Facultad de Ciencias Matemáticas D.M.A. Benjamin Ivorra

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1 Facultad de Ciencias Matemáticas D.M.A. Tendencias actuales de la matemática interdisciplinar Mathematical Modeling and Optimization Applications to industrial design problems Benjamin Ivorra

2 Industrial problem design What is the objective of a design problem? It consists in creating or modifying a product in order to obtain or improve some characteristics. i Example of products:

3 Industrial problem design What are the steps to follow in order to solve a design problem? Write a mathematical formulation of the problem Creation of a mathematical model dl Validation of the model Resolution of the problem Implementation of the result We will describe each step considering a particular We will describe each step considering a particular design problem: Design of a microfluidic mixer.

4 What is a microfluidic mixer? It s a micrometric device which objective is to mix various fluids. Here we focus on the Knight s mixer: A block of silicon containing four channels. The mixing is based on hydrodynamic properties of fluids.

5 What is a microfluidic mixer? It s a micrometric device which objective is to mix various fluids. Here we focus on the Knight s mixer: A block of silicon containing four channels. The mixing is based on hydrodynamic properties.

6 What are the applications? In pharmaceutical industry: For protein unfolding (used in drug conception, DNA identification,.): How? When the mixer mix fast enough a protein solution with a solvent (in term of concentration). The mixing time depend of the mixer shape:

7 What are the applications? In pharmaceutical industry: For protein unfolding (used in drug conception, DNA identification,.): How? When the mixer mix fast enough a protein solution with a solvent (in term of concentration). The mixing time depend of the mixer shape:

8 What are the applications? In pharmaceutical industry: For protein unfolding (used in drug conception, DNA identification,.): How? When the mixer mix fast enough a protein solution with a solvent (in term of concentration). The mixing time depend of the mixer shape:

9 Example: Design ofprotein a microfluidic mixer What are the applications? In pharmaceutics industry: For protein unfolding (used conception) in drug conception): Solvent Solvent How? When the mixer mix fast enough a protein solution with a solvent (in term of concentration). Parts of the Shape The mixing timethedepend of the mixer shape: influencing mixing time (other parts are fixed) fixed).

10 Step 1 What is the design problem? Find the best mixer shape producing the lowest mixing time. Step 2 Traducing this problem into an optimization problem Problem parameter x: mixer shape in the set Ω. Function to optimize f: mixing time. Theassociated optimization problem is of the form: min f(x) x Ω

11 Step 3 Modelization We first fix the parameter (the mixer shape). We need to find a mathematical ti model lthat t models the desired physical phenomenons: The protein concentration evolution. The mixer flow evolution. Navier Stokes (flow) ( ) + Convection Diffusion (concentration) Equations (P.D.E.) + Boundary conditions

12 Step 3 Modelization We first fix the parameters (the mixer shape). We need to find a mathematical ti model lthat t models the desired physical phenomenon: The protein concentration evolution. The mixer flow evolution. Navier Stokes (flow) ( ) + Convection Diffusion (concentration) Equations (P.D.E.) + Boundary conditions

13 Step 3 Modelization We choose an adequate numerical model (here FEM). Sometimes there aresome tips to savecomputational times: Instead of studying a 3D mixer we can consider a 2D mixer. Due to symmetrical properties, we can study only a half of the mixer.

studying a 3D mixer we can consider a 2D

14 Step 3 Modelization We choose an adequate numerical model (here FEM). Sometimes there aresome tips to savecomputational times: Instead of studying a 3D mixer we can consider a 2D mixer. Due to symmetrical properties, we can study only the half of the mixer.

15 Step 3 Modelization We choose an adequate numerical model (here FEM). Sometimes there aresome tips to savecomputational times: Instead of studying a 3D mixer we can consider a 2D mixer. Due to symmetrical properties, we can study only a half of the mixer.

16 Step 3 Modelization We choose an adequate numerical model (here FEM). Sometimes there aresome tips to savecomputational times: Instead of studying a 3D mixer we can consider a 2D mixer. Due to symmetrical properties, we can study only a half of the mixer.

17 Step 3 Modelization We choose an adequate numerical model (here FEM). Sometimes there aresome tips to savecomputational times: Instead of studying a 3D mixer we can consider a 2D mixer. We can now compute the interesting mixer characteristic: The mixing time. Due to symmetrical properties, we can study only a half of the mixer.

