Coursework examples

Modelling problems suggested by students in a brainstorming exercise

The first lecture of this on-line modelling course is titled “Mathematics and applications, an overview”which presents a broad survey on the role of mathematical models in industry, society and applied science. The lecture is followed by an exercise assignment where the students are advised to think about their past experience,their curiosity in science and technology, recent news in newspapers and TV, their hobbies, summer job etc and come up with an idea where mathematics, modeling and computation could be used in real worldproblem solving. The output from this exercise is a variety of project proposals accumulated over years 2002-2014. A selection of somewhat edited answers is given below. The idea is the problem formulation and not so much the possibility of finding a solution at this point. The questions might indeed be challenging tasks, even impossible or requiring extended research.

Year 2002

Lava flow. Given the geometry of a volcano, forecast the flow pattern and route of molten lava after an eruption. The temperature and volume of the eruption are given and also the temperature-dependence of viscosity of the lava flow.

Dressing kids. In Nordic Country in Winter time when young parents take their two children ( boy 1 and girl 4 years) outdoors, they have to dress the kids very carefully and of course themselves as well.

Dressing an 1-4 year old is tough job and takes time. The dressing is done indoors. If the parent puts his/her outdoor attire on first he/she will be hot and suffer from sweating while handling the kids.

If the young boy is dressed first he has to wait until the two others are ready and he will turn angry when he has to wait in full clothes. The 4 year of girl is perhaps a bit more patient than the 1 year old.

Assume that you know how each of these persons (mom/dad, I year old, 4 year old) will suffer and turn impatient being in full clothes inside (an increasing function of worsening temper). Construct a mathematical model to investigate what is best order of scheduling the job.

Hand weaving. Weaving/knitting a sweater at home is still a favorite form of handicraft or even a hobby for some. There are ready-made models that give detailed instructions of the shape. The problem is how much of the thread you need to buy to weave a given shape.

The weaving is dome with metal/plastic sticks that have various thickness (perhaps 2-6 mm). The threads are also of various thickness. There are some alternative weaving formulas, essentially defining the geometric structure/ornament of the weave.

If these factors are given, construct a model to estimate the demand of thread (how many meters per cm2).

Fishing line. A certain method of salmon fishing (fly-fishing) is based on throwing a fishing line with a light bait over a long distance and letting the line to float on water surface. The interest in this project is to know how the fishing line (assume 30 meters long) will behave on a water surface of a flowing river. Both horizontal movements drift and vertical movements (partial sinking) are parts of the problem.

2003

Paths in a Park. Parks in big cities are designed by architects, who decide, among other things the paved paths that pedestrians are supposed to use when walking across the park.

If the marked pedestrian routes are too complex busy people tend to take shortcuts across the lawn by walking “on the hypotenuse” . The part has several entrance gates. Suggest an algorithm to help to design the path network that would minimize the temptation to cross through the lawns.

Snow-blower. Snow-blower operator has to keep clean a 10km long road. When he starts, the road is already covered by 10 cm of snow. New show is falling down at k cm/hour. The length of the road and the width of the machine is 3 meters. The machine can remove b cubic meters of snow per minute and the maximum speed is 20km/hour. Construct a model that describes the snow-removal operation.

Delivery network. Delivery truck fleet transport route and schedule optimization

Pension system. Renewal/revision of pension system as a result of increasing life expectancy.

Elections. Study how in national elections the voting of various age groups are distributed between parties.

Cooling of Coffee Cup. A coffee cup of cylindrical shape is filled with hot coffee. The coefficient of thermal conduction of the cup walls and bottom are given. Model the cooling of the coffee (1) when the coffee is mixed continuously or (2) not mixed at all.

Space debris. The space outside our planet is littered by about 100 000 objects that are circulating the earth. Only about 10 percent of the objects are monitored so their exact locations are known. Regarding the smaller object the sizes and the velocities are random, varying according to a given statistical distribution. A rocket launch should take the debris into account and try to minimize the possibility of collision. Can you think about a strategy for maximizing the safety.

Speed in downhill skiing. You are given the shape of a ski-slope when an alpinist is going to start his/her race. You are given the (1) initial velocity (2) mass (3) air resistance (4) the friction between the skis and the track. The slope is described by a continuous monotone function. Compute the velocity of the skier throughout the track.

Loudspeakers. Locations of loudspeakers in a room that is used to listen to music.

Wire in space. Assume that one could attach a very long steel wire of 8 mm diameter to a space station floating at the height of 100 km. The steel has density of 7800kg/m^3 and the pulling force needed to break the steel is 360N/mm^2. How long the wire could be before it would break because of its own weight?

Tissue modeling. Human tissues are interesting materials with specific behavior. Example is cartilage which is the substance found in human joints between adjacent bones. It is attached to the bones and acting as a shock absorbing buffer zone and typically covered by lubricating liquid.

The cartilage can be modelled by an elastic porous material containing water. A layer of 1cm thickness of such material is placed between two metal plates and a pressing force is applied. Study the behavior, deformation etc of the material.

