CAE Sun Simulation - Thermo Structural Coupling

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IN

DEGREE PROJECT MATHEMATICS, SECOND CYCLE, 30 CREDITS

STOCKHOLM SWEDEN 2020,

CAE Sun Simulation -

Thermo Structural Coupling

KRISTIN STAFFAS

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES

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CAE Sun Simulation -

Thermo Structural Coupling

KRISTIN STAFFAS

Degree Projects in Scientific Computing (30 ECTS credits)

Degree Programme in Applied and Computational Mathematics (120 credits) KTH Royal Institute of Technology year 2020

Supervisors at Volvo Cars: Renaud Gutkin, Björn Ratama Supervisor at KTH: Patrick Henning

Examiner at KTH: Michael Hanke

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TRITA-SCI-GRU 2020:373 MAT-E 2020:085

Royal Institute of Technology School of Engineering Sciences KTH SCI

SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

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Sammanfattning

CAE solsimulering – Termo-strukturell koppling

Deformation av material till följd av hög värmelast är ofta en utmaning inom fordonsdesign. Materialet i kupén exponeras både för direkt solljus och den värme som ackumuleras i det stängda utrymmet. För att få förståelse för hur denna process påverkar hållbarheten är numerisk analys ett viktigt verktyg.

Föreliggande examensarbete undersöker om det är möjligt att, med väderdata och kunskap om hur en bil används, simulera deformationsprocessen i trimdelarna som uppkommer av värmexponering från solen.

För att besvara denna fråga har två simuleringsmetoder jämförts: Den första metoden är att endast titta på materialets åldrande över tiden. Den används idag på Volvo Cars, men har befunnits vara otillräcklig utifrån ett antal aspekter.

Den andra metoden inkluderar effekterna av de elastiska deformationerna av materialet. Denna metod har inte tidigare prövats i ett liknande sammanhang men bedömdes intressant då den modellerar mer verklighetstroget. Dessutom undersöktes effekterna av att modellera bilen enligt ett virtuellt kundbeteen- de snarare än som stillastående. De huvudsakliga simuleringsverktygen som använts här är TAITherm och Abaqus.

Resultaten visar att modellering av fordonet med en tänkt användarrutin ger en lägre medeltemperatur eftersom bilen kyls ner när den körs. Resultatet av jämförelsen mellan de två metoderna visar att inkludering av effekterna av de elastiska deformationerna bidrar till en bättre modellering av åldrandet av trimdelarna till följd av värmexponering från solen.

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2.1 Background Theory . . . 4 2.2 A Closer Look on FDM Discretization . . . 6 3 Methodology for the Experimental Approach 12 3.1 General Methodology . . . 12 3.2 Experimental Setup . . . 13 4 Study of Parameters to Improve Efficiency 21 4.1 Parameter study results . . . 22 4.2 Summarizing discussion . . . 31 5 Fully Coupled Thermal Structural Analysis 33 5.1 Virtual customer behaviour . . . 33 5.2 Year-long simulation . . . 35

6 Results 36

6.1 Thermal Results . . . 37 6.2 Coupled Thermal Structural Results . . . 40

7 Discussion 45

8 Conclusions 47

8.1 Future work . . . 47 8.2 Acknowledgements . . . 48

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vi CONTENTS

Bibliography 49

A Thermal results 50

B Some Results of the Structural Analysis 51 B.1 Displacement of the Control Nodes . . . 51 B.2 Strain of the Control Nodes . . . 54

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Chapter 1 Introduction

1.1 Background

When designing any vehicle, material choice is of great importance. It will affect many properties of the finished product, such as the performance, the aesthetics and the durability [1]. In car manufacturing specifically, the imposed heat from the sun must be taken into consideration when choosing the material for the trim parts. The trim - i.e. the finishing parts of the car’s interior - are in many ways what costumers are seeing the most of throughout the ownership of the vehicle, and it is an important part of the customer ex- perience. For this reason, the trim is arguably very sensitive to damage that compromise the design aspects. The trim parts can be seen in Figure 1.1.

When a car is parked in the sun, the trim is exposed to very high temperature due to both direct sunlight and heat build up in the cabin and, it is very common that deformations are formed over time due to the heavy thermal load. To avoid this, one can improve the design of the parts or choose a more resistant and durable material for the manufacturing. In order to do this one must know what temperatures the trim must be able to withstand. A numerical simulation of the car being exposed to the sun would be an advantageous approach to this problem as it does not require any physical prototype and the costs can therefore be kept low.

However, modeling such an extensive process is computationally very heavy.

Without some degree of assumptions and simplifications being made, the process would not be applicable to commercial computation. To determine suitable simplifications to the parameters used in a computation and still main-

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2 CHAPTER 1. INTRODUCTION

Figure 1.1: The trim parts of the car highlighted

taining accuracy in the results is a complex process. It requires a deep un- derstanding of the problem and the numerical methods used to produce the results.

Physical experiments are the main method used at Volvo today to design trim parts that do not deform in a way that compromises the design when exposed to high temperatures. However, attempts to turn to computer simulations have been made. The approach taken in those attempts has been to only look at the aging of the material, and not the reversible deformation. The mean temperature of the parts are applied as constant heat load over the entire model for a structural analysis. The cycle is then accelerated with the Arrhenius equation [2]. The simulation models a car during three years. With this approach, not all the deformations that have been observed in the physical experiments appear in the results. Even so, the results of those studies constitute a useful starting point for subsequent analyses of which this work is an initial step.

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CHAPTER 1. INTRODUCTION 3

1.2 Research Question

The research question of this thesis is "Is it possible to, with weather data and knowledge of how a car is used, simulate the deformation process of the trim parts due to heat exposure from the sun?". To answer this, the work was set up on two pillars:

• Looking at the problem from a theoretical standpoint: to investigate if the CAE software available at Volvo is suitable for solving the types of differential equations present when performing coupled thermal structural analyses.

