyle {b_j} $$ and $$\displaystyle \overline   $$ for $$\displaystyle j=1,2,3 $$

a) Polynomial Family
Satisfying the CBS for the zero term:

Satisfying the CBS for the remaining terms:

The CBS family solution:

Plot of basis functions and CBS basis functions vs. the domain of -2 to 4:

b) Cosine Family
Satisfying the CBS for the zero term:

Satisfying the CBS for the remaining terms:

The CBS family solution:

Plot of basis functions and CBS basis functions vs. the domain of -2 to 4:

c) Sine Family
Satisfying the CBS for the zero term:

Satisfying the CBS for the remaining terms:

The CBS family solution:

Plot of basis functions and CBS basis functions vs. the domain of -2 to 4:

d) Fourier Family
Satisfying the CBS for the zero term:

Satisfying the CBS for the remaining terms:

The CBS family solution:

Plot of basis functions and CBS basis functions vs. the domain of -2 to 4:

e) Exponential Family
Satisfying the CBS for the zero term:

Satisfying the CBS for the remaining terms:

The CBS family solution:

Plot of basis functions and CBS basis functions vs. the domain of -2 to 4:

Conclusion
It seems that the strength or effective representation of the basis family would be expected to follow the trend: (exponential family) << (polynomial family)<< (sine family) ~ (cosine family) < (fourier family). Although this trend may not follow for every case(small j's and linear or simple curvature), it demonstrates that as j becomes significantly large, the fourier family of basis functions tends to become more accurate.

Part 2: Show that the exponential family is linearly independent on the domain
To be linearly independent, the gram matrix of the basis functions must not equal zero:

The gram matrix consists of the dot products of basis functions as follows:

However, with a given domain, the Gram Matrix can also be written as:

where,

A few examples:

etc...

The gram matrix of the exponential family becomes:

The determinate of the exponential gram matrix does not zero:

Therefore, the exponential family of basis functions is linearly independent on the domain

=Problem 4.6: Elastic bar with a variable distributed spring force p(x)=

Given: bar of length l, cross-sectional area A(x), Young’s modulus E(x) with body force b(x)
Consider an elastic bar with a variable distributed spring p(x) along its length as shown in Figure 1. The distributed spring imposes an axial force on the bar in proportion to the displacement. Consider a bar of length l, cross-sectional area A(x), Young’s modulus E(x) with body force b(x) and boundary conditions as shown in Figure 1

a. Derivation of Strong Form
Let $$\displaystyle \vec{n}(x) $$ be defined as the unit vector having positive orientation in the direction of $$ \sigma $$ on the element we are considering. This means that the sign convention of our unit vectors will be

Which corresponds to the convention depicted in the free-body diagram shown above. Let us also define $$ \displaystyle \sigma(x) $$ as the force for a given area as,

Upon the rearrangement of terms we arrive at

Classical mechanics allows us to write the sum of the forces in static system is zero

Using the free-body diagram, let us now consider all of the forces acting on the elastic bar $$\displaystyle b(x),p(x), N(x), $$ and $$\displaystyle  N(x+dx) $$. Balancing the forces acting on our finite element results in the following relation,

Next we will substitute (1.5) and (1.6) into (1.10) to get

Recall that a Taylor's Series expansion (See:Taylor Series) about $$ a $$ near $$ x $$ has the following form:

Next we will need to determine a relationship for the displaced area and stress, namely $$\displaystyle \sigma(x+dx)$$ and $$ \displaystyle   A(x+dx)$$. To accomplish this we will performing a Taylor Series expansion choosing $$\displaystyle a=x+dx $$ and neglect all higher order terms.