18 Step 4 Model Validation We validate the model comparing with experimental data. Here the model was validated considering this mixer shape ( optimized i experimentally):

19 Step 5 Parameterization We want integrate the parameters (the mixer shape) in the previous model. We must take into account technical restrictions: The mixer must be plugged to other devices: The mixer is grabbed into a silicon piece using an automated Laser: We can only grab a limited number of geometrical forms with a minimum size:

20 Step 5 Parameterization We want integrate the parameters (the mixer shape) in the previous model. We must take into account technical restrictions: The mixer must be plugged to other devices: The mixer is grabbed into a silicon piece using an automated Laser: We can only grab a limited number of geometrical forms with a minimum size:

21 Step 5 Parameterization We want integrate the parameters (the mixer shape) in the previous model. We must take into account technical restrictions: The mixer must be plugged to other devices: The mixer is grabbed into a silicon piece using an automated Laser: We can only grab a limited number of geometrical forms with a minimum size:

22 Step 5 Parameterization We want integrate the parameters (the mixer shape) in the previous model. We must take into account technical restrictions: The mixer must be plugged to other devices: The mixer is grabbed into a silicon piece using an automated Laser: We can only grab a limited number of geometrical forms with a minimum size:

23 Step 5 Parameterization In this case, the shape is built univocally using the 12 variables and interpolation techniques. The parameter x, associated to a mixer shape, is of the form: i i min max i 1 x= ( x,...,, x ) Ω= [ x, x ] i= = All components of our optimization problem are now well defined: dfi d min f(x) x Ω

24 Step 5 Parameterization In this case, the shape is built univocally using the 12 variables and interpolation techniques. The parameter x, associated to a mixer shape, is of the form: i i min max i 1 x= ( x,...,, x ) Ω= [ x, x ] i= = Allows to compute f(x) All components of our optimization problem are now well defined: dfi d min f(x) x Ω

25 Step 6 Optimization We are now interested to solve: i f(x) min x Ω A first study of the properties of f must be done in order to choose an adapted optimization method: Geometrical properties of f: convexity, convex hull, high number of local minima, flat Numerical properties of f: high computational time, exact/automated/incomplete gradient, approximation by low cost functions

26 Step 6 Optimization We are now interested to solve: i f(x) min x Ω A first study of the properties of f must be done in order to choose an adapted optimization method: Geometrical properties of f: convexity, convex hull, high number of local minima, flat Numerical properties of f: high computational time, exact/automated/incomplete/ gradient, approximation by low cost functions

27 Step 6 Optimization We are now interested to solve: i f(x) min x Ω A first study of the properties of f must be done in order to choose an adapted optimization method: Geometrical properties of f: convexity, convex hull, high number of local minima, flat Numerical properties of f: high computational time, exact/automated/incomplete gradient, approximation by low cost functions

28 Step 6 Optimization Here f is non convex with several minima We need to use a global optimization method. The gradient of f computed using a coarse mesh can be used as an approximation of the full gradient. We can use gradient based method. HbidS Hybrid Semi Deterministic Dt iiti method

29 Step 6 Optimization Here f is non convex with several minima We need to use a global optimization method. The gradient of f computed using a coarse mesh can be used as an approximation of the full gradient. We can use gradient based method. HbidS Hybrid Semi Deterministic Dt iiti method

30 Step 6 Optimization Here f is non convex with several minima We need to use a global optimization method. The gradient of f computed using a coarse mesh can be used as an approximation of the full gradient. We can use gradient based method. HbidS Hybrid Semi Deterministic Dt iiti method

31 Step 6 Optimization Solution (shape) provided by the optimization method: We then study the numerical properties (stability in function of the parameters) of this solution: It s Importantbefore the implementation step.

32 Step 7 Implementation and validation of the solution The mixer corresponding to the solution is built (some differences appear between the exact solution and the experimental prototype):

33 Step 7 Implementation and validation of the solution The optimized and initial mixer mixing times are then compared: Initial Optimized Vs 5µs 1µs

34 Conclusions We have seen how to incorporate two important mathematical areas (modeling and optimization) into an industrial process: Mathematical sciences can be used as a primordial professional tool! The applications are numerous and varied due to the increasing demand of product improvement (Especially in term of energy consumption: Engine, Electric devices, ). There is also a high demand of development of new techniques (theoretical and numerical) for modeling and optimization!

Mathematical modeling for protein folding devices. Applications to high pressure processing and microfluidic mixers.

Mathematical modeling for protein folding devices. Applications to high pressure processing and microfluidic mixers. Ivorra Benjamin 1 & Angel Manuel Ramos 1 Microlfuidic mixer: Juana López Redondo 2 &