2004

In situ cleaning of soil. In some case a contaminated soil can be cleaned by biological methods by using micro-organism that can metabolize the contaminant and hence reduce the harmful substance. Mathematical modelling of the process of biochemical purification would involve topics like (1) modeling the soil as a porous medium (2) modeling the hydrogeological processes of water flow in the soil (3) modeling the geochemical processes and the resulting thermal effects (4) growth and transport of the microbial mass.

On/off switching of lights. In the dark period in the Nordic countries (especially October-March) one must have electric light indoors. Energy conscious people tend to switch off the light when they leave the room and often this means numerous on/off changes on the same day. Switching the bulb will cause aging that shortens the life-time of the bulb, as of course the continuous on-state (and understandable a little during the off-state also).

Using a probabilistic model study the benefit of switching the light off (and saving some power) in case you will be away from the room for a certain short time. Many parameters of course will be needed (price of electricity, factors effecting the bulb life time).

Pole vault. Initial position and speed of the athlete at the beginning of the jump is given. Investigate if it is possible to model the effect of the pole that at first is twisted into a curved shape and then the spring-tension type force is throwing the athlete upwards.

2005

Bowling. In bowling one wants to knock down as many of the ten pins as possible. Moreover the result of a throw of the ball is almost completely determined by the first hit of the ball onto the first pin that it meets. At the moment of impact the ball has (1) a position (2) direction of movement and (3) the rotation speed of the spin (4) the angle of the spin axis. The axis of the spin may also be tilted to an angle.

The modelling problem is to study the ball-pin impact and the resulting trajectory of the pin after the hit. If a solution for an optimal hit is found then the challenge for the player is how to generate such a hit !!??

Design and rules at car parking area. Assigning a tournament series structure in a two player or two-team game (chess, go, backgammon football) to find the winner.

Color Blind Vision. Finding a color-filter that would transform an ordinary color picture into a form how a color-blind individual would see it.

Chemical recycling. Modelling the recycling of chemicals in pulp factory consisting of several phases and feedback loops (cooking, washing, evaporation, deligneation, separation stages, oxygenation..)

Dynamics of a harpoon shot in underwater fishing. A pier at lake shore is constructed by attaching two parallel 10m long steel bars at concrete supports separated by 2 meters. Hence a 8m section is hanging freely over the water. Model the mechanics and forces on the supports if there is 200kg load at the end of the pier.

Savings interest policy. Five different banks offer a saving policy with different condition of interest calculation. The interest rate may depend on the balance of the account with different conditions, some time on the minimum level of a month etc. You should compare these policies to find the best one for a given saving plan.

Sports wear laundry. A student belongs to a floor-ball team that has regular training schedule and tournaments. The team has two sets of official tournament clothes (dark, light). In some of the weekly training sessions a free choice of uniform is also allowed. This generates high volume of laundry. The problem is to find a schedule for doing the laundry so that one has appropriate uniforms available when needed.

Pit strategy. Optimal Pit-stop strategy in Formula 1

Distillation column. Distillation is a physicochemical process where various liquid components are separated based on the varying evaporation temperatures. The liquid is heated and vapor fractions are collected by allowing the vapor to condense back to liquid phase.

In a big scale industrial distillation plant the heating requires a lot of energy and is the main cost of the operation. One possibility would be to apply vacuum (low pressure) since this will lower the boiling temperature and hence cause saving of energy.

The problem is to compare the cost of maintaining the vacuum to the energy saving attained in this way. Can a mathematical model help?

Gasoline prize. What gasoline prize must be at the most in order that the cost of using your own car is competitive compared to public transportation by bus?

Pizza and TV. A student want to enjoy a pizza while watching his favorite TV show The Simpsons.He will use a kitchen oven to heat the pizza. He would like to have the Pizza ready a few minutes before the show so the it cool a little. He wants to use mathematical model to compute the time when to switch the oven on and when to put the pizza into the oven.

Assuming that the thermal/electric parameters of the oven are known and the shape, size, mass and approximate heat capacity and conductivity of the pizza material are given, build a mathematical model to help the student.

Ceramic tiles. A company is manufacturing special ceramic tiles that can withstand very high temperatures and are used in building furnaces etc. Manufacturing the tiles is done by a process where the temperature of the material is raised slowly over a period of perhaps up to 20-40 hours. The heating is needed to remove all water from the clay-like mass. Slow raise of the temperature is to avoid buildup of tensions due to temperature gradient. Tensions would cause cracks.

Model is needed to describe the transport of water through the outer surface of the tile. Model could be used to optimize the heating schedule which up to now is based on empirical experience and trial-error method.

Sky gliding. Aerodynamic model of sky gliding including the mechanics and the control actions.

Acoustic surgery. Ultrasound can be used in medicine in surgical operations. An example is a method where powerful ultrasound is sent through human tissues of abdomen so that the wave is focused inside the kidney or bladder where sometimes some unpleasant stones are found. An powerful and focused ultrasound can break the kidney stone.