Taking the product of (1.13) and (1.14) gives us

Substitute (1.15) into (1.11) results in

This can be simplified by canceling the like terms and dividing through by $$ \displaystyle dx $$ as shown below,

to get the following simplified relation,

Next we must apply the product rule in the reverse directions. Meaning we will consider (1.19) the expansion obtained by applying the product rule and recombine terms to get the unexpanded form shown here,

tensile stress according to Hooke's Law

Substituting (1.22) into our simplified and unexpanded form of our force balance shown in (1.20) we arrive at the solution,
 * {| style="width:100%" border="0"

$$  \displaystyle \frac{d}[E(x)\cdot A(x)\frac] + b(x)-p(x) = 0 $$     (4.6.19) Standard form of the strong form can be written as
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

b. Derivation of weak form
Now strong form can be written as
 * {| style="width:100%" border="0"

$$  \displaystyle P\left( u \right) = 0 $$     (4.6.21)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

Where
 * {| style="width:100%" border="0"

$$  \displaystyle p\left( u \right) = \frac{d}\left[ {E\left( x \right)A\left( x \right)\frac} \right] + \left( {b\left( x \right) - p\left( x \right)} \right) $$     (4.6.22)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

Let $${w\left( x \right)}$$ is a weight function, then writing Weighted Residual Form
 * {| style="width:100%" border="0"

$$  \displaystyle \int\limits_0^l {w\left( x \right)\left[ {P\left( u \right)} \right]dx = 0} $$     (4.6.23)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

 * {| style="width:100%" border="0"

$$  \displaystyle \int\limits_0^l {w\left( x \right)\left[ {\frac{d}\left[ {E\left( x \right)A\left( x \right)\frac} \right] + \left( {b\left( x \right) - p\left( x \right)} \right)} \right]dx = 0} $$     (4.6.24)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

Writing $$w\left( x \right) = w,E\left( x \right) = E,A\left( x \right) = A,b\left( x \right) = b,p\left( x \right) = p$$
 * {| style="width:100%" border="0"

$$  \displaystyle \int\limits_0^l {w\left[ {\frac{d}\left[ {EA\frac} \right] + \left( {b - p} \right)} \right]dx = 0} $$     (4.6.25)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

Expanding integral
 * {| style="width:100%" border="0"

$$  \displaystyle \int\limits_0^l {w\frac{d}\left[ {EA\frac} \right]dx} + \int\limits_0^l {w\left( {b - p} \right)dx}  = 0 $$     (4.6.26)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

 * {| style="width:100%" border="0"

$$  \displaystyle \left[ {wEA\frac} \right]_0^l - \int\limits_0^l {EA\frac\fracdx} + \int\limits_0^l {w\left( {b - p} \right)dx}  = 0 $$     (4.6.27)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

Now assuming $$w\left( 0 \right) = 0$$ and also from natural boundary condition $$E(l)\frac = \bar t$$ above integral will take form as
 * {| style="width:100%" border="0"

$$  \displaystyle w\left( l \right)A\bar t - 0 - \int\limits_0^l {EA\frac\fracdx} + \int\limits_0^l {w\left( {b - p} \right)dx}  = 0 $$     (4.6.28)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

Rearranging terms

 * {| style="width:100%" border="0"

$$  \displaystyle \int\limits_0^l {EA\frac\fracdx} = \int\limits_0^l {w\left( {b - p} \right)dx}  + w\left( l \right)A\bar t $$ (4.6.29)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

Now writing in standard format

Reference
=Problem 4.7: Calculix User's Guide Containing Illustrative Examples For Building FEA FBDs=

First Step: Download
Calculix is a open source finite element suite. This software contains its own pre- and post processing algorithms which enables the user to prepare and solve a model after defining the appropriate boundary conditions. The solver, CCX, was written and is maintained by Guido Dhondt. The pre- and post processor, CGX, was written and is maintained by Klaus Wittig. Because CalculiX is open source users can download the latest version from Convergent Mechanical Solution's website. Convergent has done an excellent job integrating an installation wizard with the distribution which makes the installation process a breeze for experts and novice alike.

 * Users intending on downloading CalculiX with a Windows operating system should visit: http://www.bconverged.com/products.php
 * The built-in GUI makes it simple to follow the on screen prompts for a quick pain-free installation.
 * Users will be prompted to select a working directory which will be used as the starting file path whenever opening CalculiX editor.
 * The source code, Linux builds, documentation, tests, and other resources for Calculix : http://www.calculix.de/

Second Step: Test
Before starting Calculix we need to accomplish one more step.