• A numerical experiment in which a model of a Volvo V60 was exposed to measured weather conditions and the deformations were calculated from the obtained heat load.

In the theoretical approach, the algorithms that are used by the CAE software were examined to determine weather they are appropriate to use in this context. This was achieved by looking closer at the discreatization and numerical method of choice for each software and its compatibility with the partial differential equation that describes the physical phenomenon in the problem.

In the experimental approach, a model of Volvo V60 was set up and used to run simulations with the 3D thermal simulation software TAITherm [3] and the finite element solver Abaqus [4].

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Chapter 2 Theoretical Approach

2.1 Background Theory

Heat transfer, as well as many other physical phenomena can be modeled with partial differential equations, i.e. a differential equation that contains unknown multi variable functions and their partial derivatives. These types of equations are for the most part impossible to solve analytically and require numerical approximation. The basics of handling partial differential equations can be read in [5].

Numerical approximation requires digital computers for implementation. Ex- act solution of a continuous partial differential equation would require infinitely small steps and infinitely high dimension in the solution space. Since this is not feasible for classical computers, the first step is to transfer the continuous data into its discrete counterpart. This process is called discretization.

Whenever some data is discretized, there will always be a discretization error. The more points one chooses to have in the discretized data the smaller this error will be. However, this will also increase the computation time, as the computer needs to evaluate more points in the approximation. A lot of the work when using numerical approximation lies in finding a suitable discretization which keeps the error under a level considered acceptable for that specific problem as well as keeping the run time reasonably low.

When discretizing in time specifically there is, as well as the discretization error, the risk of instability. Explicit methods tend to be more sensitive to this type of instability since they have worse energy preservation properties than implicit methods. When working with explicit methods there are specific

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CHAPTER 2. THEORETICAL APPROACH 5

theoretical conditions for the appropriate size of the time step relative to the spatial mesh size. There are also some algorithms that choose the time step size adaptively to keep a stable level of accuracy in the approximation. When discretizing in space it is common to refine the discretization on critical areas on the geomerty. A discretized point on the geomerty is called a node.

When a suitable discretization in the geometry is found, a suitable linearization must be found. These linearization methods are used to approximate the solution at each time step and each node. There are many such methods with varying quality and different properties that are suitable for different types of problems. The available data in the problem can also influence the choice of linearization method. A large part of the craft lies in finding an appropriate method that fits the modeling purposes at hand. More on the most common methods can be read in [6]. A numerical method implemented with an appropriate programming language is called an algorithm. These algorithms are iterative procedures.

The algorithms approximate the solution in a point of interest by using a numerical approximation routine. This routine is specific for each method and involve using known data from surrounding nodes in each step to approximate the next nodal value. This user routine can be visualized as a geometric arrangement, showing what data the algorithm use at each step. Such a visualization is called a stencil. Two examples of a five point stencil, in 1 and 2D, is shown in Figure 2.1.

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6 CHAPTER 2. THEORETICAL APPROACH

Figure 2.1: An example illustration of the five point stencil in one and two dimensions. The red node indicates the point of interest.

When the algorithm reaches a point on the boundary of the geometry, there will not be enough surrounding points for the algorithm to use. To solve this, there are again many different options. The boundary conditions of the model will greatly influence which approach is the most suitable. More on how to handle boundaries in the context of numerical approximation can be read in [5].

2.2 A Closer Look on FDM Discretization

One of the modes of heat transfer considered by TAITherm is heat conduction, which is modeled by the heat equation.

∂u

∂t = α(∂²u

∂x² +∂²u

∂y²)

TAITherm uses finite-difference methods (FDM) for discretization. The main idea in FDM is to reduce linear ordinary differential equations or, like in this case, non-linear partial differential equations into a system of equations that can be solved by matrix algebra. This reduction makes the process of finding a solution to the differential equation at hand ideally fitted for a digital computer,

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hence the widespread use of FDM in modern computing. The exact discretization process TAITherm uses is not available to the public but a closer look on the general FDM discretization process on a simple model will be shown.

We imagine a metallic plate of length and width L [m] and temperature T = 0 with insulated boundaries. A heat pulse of temperature T0and duration tp [s]

hits the centre of the plate at time t = 0. The heat diffusion process in 2D over this plate can be modeled with the following differential equation, assuming that that thickness of the plate is negligible.

k ∂²T (x, y, t)

∂x² +∂²T (x, y, t)

∂y²

= ρC_p∂T (x, y, t)

∂t ,

t > 0, (x, y) ∈ [0, L] × [0, L]

(2.1)

Where ρ is the density [kg/m³], Cp the heat capacity [J/kg · C] and k is the thermal conductivity of the material [J/m · s · C].

The initial condition is then

T (x, y, 0) =

(T₀, x, y = L/2

0, Otherwise (2.2)

No-flux Neumann boundary conditions model the insulated boundary of the plate. The boundary conditions are

∂T (0, y, t)

∂x = 0 t > 0, y ∈ [0, L] (2.3)

∂T (L, y, t)

∂x = 0 t > 0, y ∈ [0, L] (2.4)

∂T (x, 0, t)

∂y = 0 t > 0, x ∈ [0, L] (2.5)

∂T (x, L, t)

∂y = 0 t > 0, x ∈ [0, L] (2.6)

This means that no heat can leave the plate.

The next step is to re scale equation (2.1) - (2.6) to dimensionless form. This can be done using the four variables

T = T₀u x = Lξ y = Lη t = t_pτ

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Substituting for these variables transforms equation (2.1) into

k T₀ L²

∂²u

∂ξ² + T₀ L²

∂²u

∂η²

= ρC_pT₀ t_p

∂u

∂τ τ > 0 (ξ, η) ∈ [0, 1] × [0, 1]

(2.7)

Re arranging equation (2.7) gives the dimensionless heat equation.