This technique might be used is destroying tumors in brain as well. To focus the ultrasound one needs to compute how the acoustic waves penetrate into human tissue.

RFID antenna design. In radio frequency identification RFID system, tag-chip is attached to an item. The system consists of an integrated circuit and an antenna. The information on the chip memory can be read and tracked from a distance by a RFID reader. The design of RFID antenna can benefit from mathematical simulation of the electromagnetic field that a given design is producing.

2006

Small scale furnace. Burning wood in a small-size furnace is one approach in modern distributed energy production. This call for accurate modeling the burning process to optimize the output. Basicly one should understand the behavior of a chip or small piece of wood in high temperature and supply of air/oxygen. The process stages are: Initial heating, drying, extraction and burning of gaseous compounds and finally burning of the residual coal. Thermochemical model of this process and the effect of oxygen supply should be built.

Abolishing a space station. When a space station comes to the end of its life, it must be eliminated in a safe way by controlling its descent through the atmosphere that would destroy most of it by burning. A complete burning is not possible and one should make sure that the debris will end up in an area where it does not cause danger. Controlling the orbit by rocket thrusters is expensive and the modeling problem is about how to bring the mass down safely with minimal energy consumption.

Frying pan handle. Frying pan made of steel has a handle which is supposed to stay at low temperature when the pan is heated. However the hande sometimes becomes too hot to touch. We would like to model the rising temperature in a 20 cm long steel handle when it is attached to a frying pan at temperature 200 C.

Cockroach population. A cockroach population is harassing a dormitory kitchen. To survive the population needs water and food. Careful and frequent cleaning should help to control the infestation. Build a mathematical model describing the effect of cleanliness on the population growth.

Golf ball. Modelling the flight of a golf ball and the hit.

National electricity consumption. Modelling the electricity consumption of a country on the macro level.

Cost of car parking. A city has 5 zones of varying parking prices. Near the city center parking is expensive, further from the center the price is lower. After about 4 km the parking is free. A person has some errands to run in the middle of the day. This takes 2 hours. He/she will lose his salary from the hours of absence. You should find out is it pays to drive to a cheap parking area and walk some distance. Average speed of driving in the city is 40km/h speed of walking is 3km/h and your salary is in the range 20-60euro/h.

Election candidate matcher. In recent years so called candidate matching machines have been introduced prior to national parliamentary elections. By answering a long list of questions in an online survey you can express your opinions, preferences, likes and dislikes of various social, political and many practical questions and principles. Your answers are compared with those of available candidates representing perhaps 10-12 political parties and the answers will measure your similarity/dissimilarity with the candidates.

Parties have also a possibility to agree on blocks/coalitions in a given constituency. This means that your vote is primarily for the candidate but also for the coalition he/she belongs to. In some cases this means that your vote may help the “wrong” party. Your favorite candidate belongs to party A and the next favorite to party B. In might be better to vote B for tactical reasons. It may happen that if B loses a least favorite candidate C may win. A model is needed to find a good strategy how to apply the results from the candidate matching machine.

Home loan conditions. Banks are offering loans for home buyers with different conditions regarding the computation of interest and repayment schedule. Various models include

(1) Annuity loan where interest follows the changes in financial market

(2) Fixed interest loan

(3) Fixed interest + principal loan

(4) etc

Simulate and compare various models from the homebuyer’s point of view.

Power plant. A plant produces electricity by burning solid materials (wood, bio-waste) or crude oil. The capacity of solid material furnace A is 6MW. Two crude oil burners B, C can produce up to 4MW each. Energy production from solid biomaterial is cheap compared to 2-3 times more expensive oil. Minimum power for burner A is 2MW and the ramp-up and ramp down should be avoided whenever possible due to labor costs. Energy consumption is random alternating between 0-8MW. Model is needed to help in planning an optimal policy for running the burners A,B and C.

2007

Traffic congestion. Traffic congestion in metropolitan areas. Can toll system reduce the congestion? How to find minimal efficient level for toll payment.

Metro tunnel. A tunnel is built for an underground train line. The excavation is started from both ends of a 2km long tunnel. The problem is how to make sure that the two tunnels meet at the same place. Coordinate measurement contain always errors and these errors accumulate along the track. How to minimize the total error sum?

Benji jump. Construct a model that estimates how high the benji jumper will rise after the first swing. The jumper starts from rest. The mass of the rope is 3 times the mass of the person. The jumper keeps his/her position during the flight so the air resistance is constant. Air resistance of the rope will be neglected.

2008

Cooling an engine. An engine is surrounded by a box-like housing for safety and noise reduction purposes. The engine is cooled by air stream through the empty space between the engine and the covering box. A model is needed to estimate the intensity of the cooling air stream to control the engine temperature.

Forest planting. Tree planting is a method of efficient forest cultivation. Small trees are planted in semiregular grid formation and the planted trees ar left to grow for 30-40 year until they will be harvested. Many intermediate steps like pruning thinning and spacing the tree farm are needed to control the ideal growth. The problem is to know the ideal density of the initial planting formation. Too dense seedling will stifle the growth. Too sparse forest means excessive growth of branches and less trees per hectar. One would like to optimize the timber quality and quantity.