 * {| style="width:100%" border="0"

$$  \displaystyle [Start]--[All Programs]--[bConverged]--[Calculix]--[Test Calculix *.*] $$
 * style="width:95%" |
 * style="width:95%" |
 * }

Using Calculix

 * To open Calculix in the user friendly windows version go to

 * {| style="width:100%" border="0"

$$  \displaystyle [Start]--[All Programs]--[bConverged]--[Calculix]--[Calculix Command] $$
 * style="width:95%" |
 * style="width:95%" |
 * }

 * Doing this will open the Calculix command window.
 * The command window will automatically open in the working direction that we have defined during the installation process.
 * Next we introduce the program command line parameters which serve as options when creating or edit a file.

 * {|style="width:100%" border="0"

$$  \displaystyle Command: cgx [-a|-b|-c|-duns2d|-duns3d|-isaac2d|-isaac3d|-foam|-step|-stl]  filename $$
 * style="width:95%" |
 * style="width:95%" |
 * }

Now we proceed to show an example of a pre-built FEA problem. Examples can be download from Calculix website or you can access them once bConverged has been installed and the pre-build samples have been tested. Because of the amount of sample files CalculiX must test, something initiated by the user after installation, the time will very for the completion of the initial tests. To find the examples simply go to the program directory containing bConverged. The location will very slightly depending on if you are running a 32 or 64-bit version of windows.

Since we are in the working directory when opening CalculiX we will need to copy the appropriate examples to our working directory to open them. Or you can simply change your current directory to a directory of your choosing based on where your want the examples to be placed. In the command line type ' cgx -b sphere.fbd '. Important !!!: Always make sure that the mouse is positioned on the graph window before writing commands. This point is extremely important because failing to do so will result in commands not being entered in the command window. Initially this is somewhat cumbersome because we have trained ourselves to point the mouse to the location where we will be inserting text. After a few minutes you will get the feel for doing this. Position your command window such that you can simultaneously view your entered commands and geometric figure.

Command line $$ \displaystyle \Rightarrow $$ C:\..........\examples\basic>cgx -b sphere.fbd

By using mouse you can rotate, zoom in out and out, or reposition the figure. Rotating the object is controlled by the left mouse button, zooming in and out by the middle mouse button and translating the object is controlled by the right mouse button. Before moving on try these operations with the mouse in order to become well acquainted with how these movements orient the figure.

Learning By Example
Lets learn more by doing an example on Calculix. We can broaden our knowledge on commands by doing examples. We can take modelling disc as an example.

Building Your First Model

 * Run Calculix Command
 * Type cgx -b discmodel.fbd (You probably get an error message. Because there is no exists file name: discmodel)
 * To start with modelling we have to remember the construction procedure.
 * First we will construct nodes, lines, surfaces, and the bodies.
 * Lets look at point command.

 * {| style="width:100%" border="0"

$$  \displaystyle\begin{align} Command:\quad 'pnt' <name(char<9)>& |'!' [<x> <y> <z>]| \\ & [   ]| \\                                  &  [<P1> <P2>  ]| \\ & [<setname(containing nodes)>]
 * style="width:95%" |
 * style="width:95%" |

\end{align} $$
 * }

Creating Points

 * In order to type commands in to the command window please remember to press and release to the graph window with left button mouse button.
 * Type pnt y  -0.00000        1.00000        0.00000  (recognizing that py is a dummy with three coordinates on xyz plane)
 * Type pnt P0  -0.00000       -0.00000        0.00000
 * Type pnt P001 0.70711       -0.00000       -0.70711
 * Type pnt P003 -0.00000      -0.00000       -1.00000
 * Type pnt P005 -0.70711      -0.00000       -0.70711
 * Type pnt P006 -1.00000      -0.00000        0.00000
 * Type pnt P009 -0.70711       -0.00000       0.70711
 * Type pnt P00A 0.00000       -0.00000        1.00000
 * Type pnt P00G 0.70711       -0.00000        0.70711
 * Type pnt P00I 1.00000       -0.00000       -0.00000