α ∂²u

∂ξ² +∂²u

∂η²

= ∂u

∂τ τ > 0 (ξ, η ∈ [0, 1] × [0, 1])

(2.8)

Where α = _ρC^kt_p^P_L² is a dimensionless constant. This is shown with a dimen- sional analysis.

ktP

ρC_pL² =

_J

msC [s]

_Kg

m³

h

J KgC

i [m²]

=

J mC

= 1

The initial value, and boundary conditions are re-scaled using the same variables. This yields the following conditions.

u(ξ, η, 0) =

(1, ξ, η = 1/2

0, Otherwise (2.9)

∂u(0, η, τ )

∂ξ = 0 τ > 0, η ∈ [0, 1] (2.10)

∂u(1, η, τ )

∂ξ = 0 τ > 0, η ∈ [0, 1] (2.11)

∂u(ξ, 0, τ )

∂η = 0 τ > 0, ξ ∈ [0, 1] (2.12)

∂u(ξ, 1, τ )

∂η = 0 τ > 0, ξ ∈ [0, 1] (2.13) When the equation is re-scaled, the continuous problem can be discretized in space. The plate is, in line with finite difference, divided into N × M points as seen in Figure 2.2.

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Figure 2.2: The plate discretized into N × M points. Each grid point has the coordinates (ξi, η_j)

We can now approximate the second order spatial derivatives with central differences. This approximation has second order accuracy which means that the error term is of the same order as the step size squared.

∂²u(ξ_i, η_j, τ )

∂ξ² = u(ξ_i+1, η_j, τ ) − 2u(ξ_i, η_j, τ ) + u(ξ_i−1, η_j, τ )

2∆ξ² + O(∆ξ²)

(2.14)

∂²u(ξi, ηj, τ )

∂η² = u(ξi, ηj+1, τ ) − 2u(ξi, ηj, τ ) + u(ξi, ηj−1, τ )

2∆η² + O(∆η²)

(2.15)

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Thus, the original equation is discretized in space. For simplicity we let α = 1 and M = N from which follows that ∆ξ = ∆η ≡ h, and write the approxi- mated equation at the position i,j (u^i,j(τ ) ≈ u(ξ_i, η_j, τ ) ) as:

∂u_i,j(τ )

∂τ = u_i+1,j(τ ) + u_i,j+1(τ ) − 4u_i,j(τ ) + u_i−1,j(τ ) + u_i,j−1(τ )

2h² (2.16)

This approximation routine visualized as a stencil corresponds to the two di- mensional example in Figure 2.1; for the approximation of each grid point the algorithm will use data from all of its orthogonal neighbours. As briefly discussed in Section 2.1, this will not work as the algorithm reaches the boundary of the domain. In this case, a suitable approach to solve this is with ghost grid points. These are imaginary grid points outside the domain. We look at the ξ = 0 boundary to further elaborate. The boundary condition on the first derivative is approximated with central difference as

∂u_0,j(τ )

∂ξ = u_1,j(τ ) − u−1,j(τ )

2h = 0 (2.17)

In reality, i = −1 is not defined on our domain, however since the value of this derivative is known from the no-flux boundary conditions we can state that

u−1,j(τ ) = u_1,j(τ ) (2.18) Thus, at the ξ = 0 boundary the equation to be solved is

∂u_0,j(τ )

∂τ = 2u_1,j(τ ) + u_0,j+1(τ ) − 4u_0,j(τ ) + u_0,j−1(τ )

2h² (2.19)

The same principle applies to the three other boundaries. This yields the following equations:

∂u_N,j(τ )

∂τ = u_N,j+1(τ ) − 4u_N,j(τ ) + 2u_{N −1,j}(τ ) + u_N,j−1(τ )

2h² (2.20)

∂u_i,0(τ )

∂τ = u_i+1,0(τ ) + 2u_i,1(τ ) − 4u_i,0(τ ) + u_i−1,0(τ )

2h² (2.21)

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∂u_i,N(τ )

∂τ = u_i+1,N(τ ) − 4u_i,N(τ ) + u_i−1,N(τ ) + 2u_{i,N −1}(τ )

2h² (2.22)

In the corners of the plate, two boundary conditions are applied. The same principle is applied here and it gives the equations that approximate the temperatures at the four corners:

∂u0,0(τ )

∂τ = 2u1,0(τ ) + 2u0,1(τ ) − 4u0,0(τ )

2h² (2.23)

∂u_0,M(τ )

∂τ = 2u_1,M(τ ) − 4u_0,M(τ ) + 2u_{0,M −1}(τ )

2h² (2.24)

∂u_N,M(τ )

∂τ = −4u_N,M(τ ) + 2u_{N −1,M}(τ ) + 2u_{N,M −1}(τ )

2h² (2.25)

∂u_N,0(τ )

∂τ = 2u_N,1(τ ) − 4u_N,0(τ ) + 2u_{N −1,0}(τ )

2h² (2.26)

Now, the full problem is semi-discretized in space. It is lastly solved by choosing an appropriate numerical method that iterates through the time. TAITherm’s method choice is not shared publicly. However it is known that it ensures second order accuracy in time.

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Chapter 3 Methodology for the Experimen- tal Approach

3.1 General Methodology

In order to simulate the heat distribution on, and displacement of the trim nu- merically, two main software were used. TAItherm, a multi-bounce 3D thermal simulation program was used to calculate the thermal load at the nodes of the model of the car from the weather data. TAITherm is specifically de- signed to solve heat transfer problems on complicated geometries. It takes into account heat conduction, convection and radiation which in this case is very important, and it also provides a way to model solar impact. The output from TAITherm is then used to model the nodal displacements with the solver Abaqus, which is a commercial software package for finite element simulations. Here it is an appropriate choice as it enables a creep analysis to see the permanent deformations caused by the imposed solar heat. An overview of the general course of analysis can be seen in Figure 3.1.