District heating. In Nordic countries so called district heating means a system where a big thermal power plant is producing hot water and this is carried to perhaps several hundred apartment via a network of underground pipes. The pipes are well insulated but anyhow here are thermal losses. One of the design problems is the correct size of the pipes. If the pipe diameter is too big, the flow velocity is low and this increases the heat loss. A small pipe means losses due to bigger flow resistance. Large pipes cost more etc. A model is needed to compute optimal diameter of pipes in various parts of the network.

Water network. Designing water delivery network in an urban area is a comparable problem. Instead of temperature losses here the key factor is pressure that should be maintained throughout the network.

Milk can. If a milk can forgotten on a kitchen table (instead of refrigerator) is slowly warming up to room temperature. Microbial growth in the can is increasing with temperature and once the about of microbes is exceeding a limit (perhaps 3 ppm?) the milk is spoiled.Model the process of rising milk temperature and the microbial growth.

Orienteering. In orienteering the task is to run from a given point to the next in a nature where the environment contains terrain of various type (field, open forest, dense forest, bushes, wet land etc) and also slopes and hills of varying gradient. These are marked on a map that shows the points to be visited. In orienteering competition one should quickly decide what would be the optimal route between the marked points.

Bioreactor. Modeling biochemical processes and cell populations in a bioreactor.

Urinal shape. In men’s toilet a quite general problem is the behavior of the jet of urine that tends to splatter from the walls of the urinal bowl perhaps trickle to the floor or perhaps on your shoes and trousers. Could a model help in designing the shape of the urinal bowl so that it behaves nicely (with an average user).

Steel furnace insulation. In steel making a molten iron is kept in a huge container called ladle. Due to extreme temperatures the walls of the ladle are heavily insulated by a ceramic material (magnesic tiles) that is able withstand high temperatures.

The insulation material is gradually deteriorating due to two mechanisms (1) wear that is due to chemical processes at the interface between liquid steel and the tiles of the crucible wall and (2) the mechanical stress due to sudden temperature changes. Abript temperature changes are appearing when molted steel is poured in our out.

The company would need a model to forecast how the ladle wall wear is proceeding. The schedule of perhaps hundreds of cycles of filling/emptying of the ladle is the stochastic loading of the furnace.

Color analysis in aerial photography. In color photography the colors appearing on the proto are determined not only by the actual color of the object but also about the illumination, light intensity and directions, effect due to nearby objects and the technical parameters of the camera and its light sensors. In aerial photography one would need to have the colors reproduced correctly and to eliminate the distortions caused by these external factors. Models are needed to do the color correction.

Painting a house wall. When painting the outer wall of a house the paint is applied not only for aesthetic reasons but also to protect the surface of the wooden boards. A model should be developed to describe the life time behavior of the painted wall surface. The model should incorporate effects like (1) moisture in the timber and its flow across the paint layer (2) the deterioration of the paint as a result of weather. The model should give advice for choosing thickness of the paint and the quality of the paint.

Gas price. You are filling up your car. There are two gas stations. The price in the more distant station is lower. To time to drive to the other station is stochastic variable due to unknown traffic conditions. What is the price difference that would justify to drive to the cheaper gas station?

Torrent file sharing. BirTorrent protocol means a system where a publisher places a file on a server so that distant users can download it via internet. Other users may then download the file form these second stage users etc. A certain tracker server maintains the location of available files and when a user is reaching the file the server directs the user’s demand to the “nearest” available node. When there are thousands of files in the system and thousands of users participating in the file sharing market, an interesting dynamic traffic network is generated. Several modes are needed to simulate the system, analyze the performance and capacity and help to user to choose best way to minimize waiting time etc.

Sprinklers. In dry summer months one needs to have sprinklers to take care of laws, parks, grass fields in golf courses etc. The water pressure in the network affects the quality of the watering as the pressure drops over distance. The overall performance of the sprinkler networks should be modelled and using the model one should analyze best design of a sprinkler system.

Solar system second look. Write a model on the motions of the solar system if the origin of the system is place at the center of the earth instead of the sun.

Sugar metabolism. Human body is processing sugar in the metabolic system. Main variables in the system are G1=glucose and G2=glycogen and two hormones H1= insulin and H2= glucagon. Glucose is coming from our food and it is also produced in our liver by G2 à G1 reaction. Our pancreas is producing the hormones H1, H2. Hormone H2 is controlling the speed of the glucose production in the liver.

Glucose is the fuel for the muscles. Insuline control the transport of glucose to the muscle tissue. Body has a control mechanism where the transfer of glucose from blood into the muscle tissue with a speed that is dependent on the blood sugar level and a certain equilibrium value. Write an elementary dynamical model of this system.

Network of ports. National logistic system consists of 5-6 ports on the sea front and road a rail network serving the ports. The system should be able to maintain a flow of goods (containers, trucks). Weather may cause random disturbance due to heavy winds and ice conditions in the winter. They would need modelling tools to analyze the overall ability of the system to function under the uncertain conditions with temporary disturbances.