 * Even though you have entered the points into the console you will not be able to see the point attributes until you enter a command to display them.
 * To see the points with attributes(labels) enter the following command.
 * Type plot pa all (get the same view with graph)
 * The general command used here is plot and has all of the available options shown here.
 * {| style="width:100%" border="0"

$$  \displaystyle\begin{align} Command: 'plot' & ['n'|'e'|'f'|'p'|'l'|'s'|'b'|'S'|'L']&['a'|'d'|'p'|'q'|'v'] -> \\ & 'w'|'k'|'r'|'g'|'b'|'y'|'m'
 * style="width:95%" |
 * style="width:95%" |

\end{align} $$
 * }

Creating Lines

 * Now we will look at how to generate lines.
 * Type line L001 P00I P001 p0 4
 * This will create the line connecting p001 to p00I.
 * To show this line
 * Type plus la all
 * The plus command is used when one wants to retain the previous plot and add addition
 * {| style="width:100%" border="0"

$$  \displaystyle Command: 'line' <name(char<9)>|'!' <p1> <p2> <cp|seq>
 * style="width:95%" |
 * style="width:95%" |

$$
 * }

 * As you see above we just created arc for a portion of disc.
 * Even if the command's name is line we also create arcs with the same command.
 * L001 is a dummy name.
 * P00I and P001 are limits.
 * p0 is the center.
 * And 4 is the number of divisions.
 * To see the line attributes use the following:

 * {| style="width:100%" border="0"

$$  \displaystyle\begin{align} Command: 'plus'  & ['n'|'e'|'f'|'p'|'l'|'s'|'b'|'S'|'L']&['a'|'d'|'p'|'q'|'v'] -> \\ & 'w'|'k'|'r'|'g'|'b'|'y'|'m'|'i'
 * style="width:95%" |
 * style="width:95%" |

\end{align} $$
 * }

 * Plus performs the same job as plot.
 * Plus retains all previous visuals along with the initial plot filters.
 * Type line L002 P001 P003 p0 4
 * Type line L003 P003 p0 8
 * Type line L004 p0 P00I 8
 * Type line L005 P003 P005 p0 4
 * Type line L006 P005 P006 p0 4
 * Type line L007 P006 p0 8
 * Type line L009 P006 P009 p0 4
 * Type line L00A P009 P00A p0 4
 * Type line L00C P00A p0 8
 * Type line L00G P00A P00G p0 4
 * Type line L00I P00G P00I p0 4

Creating Surfaces
To finish the model we need to add surfaces. To do so we will use the following generalized syntax.

 * {| style="width:100%" border="0"

$$  \displaystyle\begin{align} Command: 'gsur' &<name(char<9)>|'!' '+|-' 'BLEND| ' '+|-' <line|lcmb> '+|-' -> \\ &<line|lcmb>
 * style="width:95%" |
 * style="width:95%" |

\end{align} $$
 * }

 * This keyword is used to define or redefine a surface.
 * Surfaces are defined with boundary lines.
 * Type gsur A001 + BLEND - L003 - L002 - L001 - L004
 * Type gsur A002 + BLEND - L007 - L006 - L005 + L003
 * Type gsur A003 + BLEND - L00C - L00A - L009 + L007
 * Type gsur A004 + BLEND + L004 - L00I - L00G + L00C

Creating Meshes

 * In the final part of this example we will create the meshes.
 * The options tr3, qu4, and qu8 are for meshing surfaces.
 * The options he8 and he20 are used to mesh bodies.