To interpret the results the pre, and post processors Ansa [7] and Meta [8] as well as Matlab were used.

The model used is that of a Volvo V60 which is a car that has been in produc- tion at Volvo for some years now. This specific model was chosen since it has previously been used in a physical experiment of the same nature. This provides an idea of the desired results from a computed simulation. This thesis work focuses on creating a simulation closer to reality, with less assumptions

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CHAPTER 3. METHODOLOGY FOR THE EXPERIMENTAL APPROACH 13

Figure 3.1: A flow chart representing the general course of analysis.

being made than in previous attempts at Volvo. The heat load in the structural analysis is therefore treated as a time dependent function over the geometry of the model. To analyze the results, they were compared to those of the original method used at Volvo for these simulations.

The deformations on the trim due to high temperatures from sun exposure develop over time. For this reason, the model simulates a car that is exposed to the sun for one year. To make this process more realistic, a virtual customer behaviour was created. This is further explained in section 5.1.

The data used mainly consists of measurements previously used and taken by Volvo cars. To simulate a realistic weather scenario for the car to be exposed to, data has been taken from a place outside Pheonix in the United States.

3.2 Experimental Setup

The aim of this experiment is to be able to observe the deformations from the solar heat on the trim parts of the car through numerical simulations. To be able to do this, the simulation is split up into two parts: The calculation of the thermal load using TAITherm and the calculation of the displacement it entails using Abaqus.

For the first part, the model of the entire car is needed since the high temperatures the trim parts are exposed to results from the fact that they are enclosed in the cabin. The geometry constructed with CAD was discretized into elements that together form a mesh. The fixed mesh is shown on the model in Figure 3.2. More on how the coarseness of the mesh was set can be found in Chapter 4.

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14 CHAPTER 3. METHODOLOGY FOR THE EXPERIMENTAL APPROACH

Figure 3.2: The full meshed model

TAITherm provides a functionality called Patches. The patches are made out of a second surface grid on the model that is used to determine which surfaces are affected by the radiation from each other. These surfaces are larger than the area of a single element, and it can be beneficial to take advantage of that when calculating the radiation. The coarseness of the patches are determined by the user and their main purpose is to decrease the run time by, for each time step, only solving the heat transfer due to radiation [9] once per patch instead of once per element. This means that with respect to radiation the cell count is decreased by a factor determined by the patch size. Figure 3.3 shows the patches on the model. This functionality is only used in the heat transfer analysis with TAITherm, it is discarded when moving on to the structural part.

The patches are introduced since there are small variations in the radiation in this setup. More on how the patch size was set and how it affects the thermal solution can be found in Chapter 4.

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Figure 3.3: The patch grid displayed on the model

The simulation time must also be discretized, this is done through dividing it into time steps. Large time steps yields a shorter run time of the simulation but a larger risk of errors in the result. More on how the time step size was chosen can be found in Chapter 4.

Lastly, to model the vehicle’s sun exposure, weather data from previous measurements taken by Volvo were used. This data includes the ambient temperature, the wind speed and direction, the humidity and the solar radiation.

Measurements like these from Phoenix in the USA were used in this experiment.

For the structural part of the simulation using Abaqus, a more stripped down model can be used. It includes only the parts that we are interested in, i.e the trim. The trim parts from the same model used in the thermal analysis were used for the structural analysis. The model of the trim can be seen in Figure 3.4 and a closer view of the meshing, which is the same as in the thermal part, can be seen in Figure 3.5.

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Figure 3.4: The scaled down model including only the trim parts

Figure 3.5: A zoomed in view of the meshed trim

The boundary conditions were set according to the physical screws and at- tachments of the Volvo V60. They are set to allow deformations along the x y

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and z axis but prevent translation and rotation for each part in the model. The boundary conditions can be seen in Figure 3.6.

Figure 3.6: The boundary conditions on the model marked in blue.

To study the deformation of a geometry from a prescribed load it is necessary to know data about the material of the studied geometry. Therefore a material has to be prescribed to each part of the model. For simplicity, the entire trim is assumed to be made out of two types of plastic, here referred to as A and B.

In reality, these two materials make up a large part of the trim. However, there are some parts that are made of for example leather or brushed steel. These parts have been assigned plastic A. The material chosen for each part can be seen in Figure 3.7.

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Figure 3.7: The materials of the trim model. Plastic A is shown in red, and plastic B is shown in beige.

Properties of the two materials used in the thermal and structural analysis are shown in Tables 3.1 and 3.2 respectively.

Table 3.1: Properties of the two materials used in the TAITherm model.

Plastic A Plastic B

Density [kg/m³] 1040 1138

Thermal conductivity [W/(m K)] 0.189 0.3 Specific heat [J/K/kg] 1356.0 1019.0

Stress-strain curves obtained from tensile testing of the two polymers are shown in Figures 3.8 and 3.9.

In the structural analysis, the prescribed load is the nodal temperatures obtained from TaiTherm in the previous part of the experiment. See more about the time step in used in the structural analysis in Section 5.2.

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Table 3.2: Properties of the two materials used in the Abaqus model.

Plastic A Plastic B

Density [t/mm³] 1.04e-9 1.138e-9

Thermal expansion [1/K] 9.5e-5 24e-6

E-modulus [MPa] Temp. [°C] E-modulus Temp. [°C] E-modulus

23 2200 23 3138

40 2175 - -

60 1750 60 2106

80 1360 80 2067

Figure 3.8: Stress-strain curves for plastic A at different temperatures

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Figure 3.9: Stress-strain curves for plastic B at different temperatures

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Chapter 4 Study of Parameters to Improve Efficiency

Before the main part of the numerical experiment was conducted, a pre-study on how to reduce the run time was made.

The final simulation consists of two parts, calculation of the thermal load and the calculation of the nodal displacement. The former is heavier in CPU-time since it involves calculating many modes of heat transfer at each time step.