2009

Shock absorber. Shock absorber of a bike consist of a spring and a hydraulic component coupled together. Off road racing the terrain will change during the race and the shock absorber should preferably be adjustable or perhaps have automatic adaptation according to the profile of the track. To design the adjustable shock absorber one would need a model which can simulate the behavior.

Melting ice. When ace cube in a glass is melting, the size of the cube is reducing, the water near the surface is cooling since the water will not get mixed immediately. Moreover part of the ice cube is above the surface. By making simple assumption about heat transfer in the ice-water and ice-air interface, build a model about a melting ice cube.

Size of emergency unit. How to optimize the size of the emergency unit serving of a large urban area? Big number of emergencies are happening daily. The frequency depends on the hour of the day and location. Statistics about past years are available.

Rainbow. The physical principles explaining rainbow are well understood. The main factor is refraction of light on drops of water. The rainbow is a result of millions of these events and their statistical distribution. A mathematical model could be used to investigate some hypothesis (1) what is the effect of the size distribution and density of the water drops (2) does the model explain what happens if we have small ice chrystals instead of water drops (3) how does the result change if one takes into account hits of the light ray on multiple consecutive drops, multiple reflections etc.

How does a catalyst behave? Catalysis is a chemical phenomenon where the surface of a certain compound (catalyst) is facilitating a chemical reaction. In a simple example of reaction A + B à AB the molecule A perhaps is attached to the surface of the catalyst and this helps or enables molecule B to connect with A. In chemical terms A is adsorbed on the surface of the catalyst. When the reaction output AB is detached from the surface, this is called desorption. To understand the catalytic reaction we should be able to model and quantitatively understand the adsorption and desorption speeds.

Dynamics of a fire in a building. A model is needed to forecast how a fire would spread in the building. Such a model would help in safety assessment of architectural design and planning of evacuation routes.

Sludge. Modelling of active sludge process in biological waste water treatment. The phenomenon contains several components. Microbiological processes, bio-population growth, microbe metabolism and reaction kinetics, catalyzing and inhibiting enzymes, substrate concentration and composition, temperature, exothermal/endothermal processes, flocculation/aggregation of sludge, floc-descent/sedimentation etc. Exact modelling of the complex process is said to be “beyond computation” but certain qualitative partial models are likely to be useful.

Hunting permits. Hunting and wildlife authorities control (for instance) the elk-hunting intensity by issuing hunting permits in limited numbers in order to maintain a sustainable population size. A model is needed to compute, how many permits each year can be approved, when the variability of the natural breeding, uncertainty in game counting data, statistics from past years etc are taken into account.

Cell sample cultivation. In a biological laboratory one wants to grow and essentially multiply a given cell sample for biochemical measurements. Cell growth is very sensitive and it depends on the chemical composition of the substrate, temperature and effect of other cell species lie bacterial that are hostile for the cells that one wants to cultivate. For this reason some antibacterial substance (antibiotics) is also added to the solution. The cultivation process may take weeks. A model should be built to simulate the growth of the cell population. An optimal composition and correct temperature control could be found with the help of such model.

Cleaning in dairy process. Several billion liters of milk is taken to dairies annually in our country. The system consists of a complex network of tanks, pipes, pumps, filters, heating/cooling units etc. In order to maintain extreme hygiene the system must be “washed” regularly. Wash-procedure may have several stages and several substances: water, alkaline detergent, acidic detergent etc. Numerous washing recipes have been programmed. The frequency, intensity and length of the washing periods are based on past practices and one can suspect that there is considerable “overkill” excessive washing justr to guarantee the safety. Can one build a model about the effect of various washing stages and methods so that the model could be used to find a better scheduling and intensity for the washing programs.

2010

Frisbee dynamics. Frisbee golf is a past time for youngsters but also a sport with rules etc. There are several types of Frisbee discs (putter, mid-range, driver) that differ by their design and material properties, surface texture are different. A model should be developed to explain the aerodynamic properties of a given design.

Lake crab population. Finnish lakes inhabit crab population and they are heavily harvested for culinary purposed. To maintain the population the crab catching is regulated. One is allowed to catch them only during a given period of about two months. There is also a minimum allowed size, smaller must be released.

In spite of these regulations in some lakes the crab population may suffer from a phenomenon where the average size decrease so that majority of the crabs are dwarfs. A model is needed to explain this phenomenon. For instance it might reveal whether some special feeding program might help or perhaps make increase dwarfing?

Bottle as organ pipe. It is known that by blowing air perpendicular to the opening of a glass bottle one can generate sound – like whistle or an organ pipe. By addig water one can adjust the frequency of the sound. It would be nice to have a model to compute the pitch of such a bottle when the shape is given.

Texas Holdem robot. I would like to have a model that would help to play poker.