 * {| style="width:100%" border="0"

$$  \displaystyle\begin{align} Command: 'elty' &[ ] ['be2'|'be3'|'tr3'|'tr3u'|'tr6'|'qu4'|-> \\ &'qu8'|'he8'|'he8f'||'he8i'|'he8r'|'he20'|'he20r']
 * style="width:95%" |
 * style="width:95%" |

\end{align} $$
 * }

 * Type elty all QU4
 * Type mesh all
 * We just have meshed the surface with an elements of qu4.

 * Type plot m all

Basic Models
Shown in the figures below are the example models that come installed with the CalculiX suite. We were able to regenerate these models by applying similar commands to the one present in the above tutorial. After playing around with this software we were finally able to obtain the right combination of commands the display the colorful models with attributes shown below.

Turbine: Illustrating the Use of Colors and Surface Element Attribute Specifications

=Problem 4.8: The Mass Matrix in Weak Form Using F&B problem 3.4=

Given: PDE and Corresponding Parameters
where the weak form is A trial solution for u(x) is suggested as and w(x) as

Find: The Mass Matrix in Weak Form
with n=3 and for dynamics:

Find $$\underline {F}(t)$$

State the Weak Form of the dynamic PDE for elasticity
Our given basis is The discretized forms of our trial solution and our weighting functions respectively are

Substitute back the operator definitions
=Contributing Members & Referenced Lecture=


text: File:Gantt chart finaal Lightspeed.pdf

title: File:Gantt chart finaal Lightspeed.pdf


text: File:PVA - lightspeed.pdf

title: File:PVA - lightspeed.pdf


text:
Talk:How going open with Moodle can save staff time

Feel free to add more ideas about pros and cons of an open educational approach as a LMS time saver for teachers.

Time-saving advantages of an open approach
Two types of openness (point out to the world and allow the world in) are involved in these examples of how an open educational approach with learning management systems (LMS), such as Moodle, can save a teacher time:
 * 1) Provide links in an LMS site to open educational resources (point the LMS site out to world)
 * 2) Linking to external open educational resources (especially if they are not time-bound or unit-specific) from a LMS site minimises the need to export/import LMS content each time a new unit instance is offered.
 * 3) Using open educational material (e.g., in a open educational wiki environment) allows and encourages students and others to correct errors and to improve and peer review the material.
 * 4) Using educational resources allows for teaching in a wider variety of contexts (e.g., teaching/sharing in a non-university role or wanting to use the material in different institutions and/or learning management systems).
 * 5) Make an LMS site openly accessible and its content openly licensed (open the LMS site to the world)
 * 6) Making an LMS site open to the world provides free advertising.
 * 7) Making an LMS site openly accessible allows students to start early and finish late without the instructor having to give additional permissions and access.

Risks or problems with using an open approach

 * 1) The defaults are often closed, however there is a momentum towards openness in education.
 * 2) Academic institutions are often ambiguous or hesitant about allowing open release of teaching materials and  make claims of copyright of teaching materials.


text:
User:Eml5526.s11.team3.akj/Homework 4

=Problem4.3 Quadratic Trial Solution of Weak Form for the heat conduction problem in a circular plate =

Given:Strong form for the heat conduction problem in a circular plate
The Strong form for the heat conduction problem in a circular plate is

Where R is the radius of the plate, s is the heat source per unit length along the plate radius, T is the temperature and k is the conductivity, and k,s and R are given

a) Construct the weak form for the given Strong form
Construct the weak form for the above strong form.

b) Solution to the weak form
Use the quadratic trial (candidate) solution of the form $$\alpha _{0}+\alpha _{1}r+\alpha _{2}r^{2}$$  and weight function of the same form to obtain the solution of the weak form.

c) Solution to the strong form
Solve the differential equation with the boundary conditions ans show that the temperature distribution along the radius is given by

a) Weak form of the Strong form for the heat conduction in a circular plate
Multiplying the given strong from with the weight function w and integrating it over the interval $$0< r\leq R$$  we get,

Integrating first term of Equation 4.3.3 by parts:

We know that the weight function should vanish at $$r=R$$  i.e it should satisfy the homogeneous essential boundary condition  $$(w(R)=0)$$  and from the natural boundary condition  $$\frac{\mathrm{d} T}{\mathrm{d} r}(r=0)=0$$

hence substituting the above boundary conditions and rearranging the terms we get,

hence the weak form for the strong form in Eq.4.3.1 is

b) Solution of the weak form using quadratic trial solution
Given the trial solution is of the form $$T=\alpha _{0}+\alpha _{1}r+\alpha _{2}r^{2}$$  and the weight function is also of the same form and can be written as  $$w=\beta _{0}+\beta _{1}r+\beta _{2}r^{2}$$

From the given essential boundary condition we can write,

differentiating the trial solution of T we get

given natural boundary condition as $$\frac{\mathrm{d} T}{\mathrm{d} r}(r=0)=0$$ hence, $$\alpha_{1}=0$$ hence the trial solution is such that

from the homogeneous essential boundary condition we know $$w(R)=0$$ hence $$\beta _{0}+\beta _{1}R+\beta _{2}R^{2}=0$$ hence the weighting function is such that

substituting Eq 4..9 and 4.3.10 into the weak form derived in Eq 4.3.6 we get,

Integrating the left hand side of Eq 4.3.11 we get

Integrating the right hand side of Eq 4.3.11 we get

Hence combining Eq 4.3.13 and Eq 4.3.12 we can write,

Rearranging the terms in the above equation we get,

The above equation should be true for any value of $$\beta_{1}$$  and  $$\beta_{2}$$  hence equating each of  $$\beta_{1}$$  and  $$\beta_{2}$$  to zero we get, both the equations yield the same result

Substituting $$\alpha_{2}$$  into Eq 4.3.7 we get,  $$\alpha_{0}=\frac{sR^{2}}{4k}$$ Hence the trial solution to the weak form in Eq 4.3.6 is

c) Solution of the strong form
The Strong form for the heat conduction problem in a circular plate from Eq 4.3.1 is

Rearranging the terms in the strong form we get,

integrating the above Eq we get,

using the natural boundary condition we get, $$c_{1}=0$$ Agian integrating Eq 4.3.19 we get,

Applying essential boundary condition $$(T(r=R)=0)$$  we get,  $$c_{2}=\frac{R^{2}s}{4k}$$

Hence the temperature distribution of the circular plate along the radius is given by

Note:- From Eq 4.3.17 and Eq 4.3.21 it is observed that the solution to the weak form and the strong form are the same in this case

=Problem 4.8 Find mass matrix and force matrix=

Given : The strong form and a cubic trial solution
The strong form is

Given $$A=1$$ ,  $$E=2$$  ,$$\tilde{m}=3$$. The cubic trial solution is given as $$u^{h}(x)=\alpha _{0}+\alpha _{1}(x-3)+\alpha _{2}(x-3)^{2}+\alpha _{3}(x-3)^{3}$$

Where the basis function is given by  $$\left\{b_{i}\right\}=\left\{1,(x-3),(x-3)^{2},(x-3)^{3};i=0,1,2,3\right\}$$

a) Mass Matrix
Mass matrix $$\mathbf{\tilde{M}}$$

b) Force Matrix
Force matrix $$\mathbf{F}$$  for dynamics with  $$u^{h}(\beta ,t)=g(t)=sin(2t)$$

a) Mass Matrix
The mas matrix $$\mathbf{\tilde{M}}$$  is given by

where $$\mathbf{M_{EE}}=[M_{00}]$$ ,  $$M_{00}=\tilde{M}_{00}$$  and  $$\tilde{M}_{00}=\tilde{m}(b_{0},b_{0})$$

$$\tilde{m}(b_{0},b_{0})=\int_{1}^{3}b_{0}\bar{m}b_{0}dx$$ as given in Lecture 19-1

Substituting for $$b_{0}\quad and\quad \bar{m} $$  we get $$\tilde{m}(b_{0},b_{0})=\int_{1}^{3}3dx$$ hence