To avoid excessive simulation time when computing the results, a study on how the CPU-time and the results vary when changing some discretization parameters of the setup was made. The parameters investigated in this study are

• The time step

• The coarseness of the mesh

• The patch size

In addition, the benefits of parallelizing the computations on different processes on the server was investigated.

To look closer at these different cases, calculations simulating a car in Phoenix, USA during 12 hours, from 08.00 to 20.00 the 8^thof August 2018, only varying one parameter at the time was made. During this part of the experiment, the car is stationary throughout the simulation. The results are shown in section 4.1.

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22 CHAPTER 4. STUDY OF PARAMETERS TO IMPROVE EFFICIENCY

4.1 Parameter study results

To investigate each parameter’s influence on CPU-time and results a set of simulations for each one, where only the parameter tested was altered in between runs, was made. The other parameters were kept as in Volvo’s current model.

The results and the run-time were then compared to those of the original model at four control elements. One at the roof, one on the hood, one on the dashboard and lastly, the interior air. The interior air is a fluid node used by TAITherm to model the air inside the cabin. It is needed to calculate the convection between elements. The control elements can be seen in Figure 4.1.

(a) Roof element (b) Hood element

(c) Dashboard element

Figure 4.1: Control elements for comparison of thermal results.

The parameters used in the current model at Volvo are shown in table 4.1.

Table 4.1: The parameter used in the current model

Step size [min] 10

Patch size [number of elements per patch] 10 Mesh coarseness [Number of elements in model] 362827

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CHAPTER 4. STUDY OF PARAMETERS TO IMPROVE EFFICIENCY 23

The first set of simulations was carried out to investigate the time step’s influence on the results and the CPU-time. Figure 4.2 shows the thermal results on the four control elements as a function of time for each simulation run. Figure 4.3 shows the computation time for the different cases.

Figure 4.2: The temperature of the control elements as a function of time, run with different time steps.

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Figure 4.3: The CPU-time as a function of time-step size.

One can see that the results for the two control elements on the outside of the car, the roof and the hood, get less accurate with a longer time step. The reasons for this was not investigated further, as this thesis work is focused on the trim parts. The results for the two control elements that are inside the car however, does seem to hold some accuracy even with a coarser time step. It is clear that having even a slightly larger time-step size is significantly reducing the run time. Figure 4.4 shows the computation time for the different cases on a log-log plot, where it is clear that the CPU-time is proportional to 1/∆t.

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Figure 4.4: The CPU-time as a function of time-step size on a log-log plot

The second set of simulations was carried out to investigate the patch size’s influence on the results and the CPU-time. Figure 4.5 shows the thermal results on the four control elements as a function of time for each simulation run.

Figure 4.6 shows the computation time for the different cases.

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Figure 4.5: The temperature of the control elements as a function of time, run with different patch sizes.

Figure 4.6: The CPU-time as a function of the patch size.

In this case, we can see that changing the patch size does not have a very big influence on the results, it scales them slightly. However, as shown in figure

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4.6, having larger patches than 10 elements per patch does not result in any improvements in CPU time.

The third set of simulations was carried out to investigate the mesh coarseness’

influence on the results and the CPU-time. Figure 4.7 shows the thermal results on the four control elements as a function of time for each simulation run.

Figure 4.8 shows the computation time for the different cases.

Figure 4.7: The temperature of the control elements as a function of time, run with different mesh sizes.

Here, it appears to be very little difference in the results between the different runs and the computation time is heavily influenced. In Figure 4.8 there is an unexpected drop in CPU-time at 14 processes. The reason for this was not investigated, but it is likely to be a result of improved cache locality. Despite these results, there are some specifications that need to be met when meshing such a complex geometry as this, other than just element size. Re-meshing the whole model correctly many times are not within the scope of this thesis work. Therefore this test was mainly used as an indication on how the CPU- time varies with the number of elements, rather than using this exact meshing in the final simulation. Figure 4.9 shows the computation time for the different cases on a log-log plot, where it is clear that the CPU-time is proportional to the number of elements to the power of two.

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Figure 4.8: The CPU-time as a function of the total number of elements in the model.

Figure 4.9: CPU-time as a function of the number of elements in the model, on a log-log plot

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Based on this result, a new model with fewer elements was created, which is used for increased efficiency in the main experiment.

In the fourth simulation iteration, the parameters from each individual test that showed a decrease in CPU-time while still keeping the results similar to those of the current model used by Volvo, were all applied to the model. The parameters used along with the total CPU-time are shown in table 4.2.

Table 4.2

Current model New adapted model

Step size [min] 10 40

Patch size [Number of el. per patch] 10 10

Mesh coarseness [Number of nodes] 362827 319078

Total CPU-time [h] 4.0 1.7

The thermal results as a function of time for four new control elements; one on the roof, the dashboard, the a-pillar and the interior air, are shown in figure 4.10

Figure 4.10: The temperature of the control elements as a function of time, run with a combination of the parameters from the previous experiments.

One can see that the simulation using the mesh altered quickly to reduce the

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number of elements does not provide a very good result on the a-pillar. How- ever, the custom model that hold the mesh quality that is needed to produce a better result was more successful. With this analysis we have been able to reduce the CPU-time with 2.3 h while maintaining results that gives a good indication of the temperature of the trim parts throughout the simulated time.

This corresponds to a reduction of the CPU-time of 59 %.

As for parallelization of the calculations, a set of simulations paralellized on a different number of processes, but with equal parameters, were made. The number of processes along with the total CPU-time are shown in figure 4.11.

Figure 4.11: The CPU-time as a function of the number of processes.

The thermal results are the same for each simulation since they all have the same parameters applied to them. One can see that the run time is heavily reduced as the number of processes increases.