Bicycle riding and hills. The route in a bicycle race has a profile with varying elevation and gradient (uphill/downhill). The energy consumption of the driver depends on the slope, the speed, the used gear of the bike and the air resistance. A model is needed to help in finding a good way to balance the speed so that energy consumption and the total driving time are minimized.

Measuring a javelin throw. A javelin throw is done behind a circular line and javelin may land within a wide sector. The length of the throw should be measured from the hit point to closest point of this circular arc. Measuring used to be done by a long steel band with centimeter marks. This method is clumsy, slow and imprecise. The measuring tape should be moved sideways, one should guarantee correct tension and direct line, to identify the closest point on the arc etc.

For this reason in a modern competition an electromagnetic distance meter and a device for measuring angles (tachymeter) are used. Tachymeter is placed at the side of the field and it measures the distance and angle to the hit point and the center point of the arc. From these measurements and two control measurements the correct distance should be calculated.

2011

Keyboard. Analyzing the performance of various keyboard setting (qwerty, dvorak, ..) using linguistic data of frequencies of words, syllables, letters, etc.

Sound insulation. Modelling energy absorption and damping of sound waves in buildings, various wall-structures, layers etc.

De-growth ideology. Modelling the effect in the system of national or global economy if the idea of maximizing efficiency and profitability in economy would be replaced by the idea of de-growth, moderate profit margins, zero-sum economy, revaluation of immaterial vs material production etc.

Office lighting. In Nordic countries daylight decreases in Autumn and Winter and artificial illumination is deeded. In the middle of the day natural sunlight through windows can be used but indirectly by using reflective blind elements that illuminate the ceiling. Model the effect of such indirect lighting.

Portfolio. Investment portfolio for forest industry of varying forest types

Insurance. Design and pricing of a new insurance product

Banks and Greece. Modeling the risk scenario of banks who have financed Greek banks taking into account the possibility of debt restructuring, bank capitalization criteria, solvency constraints etc.

Post man. Route optimization of a postman including the size of his/her carriage.

Bioleaching pool. Open pit mine contains a huge rectangular pool (50m x 100mx 1,5m) where a mixture of ground ore powder, strong acid, water and microbial substance are kept. The bottom of the pool is a plastic sheet. There is leak somewhere at the bottom. This hole should be found. Emptying the pool is too expensive. If the water is removed the solids will form a sediment covering the bottom so the leak cannot be seen. Can a mathematical model help?

Droplets in heat exchanger. Heats exchanger consists of vertical slits between metal plates that separate hot vapor and cold water. The performance depends on how effectively heat is transferred through the plate. In the hot stream vapor will condense on the metal surface and the attached droplets form an insulation that reduces the heat conduction. In some condition the droplets grow big enough so that they slide down on the wall sweeping other droplets along the way. The result is a thinner water film and improved heat transfer. This behavior of droplet formation and water film should be modelled.

Popcorn. Generate a model that explains the “pop” of the popcorn

Golf bat angle. Ball is hit by a golf club that has 44 degree tilt angle. Model should be built to understand what happens to the ball.

Beer glass. During the time when a beer glass is consumed, the remaining beer gets warmer (unless one drinks very fast). Assume that the beer is leaving the glass with constant speed. Study if the shape of the glass (like decreasing diameter) has an effect on the temperature of the remaining beer when the glass is becoming almost empty.

Concrete for winter conditions. In building industry the concrete structures must withstand low temperatures (-30 and below) and strong temperature variations. Concrete will endure freezing if it has suitably porous structure. These are generated by adding suitable substance like soap water into the mixture. Sufficient density of small pores guarantee that internal pressure from freezing of the residual moisture is relaxed internally without cracking the material. A model is needed to understand the formation of pores.

Accelerated testing of material safety. In nuclear reactor special metal alloys are needed that can tolerate extended irradiation with neutrons that are hitting the atoms of the metal causing atom scale deformation that in the long run will affect the metallurgic properties and safety. An atomic level model is needed to simulate and understand how the hitting neutrons behave. The materials are required to have life span of 50-100 years and the testing cannot afford such a long time. A challenge is how to organize a test that can simulate in 100 hours the effect of 100 years of neutron radiation.

Maintenance schedule. A lifting crane is a key resource in mechanical workshops. A breaking crane is a major catastrophe. A crane is a device consisting of several components that are subjected to heavy randomly varying loads. Maintenance checks must be scheduled to do repairs before the contraption breaks. Some statistics of past damages are available and some diagnostic measurements from sensors and performance inspections are recorded. A model should be built to anticipate maintenance operations.

Heat pump. A modern alternative of house heating is so called air-water-heat-pump, which takes heat energy from outside air (in other applications from underground soil, lake water etc) and brings the air inside the house to heat the air and tap water. The system is based of thermal physics (Carnot process, reverse idea of refrigerator). A model is needed to calculate the performance and economic parameters of the heating system under random weather temperatures etc…

Forecast model for health risks. Hospitals and health centers have large data bases about health data and history of treatments, symptoms and medical interventions regarding huge number of people. Can this data be used to build a forecast model for the risk of appearance of musculo-skeletal problems.