Similarly $$\mathbf{M_{EF}}=[M_{0j};j=1,2,3]$$ ,  $$M_{0j}=\tilde{M}_{0j}$$  and  $$\tilde{M}_{0j}=\tilde{m}(b_{0},b_{j})$$

Clearly we can see that $$\mathbf{M_{EF}}$$  is row matrix

hence we can write $$\tilde{m}(b_{0},b_{1})=\int_{1}^{3}3(x-3)dx$$ which on integration gives $$\tilde{m}(b_{0},b_{1})=-6$$

similarly $$\tilde{m}(b_{0},b_{2})=\int_{1}^{3}3(x-3)^{2}dx=8$$

and $$\tilde{m}(b_{0},b_{3})=\int_{1}^{3}3(x-3)^{3}dx=-12$$

hence

it is observed that $$\mathbf{M_{EF}}=\mathbf{M_{FE}^{T}}$$  hence  $$\mathbf{M_{FE}}$$  is column matrix given by

and $$\mathbf{M_{FF}}=[M_{ij};i,j=1,2,3]$$,  $$M_{ij}=\tilde{M}_{ij}$$  and  $$\tilde{M}_{ij}=\tilde{m}(b_{i},b_{j})$$ where $$\tilde{m}(b_{1},b_{1})=\int_{1}^{3}(x-3)3(x-3)dx=8$$ $$\tilde{m}(b_{1},b_{2})=\int_{1}^{3}(x-3)3(x-3)^{2}dx=-12$$ $$\tilde{m}(b_{1},b_{3})=\int_{1}^{3}(x-3)3(x-3)^{3}dx=\frac{96}{5}$$ $$\tilde{m}(b_{2},b_{1})=\int_{1}^{3}(x-3)^{2}3(x-3)dx=-12$$ $$\tilde{m}(b_{2},b_{2})=\int_{1}^{3}(x-3)^{2}3(x-3)^{2}dx=\frac{96}{5}$$ $$\tilde{m}(b_{2},b_{3})=\int_{1}^{3}(x-3)^{2}3(x-3)^{3}dx=-32$$ $$\tilde{m}(b_{3},b_{1})=\int_{1}^{3}(x-3)^{3}3(x-3)dx=\frac{96}{5}$$ $$\tilde{m}(b_{3},b_{2})=\int_{1}^{3}(x-3)^{3}3(x-3)^{2}dx=-32$$ $$\tilde{m}(b_{3},b_{3})=\int_{1}^{3}(x-3)^{3}3(x-3)^{3}dx=\frac{384}{7}$$

hence

Substituting Eq 4.8.3,Eq 4.8.4,Eq 4.8.5 and Eq 4.8.6 into Eq 4.8.2 we get

b) Force Matrix
The force matrix is given by

where $$\mathbf{\tilde{F}}=\left \{ \frac{\mathbf{F_{E}}}{\mathbf{F_{F}}} \right \}$$ and $$\mathbf{F_{F}}=\left \{ F_{i},i=1,2,3 \right \}$$ ,  $$ F_{i}=\tilde{F_{i}}$$  and  $$\tilde{F_{i}}=\tilde{f}(b_{i})$$  as given in Lecture 19-1

$$\tilde{f}(b_{i})=b_{i}(\alpha)h+\int_{1}^{3}b_{i}fdx$$

at $$\alpha$$  natural boundary condition is specified, in  this case is  $$\alpha=1$$  and  $$h$$  is the value of the of the natural boundary condition given by  $$h=0.1$$  and  $$f$$  in this problem is  $$2x$$

hence $$\tilde{f}(b_{1})=(1-3)0.1+\int_{1}^{3}(x-3)2xdx=\frac{-103}{15}$$ $$\tilde{f}(b_{2})=(1-3)^{2}0.1+\int_{1}^{3}(x-3)^{2}2xdx=\frac{42}{5}$$ $$\tilde{f}(b_{3})=(1-3)^{3}0.1+\int_{1}^{3}(x-3)^{3}2xdx=-12$$