A clearer view of the effects of the parallelization can be seen from the speed up. The speed up S is defined as the ratio of the sequential run-time T^sand the time taken by the parallel algorithm to solve the same problem on p processes T_p.

S = T_s T_p

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There was no measurement of the sequential run time, so it was estimated through extrapolation using the least square method. A plot of the speed up is shown in Figure 4.12.

Figure 4.12: The speed up as a function of the number of processes.

The curve in figure 4.12 is however not linear. This is due to how parallelization works. The total CPU-time is made up of two parts; computational time t_comp and communication time tcomm. With increasing number of processes, tcomp will decrease as each process will handle a smaller share of the entire model. On the other hand, t^comm will increase because each process require more inter-process communication during execution. Therefore, there is always an upper bound to how much one can speed up a single program using parallel computing since after a certain number of processes are used, t^comm will be the dominant part of the total CPU time.

4.2 Summarizing discussion

The results of this study provides a way to directly almost halve the run time.

This is valuable since the main part of the experiment is much more extensive and would, without any coarsening of the parameters, take several weeks to perform. However, the thermal results from these experiments are only com-

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pared on a very small sample of four reference elements of the model. To ensure a higher accuracy, a larger sample can be taken.

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Chapter 5 Fully Coupled Thermal Structural Analysis

5.1 Virtual customer behaviour

To make the process of simulating the car during one full year more realistic, a virtual customer behaviour was set up. As a car is rarely just parked in one space for a year, a user routine was imagined. For simplicity, the owner of this virtual vehicle drives to and from their workplace every weekday, and the drive takes them 30 minutes one way. On the weekends, the car is parked at their home. A workday lasts from 08.00 to 17.00. Thus, the car is driven from 07.30 to 08.00 and from 17.00 to 17.30 on weekdays.

(a) The car parked at 0 °at the workplace.

(b) The car parked at 45 °at the home.

Figure 5.1: The two heading positions considered for the virtual customer.

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34 CHAPTER 5. FULLY COUPLED THERMAL STRUCTURAL ANALYSIS

The parked car’s cardinal direction will affect which parts of the car that is exposed to direct sunlight and therefore also the deformations on the trim. As it is not in the scope of this thesis work to find the average cardinal direction of the parking spaces in Phoenix, it was simply imagined that on a compass rose, the rear of the car is 0°when parked at the workplace and 45°when parked at the home. See Figure 5.1.

This yields the weekly schedule shown in Table 5.1

Table 5.1: Weekly virtual customer schedule

Days Time State Place Heading

Mon to Fri 00.00-07.30 Parked Home 45

07.30-08.00 Driving - -

08.00-17.00 Parked Work 0

17.00-17.30 Driving - -

17.30-23.59 Parked Home 45 Sat and Sun 00.00-23.55 Parked Home 45

As the workplace is a half an hour drive from the home, it is assumed to have the same weather.

In reality, the car will cool down when driven as the apparent wind gets stronger and the AC system is used inside the cabin. To achieve a similar effect in the model, the weather data was manipulated and the sun radiation was set to zero at the relevant time steps. The wind will then cool the model down during the time steps for which there are no imposed heat from the sun. The manipulated sun radiation as a function of time for a representative Thursday to Sunday is shown in Figure 5.2.

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CHAPTER 5. FULLY COUPLED THERMAL STRUCTURAL ANALYSIS 35

Figure 5.2: The manipulated solar data

5.2 Year-long simulation

For the final experiment the aim is to compare the approach taken in this thesis work, with the approach originally used at Volvo to predict the deformations of the trim due to sun exposure. Thus, two types of simulations were conducted.

The first using the previously used method with a constant heat load over the entire trim, and the second using the method described in this work. The discretization parameters were set according to the results of the efficiency study in Chapter 4 with one exception. The time step was chosen to be 30 minutes to make the manipulation of the input simpler. The manipulated weather data discussed above was applied as well as the right heading at each time step. The setup models the car in line with the customer behavior for one full year. The weather data is from 2018.

When performing the structural analysis, the software has a built in algorithm that adapts the time step-size to achieve convergence while keeping the run time low. Due to software limitations in Abaqus, a solution with all output data obtained in the thermal analysis as the prescribed load could not be found, the input files were too large for Abaqus to handle. To reduce the amount of data, two simulations with input data every 3^rdand 4^thhour respectively, were conducted and the results are compared in Chapter 6.

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Chapter 6 Results

The numerical results were analyzed on five control nodes: One on each A- pillar, one in the middle of the dashboard (IP), one on the tailgate and lastly, one on the left waist rail. The coordinates of these nodes are shown in Table 6.1 and Figure 6.1 shows the nodes highlighted in the model.

Table 6.1: The nodal ID’s and coordinates of the five control nodes.

Part Nodal ID x-coordinate y-coordinate z-coordinate A - pillar, left 1464 1352.95 -1664.33 170.44

Tailgate 26139 4315.32 -1025.51 215.02

Waist rail, left 40772 2127.5 -1749.44 135.87 Dashboard (IP) 6058422 1459.07 -955.93 151.82 A - pillar, right 6164479 1367.95 -342.17 176.83

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CHAPTER 6. RESULTS 37

(a) Left A-pillar (b) Tailgate (c) Waist rail

(d) Dashboard (IP) (e) Right A-pillar

Figure 6.1: The five control nodes highlighted in the model.

6.1 Thermal Results

The temperature of the control nodes obtained from the year-long thermal analysis are shown as a function of time for the full year in Appendix A. Fig- ure 6.2 shows the same curve zoomed in on day 186 and 187 of the year. This corresponds to the 5^thand 6^thof July.

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38 CHAPTER 6. RESULTS

Figure 6.2: The thermal results on the five control nodes as a function of time, a closer look.

Table 6.2 shows the mean temperature on the nodes throughout the year, as well as the mean temperatures at 12.30.

Table 6.2: The mean temperature of the five control nodes.