Accident of chemical transport. A truck carrying poisonous chemical is overturn in a collision and 10 tons of the liquid escapes to the ground, which consists of layers of soil of varying porosity. Develop a model describing the spreading of the contamination in the soil.

Plow tractor routing. Farmer is plowing his land with a tractor which has 3 meter wide plow unit. The field has a shape of irregular perhaps nonconvex polygon. You should design an algorithm that would find a good route for the plow so that it would minimize plowing same area twice and minimizing the total time (assuming that the speed is lowered when the drive must take sharp turns.

Trapez artist. Model the sideways movements of a circus artist who walks on a wire holding a vertical pole to maintain balance. The model should take into account the movements of the wire, the trapez-artist and the pole.

Water pipes in a city. A new residential area for 30 000 families is being constructed. Planning of the water pipe network is part of the design. A model is needed to decide proper diameters of the pipes so that acceptable water pressure is maintained in the whole network.

Waste burning plant. Even after modern recycling some part of the residual of community waste will be burnt to generate thermal energy. The waste is very heterogeneous so the burning process is irregular and the air-feed into the burner must be regulated. Air is pumped into the system in stages (primary air feed, secondary and tertiary air feed). Some sensors can be used to monitor the burning temperature etc. A model is needed to regulate the air feed to optimize the burning.

City water purification. Purification of city water is a complex and multistage process. The raw (unclean) water is after primary filtering mixed with chemicals that cause suspended impurities to aggregate into particles. After that pressures air is used to generate bubbles that lift the particles as a foam onto the surface. The dirty foam is removed mechanically by a moving arm. Next phase is a sand-bed that is able to filter much of the remaining microscopic particles. Finally some chemicals and pH-adapters are added. A model could be built to describe each of these steps. A model of the whole purification cascade is needed to optimize the performance.

Combined energy plant. Thermoelectric energy producing plant consist of several units that can use various materials as fuel (wood, turf, coal, crude oil, waste, pellets). The units have various constraints and technical parameters. The challenge is to find optimal scheduling of the plant under changing weather/temperature conditions. Prices and efficiency of each of the fuels are different. Change from one fuel type to another may require ramp down/ramp up of the units that requires time and adds costs.

2012

Government tax-policy. Dynamics of state economy is a challenging game. If tax-percentage is raised the government is (at first?) getting more funds to the state coffin for public investments, salaries, subsidies for industry and universities etc. But higher taxes means less money for consumers to spend and this slows down the economy. Tax increase also generates pressure for salary increase and lowers work motivation. The challenge is to model this complex network of interdependent quantities and money flows to optimize the performance of the economy.

Tree planting. In modern forestry new tree-plants (perhaps 30cm high) are planted on an open clear-cut field to replace the previously harvested forest. Approximately 1800-2000 trees are planted per hectar. The total area may be anything between 5-20 hectars so a huge number of tree-plans must be transported by manual labor, walking in the field, sometimes in quite ragged topography. The delivery is done in stages. Boxes of 1000-1500 plants are distributed to the field by carrying from a truck. From the boxes the tree-planters pick 100-200 plants into a basket that is easy to carry around. The person walks around to stick the plants into the ground one-by-one and then returns to the box to fill his/her basket. The task is to find an efficient way to decide the sizes and locations of the boxes and the size of the carry-on basket to minimize the total walking distance.

Nuclear fuel rod. The radioactive fuel, typically uranium is contained in rods, which have a protective surface layer. The layer is made of zirconium which is supposed to be very safe against corrosion. The challenge is to model the oxide-formation at the zirkonium-surface under the heavily corrosive circumstances. At first oxygen atoms reach the surface easily and an oxide-layer that is growing rapidly. The oxide layer is itself protecting the surface so that less oxygen molecules reach the metal. So the growth is slowed by the accumulation of oxide layer. However when the oxide layer is aging, some minor events and secondary processes in the oxide layer generate microscopic defects which open pathways for oxygen to reach the metal surface and so the oxidation continues. Generate a model for the oxide layer growth.

Ultrasound on microbubbles. Utrasound is used in some therapeutic treatments where the ultrasound is penetrating a tissue and causing small gas filled bubbles to vibrate, break etc. The modelling challenge is to compute how a microscopic gas filled bubble suspended in a liquid will react when an ultrasound is applied. The modeling challenge is increased when the nature of the bubble-wall is taken into account. The wall may be a membrane with certain mechanistic properties. The complexity is further increase if one considerer several adjacent bubbles, border regions etc.

Glacier mass. An important environmental concern is the development of glaciers. It would be important to know the mass of ice that has been accumulated in a glacier. Due to temperature variations and weather, the total amount is changing. Rain and snowfall, melting and freezing, flowing of liquid water will affect the mass of the glacier. A model is needed to describe how given weather conditions and temperature change will affect the glacier mass evolution. More sophisticated models should also account for the extremely slow movement of the ice mass itself.