stiffness matrix matrix is given by $$\mathbf{K_{EF}}=[K_{0j};j=1,2,3]$$ ,  $$K_{0j}=\tilde{K}_{0j}$$  and  $$\tilde{K}_{0j}=\tilde{k}(b_{0},b_{j})$$ and $$\tilde{k}(b_{0},b_{j})=\int_{1}^{3}{b_{0}}'AE{b_{j}}'dx$$

but $$b_{0}=1$$  hence  $${b_{0}}'=0$$ therefore  $$\tilde{k}(b_{0},b_{j})=0$$ $$\implies\quad\mathbf{K_{EF}}=\begin{bmatrix} 0 & 0 & 0 \end{bmatrix}$$

since $$\mathbf{K_{EF}}=\mathbf{K_{FE}^{T}}$$

from Eq 4.8.1 $$g=sin(2t)$$ $$\implies \quad {g}''=-4sin(2t)$$

hence substituting Eq.4.8.5, Eq4.8.9 and Eq 4.8.10 into Eq 4.8.8 we get

title: User:Eml5526.s11.team3.akj/Homework 4


text:
User:Brigitte 123

I'm a latin student of 1 year and am on my 2nd year of learning. I would like to increase my knowledge and ability in Latin and hopefully help others beginning the language!

title: User:Brigitte 123


title: File:Fe1.s11.mtg24.djvu


text:
Talk:Psycholinguistics/Speech Errors

Research
Your citations seem to be consistent throughout the text and to the APA style standards. Some of your citations have footnotes attached to them, but some of them do not. Consistent use of a single reference format is needed as it allows readers to easily use your chapter to find more information on the topic. You cite many sources but only three had references, which made it impossible for me to tell if you were using high quality sources. The three that you did reference, however, seem to be relevant and authoritative. Many of your citations seem to come from research done before the new millennium, it would be a good idea to get more recent research since text books need to be as up to date as possible. More recent research could show how speech errors are still being analyzed or are still influencing the field.

Organization and Answering Questions
How you’ve laid out your sections seems to make sense. You introduce the idea of speech errors, then detail the different types, and then show how they support different models. Some of your sections could be a bit more detailed, though. I found myself asking a lot of questions that your chapter should have answered, especially regarding the sections about speech models. I also suggest that you have larger introduction and summary sections that are separated from the rest of the text. I’m basing this suggestion on the format of the textbook chapters I’ve seen other people working on and textbook chapters in general. It also makes sense since it allows a reader to ease their way into the material at the beginning and then collect and remember what they’ve just read once they’re finished.

Writing Style and Structure
There are a lot of awkward sentences in your chapter. I found it hard to read because it didn’t flow well and you used a lot of figures of speech and I think it would probably be the same for most readers. Having a chapter that is easy to read is important so that the reader doesn’t get frustrated while trying to understand the material. I would suggest reading your writing out loud to another person to assess whether or not a sentence should be fixed. The flow between paragraphs was also very awkward. Most of the time there was no transition from one section to the next, making the information seem disconnected. Like an essay, your chapter should have a rhythm to it that allows the reader to easily move from section to section and allow the information of each section to connect and build upon itself. If the information is disconnected, it may make it harder for the reader to understand what you’re trying to say. As far as grammar and spelling mistakes go, you’ve made a handful of spelling mistakes (usually accidentally changing the first letter of a word) and a lot of grammar mistakes. You tend to place commas in inappropriate places, or forget to use the appropriate punctuation when it is needed. There are several places in your chapter where a semicolon was needed, but instead you let the sentence run on. I have not edited your page for you, so you will have to fix these yourself.

Matt Orr 12:36, 26 February 2011 (UTC)

Poor/No Coverage of Non Indo-Aryan Languages
At least a mention of other language types is required, since the scope of the article as it stands is evidently limited to English/Indo-Aryan languages. Concepts of derivation, for instance, do not hold in other language families. Pratyeka 01:18, 5 January 2012 (UTC)

title: Talk:Psycholinguistics/Speech Errors