解析事例の紹介 (Introduction of numerical examples)

解析事例の紹介 (Introduction of numerical examples) 機械系 (Department of mechanical engineering) 倉橋　貴彦 (Takahiko Kurahashi) 1

Example of numerical simulations 有限要素法　（Finite Element Method） Example1 : 浅水波の伝播問題（Propagation problems of shallow water wave) Example2 : 物体に対する伝熱の問題 (Heat transfer problems) Example3 : 非圧縮粘性流体の流れ場の計算 (Computation of fluid field for incompressible viscose flow) Example 4 : 構造内における応力分布の評価 (Evaluation of stress distribution in structures)

1.1 Surge problem Surge phenomenon : Discontinuous wave is propagated from high wave height to low wave height by Tsumami etc. Numerical example Waveform at initial time 3 Problem : Find time history of wave form variation.

Computational result by FEM 4

1.2 Shallow water flow analysis for Tokyo bay State equation (Shallow water equation) Example of forward analysis Flow analysis in Tokyo Bay Tokyo Image diagram of shallow water flow Velocity component for x direction Velocity component for y direction Water elevation Gravity acceleration Mean water depth Finite element approximation for shallow water equation 5

Computational conditions (“Finite element mesh” and “boundary condition” and “water depth distribution”) Total number of nodes : 17,160 Total number of elements : 33,433 Water depth distribution Inflow B.C. (Main four tidal components) ; Land Boundary (Slip B.C.) ; Inflow Boundary 6

Computational result of water elevation and velocity vector in Tokyo bay Animation of water elevation Animation of velocity vector 7

1.3 Application of shallow water flow analysis to actual water quality purification problem （Example：Kita-Chiba Water conduction project at lake Teganuma in Chiba prefecture） Lake Teganuma Average value in each year 8.86m3/s 2.68m3/s Environmental standard 5mg/l Water conduction points Target point (Inside of this lake) Fig. Variation of COD concentration for each year Purpose of this project Reduction of COD concentration to environmental standard 5mg/l. Investigation Relationship of COD concentration between at conduction point and at target point Problem Find appropriate COD concentration at conduction point so as to be close to environmental standard of COD concentration at target point. ⇒ Inverse problem ( optimal control theory )

Computational result based on optimal control theory ( COD concentration at conduction points ) COD concentration on Γ1U-2 COD concentration on Γ1U-1 COD2mg/l Conduction point COD18mg/l

Distribution of COD concentration at optimal control of COD concentration at conduction points (1/2) 1 day later 2 days later 3 days later 4days later

Distribution of COD concentration at optimal control of COD concentration at conduction points (2/2) 5 days later 6 days later 7 days later

Time history of COD concentration at target point Result at without control of COD concentration at conduction points Result at optimal control of COD concentration at conduction points Target concentration(5mg/l) Target point

2.1 Heat transfer problems 1m Thermal diffusivity 0.001m2/s Distribution of initial temperature 2m 1m Boundary condition Temperature on outside boundary is set to 0 degree. 2m 13 Problem : Find time history of temperature variation

Computational result by FEM 14

2.2 Heat transfer analysis considering movement of heat source point Endmill Milling machine Aluminum plate after milling

Numerical experiment　（Machining problem of aluminum plate） Material： Aluminum Rotation speed of endmill： 1,750rpm Transferred speed of plate：120mm/min (=2mm/s) （Actual experiment：thickness8mm） Initial temperature（room temperature）　21.975℃ D=32mm 100mm 100mm Tab. Thermal properties of Aluminum

Fig ; Temperature distribution

2.3 Nondestructive testing of reinforcement corrosion shape based on FEM and adjoint equation method Observation system of temperature on concrete surface by electromagnetic induction heating Coil for electromagnetic induction heating Reinforced concrete Infrared sensor Fig, System for deterioration diagnosis in reinforced concrete (Oshita et. al. (2008)) Heat image on concrete surface is obtained by this system.

Experimental process (Heat image on concrete surface by electromagnetic induction heating) Reinforcement bars Heat image Heated reinforcement bars Coil Machine for electromagnetic induction heating [1st. step] Heat reinforcement bars by electromagnetic induction. [2nd. step] After the heating, except the coil and take heat image by infrared sensor. Infrared sensor

Examples of heat image Cavity Region of reinforcement corrosion Reinforcement bar Reinforced concrete (a)Cavity (b)Reinforcement corrosion Fig. Examples of heat image on concrete surface Problem; Find shape of reinforcement corrosion using observed temperature on concrete surface.