Coffee machine. How an ideal coffee filtering machine should be designed? How to model the penetration of water through the porous layer of fine coffee powder? How to describe the extraction of taste and aroma from the coffee grains into the hot water flowing through. What would be the ideal speed to pour the water? What shape of the filtering cone would be best?

Beer orders. A corner bub has to decide how much beer they should have in store and how to place the pre-orders for the coming weeks and months. The consumption on different days of the week have an index (Monday 0.7x, Friday 2,5x average). Months are different (relative demand January 0,5x, July 2,8x). Beer should not stay in the back room storage for too long (max 14 days?), storage capacity is limited etc. Under such constraints one should design a strategy for ordering beer into the storage.

Heat through a dirty plate. Heat exchangers are common devices in process industry, buildings etc. They consist of pipes and plates that separate the hot and cold fluid (liquid, gas, mixtures). Heat transfer takes place through the metal plate or pipe wall. The performance of the device is affected if the metal surface is contaminated by dirt. The task is to model the heat transfer through a metal plate that is covered by a layer of dirt with given thickness and material parameters.

District heating network. District heating system consists of a bit thermal plant and a large pipe network that connects large number of houses and apartments to the supply of heat from the thermal plant. The area may be a big section of a city (or even whole city?) and there are hundreds or households receiving their heating and hot water from the network. The tasks is to devise a model that will forecast what happens if the thermal plant is cut off due to maintenance or malfunction. The insulation of the pipes is very high and the heat loss via pipe walls can be neglected.

Parasite epidemics. One of the trouble in modern urban life is the recurrence of certain parasite epidemics (like cockroach, louse, flea). Epidemics of head louse are appearing frequently is some schools, kindergartens etc. A model is needed to explain the infection patterns. Assume that a city keeps record of detected louse-epidemics (locations, timing, number of cases) in the region (perhaps 50-100 schools and nursery schools) over a 3-4 year period. How such data could be analyzed and what could be learned?

Bike brakes. Study the effect of bike braking system, when front wheel brake, rear wheel brake or both are applied. The center of the mass of the driver is off the ground and due to inertia the weight balance is shifted when brake is applied, right? What would be the best way to stop the bike?

2013

Brewing cider. Apple cider is produced by letting a pressed juice to ferment in a tank. Some sugar and yeast are added. During the fermentation the yeast microbes are transforming sugar into ethanol. The tank must be carefully sealed because opening would disturb the fermentation process and oxygen should not get into the mixture. Direct measurement of the concentration of sugar and alcohol is not possible during this phase. One would need a model to simulate the biochemical process of fermentation.

Pit. An open pit mine is bases on bioleaching process where microbes are extracting metal from the ore that has been crushed into thick layers of gravel. Some acid and huge amount of water are poured into the heap of ore. The bioleaching plant consists of many huge interlinked storages of water that is flowing (or being pumped) from one process stage to the next. Key parameter of the process are concentrations of many chemicals in various parts of the process, some of them are difficult to observe. A mathematical model of the plant as a network of interconnected storage vessels would be a big help in controlling the system.

Traffic by fixed speeds. Study how the regional traffic network would work if cares would be equipped with a mechanism that would limit the speed to integer multiples of 10, that is 40km/h, 50km/h, 60km/h,…, 120km/h.

2014

Modelling the risk scenario of nuclear plant safety. A possible accident in a nuclear plant would appear as a consequence of a sequence of failures, some of which would indicate causal dependencies. The whole fault scenario would be modelled as a fault tree, with probabilities of different outcomes and causal effects. Once such complex fault tree model is set up, statistical safety analysis can be done with Monte Carlo simulation.

Ice breaking. Ice mechanics is important in arctic maritime industry of building ice breakers and planning their efficient use. Single modeling task is to study how an 1 meter thick sheet of ice on the surface of frozen sea will break when the wedge shaped front end of an ice breaker ship is pushed on top of the ice sheet.

Forest trimming. In forest management a newly planted forest is allowed to grow freely until the highest threes are perhaps 4-5 meters tall. The first thinning takes place by removing part of the smaller trees to allow the remaining to grow faster, having more light and nutrition. The second thinning takes place again after a few years. The task is to model the forest growth to find a most efficient thinning strategy.

Cross-bike jump. Cross-country bike is a vehicle with two wheels, which have a suspension system of spring-damper type. The drivers are doing wild jumps by riding over hills of different shape. Model the mechanical behavior of such bike given the speed and the shape of the hill. Drivers steering effects – of course crucial – can be forgotten first.

Newspaper delivery. When a national newspaper is printed, the printing takes several hours. The delivery to the subscribers can be started before the printing is finished. This early delivery time can be used to carry packets of the newspaper to remote locations that require longer delivery time. Modelling the logistics of newspaper delivery is the task here.

Inverse kinematics. Mechanical device consists of rotating wheel, and interconnect shafts, hinges and sliding sleeves. Such couples system transform the rotational movement into different shape of output curve. Simplest type would be one dimensional movement of piston shaft but the output curve can be more complex. The modeling task is of inverse kinematics. Given the desired output curve, construct a mechanical device that produces the desired movement.