Control node Mean Temperature [°C] Mean Temperature of peaks [°C]

A - pillar, left 32.70 69.90

Tailgate 28.90 61.70

Waist rail, left 32.20 72.80

Dashboard 35.20 82.00

A - pillar, right 34.98 80.90

The thermal results were compared to physical measurements of the air temperature inside the cabin of a Volvo V60. This measurement was taken by Volvo Cars during the summer of 2018 in Phoenix, USA. This comparison is showed in figure 6.3.

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Figure 6.3: Comparison of the modeled and measured compartment air temperature

The mean of the modeled and measured temperatures for the time period shown are 39.5 °C and 45.4°C respectively.

The error between the modeled and measured peaks, at 12.30 each day, was measured and is shown as a function of time in Figure 6.4.

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Figure 6.4: The magnitude of the error between the modeled and measured peak temperatures as a function of time.

The mean of the error between the peaks are 5.48 °C.

6.2 Coupled Thermal Structural Results

The coupling with the structural analysis was made in two versions with different input frequencies. The two input curves for an example node are shown in figure 6.5.

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Figure 6.5: The two input curves shown alongside the full thermal output.

The displacement and the strain of the control nodes as a function of time for the year long structural analysis can be seen in Appendix B.1 - B.2.

A closer look at the displacements at 12.30 of the hottest day of the year is shown in Figure 6.6.

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Figure 6.6: The displacement of the five control nodes 12.30 the hottest day of the year for the two input frequencies.

A qualitative visualization of the deformations after the full simulations for the two input frequencies and the current approach, here referred to as the oven-test, is shown in Figure 6.7.

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(a) 4h (b) 3h

(c) oven

Figure 6.7: Visualizations of the displacements on the model after the full simulations.

A closer look at these deformations on the five control nodes at the final time step can be viewed in Figure 6.8.

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Figure 6.8: The displacement of the five control nodes after the full simulations for the two input frequencies and the current approach.

The logarithmic strain after the full simulations are shown for the five control nodes in Figure 6.9.

Figure 6.9: The strain on the five control nodes after the full simulations for the two input frequencies and the current approach.

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Chapter 7 Discussion

Form studying Figures A.1 and 6.2 one can see that the results from the thermal analysis seem to be affected by the virtual customer behaviour. The temperatures of the control nodes decrease when the car is modeled as being driven.

When comparing these results to the measured temperatures in Figure 6.3 the effect of this cooling can be seen clearly. The modeled car has a lower mean temperature than in the physical test, and that is largely because of the results during the night time. After the modeled car is driven, the compartment temperature does not recover to the same level as the one in the physical car. This indicates that it is not reasonable to do these types of experiments on a car that is parked in the same position throughout the duration of the experiment, as that is not a very realistic scenario of how a car is used. Doing so produces overestimated temperatures that does not reflect the reality of most cars’ life cycles.

When studying the error between the peak temperatures and the mean value (Figure 6.4), it is important to keep in mind that the conditions for this comparison are not the same. The weather data used as input in the modeling is collected by the American Weather institute in Phoenix. The data measured by Volvo in the physical experiment is also collected in Phoenix, but not in the same locations and the headings of the two cars are not the same. The fact that the error is relatively small despite these differences is an indication that the results from the thermal analysis are a good prediction of the temperatures on the trim.

The two structural input curves shown in Figure 6.5 consist of samples from the thermal output. They slightly distort the original curve as they do not fully

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46 CHAPTER 7. DISCUSSION

capture the amplitude of the temperature peaks. This effect is called aliasing and occur when the sampling frequency is not high enough or is out of phase.

The 3 hour frequency curve captures the peaks slightly better which result in it having a marginally higher mean temperature.

The deformation and strain of the control nodes throughout the year for the two input frequencies, which can be viewed in Appendix B.1 - B.2, follow a similar pattern but are not equal. This can be expected since the input varies between the two simulations. However, as there are no measured deformations from a physical experiment of the same nature as simulated in this work, it is impossible to evaluate the accuracy of these results.

When looking closer at the deformation on the hottest day of the year seen in Figure 6.6, we can see that the difference between the two inputs are not very large. This indicates that the numerical methods used are well posed, i.e. that small differences in the input produce small differences in the output. As expected, the deformations in the simulation with input every 3 hours are larger since they model a higher mid-day temperature. The deformations at this time point of the simulations are partly reversible. When the load decreases the elastic material will revert back towards its original state. Even so, this deformation is interesting to investigate as it may cause conflict between the parts.

The parts in the car may not be free to deform due to screws and attachment points, which can cause a high strain or even cracking. The parts that are al- lowed to deform on the other hand, might put pressure on other parts causing them to age faster. This effect is not considered in the current oven approach used to simulate the deformation process.

The deformations and strains after the full simulations shown in Figures 6.7, 6.8 and 6.9 show that the oven test has the highest effect on the nodes. This is expected as the current cycle models three years of aging of the polymer parts, while the simulation cycle created for this work models one year. How- ever, the deformation and strain on the waist rail control node for the two input frequencies are approaching the same level as after the oven test, despite being modeled during a shorter time. This indicates that the waist rail could be subjected to accelerated aging as an effect of the reversible displacement that occur when the heat load is high, and that its aging is underestimated in the current approach.

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Chapter 8 Conclusions

The research question investigated in this thesis work was "Is it possible to, with weather data and knowledge of how a car is used, simulate the deformation process of the trim parts due to heat exposure from the sun?" This was answered by conducting a fully coupled thermal-structural analysis of this process during one year. The results from this numerical experiment showed the deformation of the parts both considering reversible and permanent deformation.

It can be concluded that simulating the cycle is possible. The thermal analysis produced results that when compared to measured temperatures seem to be reliable. Since the structural analysis can not in this stage be compared to any measured deformations, it is not possible to draw conclusions on the accuracy of the final results.