Computational conditions (2) （Physical and computational conditions）　 Tab. Computational conditions * Computational domain （* Unite for numerical conditions are changed to “mm”.) 【Example of measurement】 Cover depth 30mm DiameterD16 ( D=16mm) Region of reinforcement corrosion Length of corrosion 100mm Width of corrosion 1.0mm Tab. Physical constants *

Computational results (shape of reinforcement corrosion) Computational condition : Initial corrosion length : 100mm The obtained shape is quite close to the target shape. Initial shape Final shape Volume：V=9,391mm3 (Width h=2mm，Length l=100mm) Width：h=1.26mm，　Volume：V=4,980mm3 （Correct volume：V=4,708mm3) （Correct solution：Width h=1mm，Length l=100mm)

3.1 Fluid analysis around body based on FEM Characteristic inflow velocity ：U Reynolds number Re=(UL)/ν ν ：kinematic viscosity coefficient L : Characteristic length U : Characteristic inflow velocity Characteristic length：L L Vortex street occurred by difference of Reynolds number 23

Fluid analysis around circular cylinder based on FEM Reynolds numner　（Re=UL/ν=112）

3.2 Finite element analysis using fictitious domain method Difference points between present method and traditional FEM ・Two type of domain (overlap domain) ・Connectivity of physical value between two domains is carried out by interpolation Background mesh (whole domain Ω) Foreground mesh (sub domain ω) Magnified figure of overlapped region Advantage Application for moving body problem → It is not necessary to do re-meshing.

Comparison between Fictitious domain FEM and conventional FEM for flow analysis around circular cylinder Reynolds number Re = 250, Δt=0.001 Conventional FEM Fictitious domain FEM Computational model

Numerical results Pressure distribution and velocity vector at each time Fictitious Domain FEM Conventional FEM T=5 T=10 T=15 T=20

Fluid analysis around circular cylinder based on FEM using Fictitious Domain Method Reynolds numner　（Re=250, Δt=0.001）

3.3 Application of FEM to two phase flow problem in micro-channel Number of nodes : 11,991 Number of elements : 23,200 Q2 Measurement area Q2 Non-Slip B.C. on wall Q1 Q1 Experimental condition

Continuum surface force (CSF)model (Model of surface tension) Reference, J.U. Brackbill, D.B.Kothe and C.Zemach Continuum method for modeling surface tension, Journal of computational physics. 100, pp.335-354, 1992. Surface tension coefficient Interface curvature Density Coefficient for density ratio interface Fluid 2 Ex: → on interface Interface curvature Fluid 1

Numerical and experimental results 【 Magnification】×540, H=100μm H b1 b1/H = 0.243 b1/H =0.296 Case3 : Pure water50 μl/min, Ethyl acetate 10 μl/min Comparison of interface line Experimental result Numerical rsult Pure water Ethyl acetate

4.1 Evaluation of stress singularity field based on FEM Stress distribution on interface σyy σ Material 2 Singular point Interface r O r Material 1 y Purpose of this study： Evaluation of stress singularity field near interface edge of bonded structure based on FEM using singular element x Fig ; Image diagram of stress distribution at vertex on interface M.L.Williams, The stress around a fault or crack in dissimilar media, bulletin of the Seismological Society of America, Vol.49, No.2, (1959), 199-204.

Interpolation function ・・・In case of elements included singular point ・・・In the other elements Linear tetrahedron element 4 4 4 4 3 3 3 3 2 2 2 2 1 1 1 1 N1 N4 N2 N3 Akin Singular Element　（in case of λ=0.5）． 4 4 4 4 3 3 3 3 2 2 2 2 Singular point Singular point 1 1 Singular point 1 1 Singular point SN1 SN4 SN2 SN3

4.2 Application of singular element for 3D model Tab. Material properties σzz=10MPa Minimum mesh length 3mm Material 1 (Mild steel) Material 2 (Aluminium) 3mm Tab. Minimum mesh size and nodes and elements Characteristic minimum mesh length Δhmin≒ Case1 : 1μm， Case2 : 1.5μm 1mm 1mm

Comparison of stress distribution for each minimum element mesh size （θ=90°，φ=45°） Stress distribution around singular point r Interface are obtained by least square method φ=45° Case2 Δhmin≒15μm Case1 Δhmin≒8μm It is found that gradient of stress distribution is close to correct order of singularityλ=0.121.

4.3 Application of mesh free method for evaluation of Intensity of stress singularity for 3D bonded structures y z Advantage of mesh free method Mesh division is not needed. (It is not necessary to satisfy the connectivity condition of domain of integration.) 1/8model z x 6.0[mm] Fe Minimum nodal distance ≒3.9[μm] Computation conditions Tensile stress z： 10[MPa] Width of model b：0.25, 0.5, 1.0, 2.0, 4.0, 6.0, 8.0[mm] 6.0[mm] Al Background cell y Material propaties o 1.0[mm] b x

Comparison of results obtained by mesh free method and Boundary element method K1：Intensity of stress singularity for distance r from singular point vertex : Order of singularity at vertex on interface ＊ vertex＝0.121 Stress distribution for radius direction Variation of K with respect to “Width” b Intensity of stress singularity K obtained by MFM is close to that obtained by BEM. But It is seen that difference of K between MFM and BEM increases with increasing width “b”.

解析事例の紹介 (Introduction of numerical examples)