It can be concluded that this simulation method captures behaviours that the current method does not. This may be relevant when continuing the work to find a suitable way of preventing deformation of the trim parts due to sun exposure.

8.1 Future work

There are many aspects of this problem that have not been investigated in this report. When continuing with this work it would be interesting to find a virtual customer behaviour that closer reflect the ones of real Volvo owners.

The main challenge when setting up the simulation cycle was the amount

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48 CHAPTER 8. CONCLUSIONS

of data. It would help to investigate more efficient ways of handling larger amounts of input. Alternatively looking into how this cycle could be further simplified to reduce the amount of data needed, this would also reduce the CPU time.

It would be very interesting to validate the results from the structural analysis through comparison with measured results from physical testing.

8.2 Acknowledgements

I would like to extend my sincere thanks to my supervisors Renaud Gutkin and Björn Ratama for all the invaluable help and guidance through this process.

I would also like to thank Volvo Cars, specifically the CAE team for always being helpful and supportive when I needed it.

Lastly, thank you to my university supervisor Patrick Henning for helpful dis- cussions and insightful suggestions to this work.

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Bibliography

[1] Erica R.H. Fuchs et al. “Strategic materials selection in the automobile body: Economic opportunities for polymer composite design”. In: Com- posites Science and Technology 68.9 (2008), pp. 1989–2002. issn: 0266- 3538. doi: https : / / doi . org / 10 . 1016 / j . compscitech . 2008 . 01 . 015. url: http : / / www . sciencedirect . com / science/article/pii/S0266353808000316.

[2] P. K. Ramteke et al. “Thermal ageing predictions of polymeric insulation cables from Arrhenius plot using short-term test values”. In: 2010 2nd International Conference on Reliability, Safety and Hazard - Risk-Based Technologies and Physics-of-Failure Methods (ICRESH). 2010, pp. 325–

328.

[3] ThermoAnalytics. TAITherm. 2020. url: https://www.thermoanalytics.

com/taitherm (visited on 06/25/2020).

[4] Simulia. Abaqus. 2020. url: https://www.3ds.com/products- services/simulia/products/abaqus/ (visited on 06/25/2020).

[5] Lennart Edsberg. Introduction to computation and modeling for differ- ential equations. Second edition. Chichester, West Sussex: Wiley Black- well, 2016. isbn: 9781119018445.

[6] Daniel Zwillinger. Handbook of differential equations. 3. ed. San Diego:

Academic Press, 1998. isbn: 0127843957.

[7] Beta. Ansa. 2020. url: https://www.beta- cae.com/ansa.

htm (visited on 11/04/2020).

[8] Beta. Meta. 2020. url: https://www.beta- cae.com/meta.

htm (visited on 11/04/2020).

[9] Robert Siegel and John R. Howell. Thermal radiation heat transfer. 3.

ed. Washington: Hemisphere, 1992.

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Appendix A

Thermal results

This section contains a figure displaying the temperature of the control nodes obtained from the year-long thermal analysis are shown as a function of time for the full year thermal analysis.

Figure A.1: The thermal results on the five control nodes as a function of time.

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Appendix B

Some Results of the Structural Analysis

B.1 Displacement of the Control Nodes

This section contains figures displaying the displacements of the five control nodes as a function of time for the full year structural analyses.

Figure B.1: The displacement of the left A-pillar control node as a function of time for the two input frequencies.

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52 APPENDIX B. SOME RESULTS OF THE STRUCTURAL ANALYSIS

Figure B.2: The displacement of the right A-pillar control node as a function of time for the two input frequencies.

Figure B.3: The displacement of IP control node as a function of time for the two input frequencies.

[H]

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APPENDIX B. SOME RESULTS OF THE STRUCTURAL ANALYSIS 53

Figure B.4: The displacement of the Tailgate control node as a function of time for the two input frequencies.

Figure B.5: The displacement of the left Waist Rail control node as a function of time for the two input frequencies.

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B.2 Strain of the Control Nodes

This section contains figures displaying the logarithmic strain on the five control nodes as a function of time for the full year structural analyses.

Figure B.6: The strain on the left A-pillar control node as a function of time for the two input frequencies.

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APPENDIX B. SOME RESULTS OF THE STRUCTURAL ANALYSIS 55

Figure B.7: The strain on the right A-pillar control node as a function of time for the two input frequencies.

Figure B.8: The strain on IP control node as a function of time for the two input frequencies.

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Figure B.9: The strain on the Tailgate control node as a function of time for the two input frequencies.

[H]

Figure B.10: The strain on the left Waist Rail control node as a function of time for the two input frequencies.

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TRITA -SCI-GRU 2020:373

www.kth.se

CAE Sun Simulation - Thermo Structural Coupling

CAE Sun Simulation -

Thermo Structural Coupling

KRISTIN STAFFAS

CAE Sun Simulation -

Thermo Structural Coupling

KRISTIN STAFFAS

Sammanfattning

Contents

Chapter 1 Introduction

1.1 Background

1.2 Research Question

Chapter 2

Theoretical Approach

2.1 Background Theory

2.2 A Closer Look on FDM Discretization

Chapter 3

Methodology for the Experimen- tal Approach

3.1 General Methodology

3.2 Experimental Setup

Chapter 4

Study of Parameters to Improve Efficiency

4.1 Parameter study results

4.2 Summarizing discussion

Chapter 5

Fully Coupled Thermal Structural Analysis

5.1 Virtual customer behaviour

5.2 Year-long simulation

Chapter 6 Results

6.1 Thermal Results

6.2 Coupled Thermal Structural Results

Chapter 7 Discussion

Chapter 8

Conclusions

8.1 Future work

8.2 Acknowledgements

Bibliography

Appendix A

Thermal results

Appendix B

Some Results of the Structural Analysis

B.1 Displacement of the Control Nodes

B.2 Strain of the Control Nodes