THE 7th STUDENT SYMPOSIUM ON MECHANICAL AND MANUFACTURING ENGINEERING

Metal to Composite

D. Villadsen, F. Prieto Viejo, J. Mato Sanz, P. J. Jaume Camps and S. Hermansen

Department of Materials and Production, Aalborg UniversityFibigerstraede 16, DK-9220 Aalborg East, Denmark

Email: {dvilla15, jmatos18, fpriet18, pjaume18, sherma15}@student.aau.dk,Web page: http://www.mechman.mp.aau.dk/

AbstractThis work concerns redesign of a centrifugal pump. The motivation for the redesign is to change the material frommetal to composite in order to eliminate the main problem of corrosion. The material chosen to apply to the pumpis a discontinuous fiber reinforced polymer material PPS-GF40. The behavior of such a material is anisotropic innature and, as opposed to continuous fibers, difficult to design. Material modeling is done in order to characterizethis behavior, which entails fiber orientation prediction, micromechanical modeling of the stiffness and strength. Alaminate approximation is also proposed in order to improve the computational speed in the following optimization.Topology optimization is then applied in order to gain an initial design with the optimal distribution of the new material.Inadequacy in the topology optimization method used however lead to the development of further optimization models,which were applied to treat the problem with the stress in the tongue and eigenfrequencies of the pump. A final designwas proposed, which is made for injection molding, has characterized its theoretical stiffness and strength properties,eliminated corrosion and reduced the stress by a factor of 6.6.

Keywords: Centrifugal pump housing, Redesign, Discontinuous fiber reinforced composite, Material modeling,Structural optimization, Topology optimization

1. IntroductionMany pumps are designed today with metal materialssuch as steel, cast iron, bronze, etc. A problem withthese materials is that they can corrode, which severelyimpacts the mechanical performance of the material andthereby its life time. This is also the case with the pumphousing for the NB 65-200 centrifugal pump made ofcast iron, manufactured by company Grundfos.

Composites are widely used today due to their abilityto be tailored for a specific purpose. Compositescan be made out of two materials from differentmaterial groups. A common combination in engineeringapplications is plastic, reinforced with ceramic fibers.By using this combination, it is possible to eliminatethe problem with corrosion, as neither of the materialscorrode. Due to this, Grundfos proposed a project,which entails redesigning the pump housing such thatthe cast iron material can be replaced by a fiberreinforced polymer (FRP).

As Grundfos is implementing more complex manufac-turing methods, which allow for decreased manufactur-ing time, increased automation and creation of complexgeometries, it is desired to design for manufacturingwith one of these methods. The design will therefore

be developed for injection molding. Injection moldingallows for complex shapes to be manufactured by injec-tion of a molten (reinforced) plastic into a mold.

In general, composite materials can be grouped intotwo different variations; continuous fiber reinforcedcomposite materials (CFRC) and discontinuous fiberreinforced composite materials (DFRC). When usinginjection molding, the fibers are supplied to a rotatingscrew, which mixes the plastic material with the fibers.No matter if either continuous fibers or discontinuousfibers are used, the fibers will be cut into small piecesduring the injection process. This means that the designmust be made with DFRC if injection molding is used.Although the fraction of fibers and matrix is the samefor a CFRC and a DFRC, the properties will be waydifferent. CFRC are commonly used in applicationswhere superior strength or stiffness are required inknown directions, allowing for alignment of the fibersin this direction. DFRC, however, cannot achieve thesame amount of strength and stiffness, even if thematerial is completely aligned in the same direction asa CFRC. Furthermore, DFRC will have a very complexorientation state due to the injection molding process. Amodel must therefore be developed, which can predictthe orientation, stiffness and strength of the material.

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The new material utilized will exhibit a very differentbehavior compared to the previous one - most notably,it will go from being isotropic to anisotropic. There-fore, to have an idea about the new distribution ofmaterial, topology optimization will be performed onthe design. This optimization will be conducted usingthe commercial program ANSYS Workbench. However,as the topology optimization module in ANSYS haslimitations, post treatment will be done by performingdetailed design of the areas, which were not adequatefrom the topology optimization results.

1.1 Analysis of the Existing DesignThe NB 65-200 pump is a single stage, non-self-priming, end suction, volute centrifugal pump. A 3Dmodel of the pump is shown in Fig. 1.

Fig. 1 3D CAD model of the NB 65-200 pump. Left sideshows the rear of the pump with the motor flange, while theright side illustrates the side of the pump.

The flange with bolt holes shown on the left of Fig.1 is the mounting flange for the motor. The pump isconnected to a piping system through the inlet andoutlet, which are the right and top flanges, respectively,on Fig. 1. The pump housing is fixed to the ground bybolts.

In order to identify the structural critical areas of thepump, a Finite Element Analysis (FEA) is performedon the pump using the commercial program ANSYSWorkbench. The first analysis is that of a staticstructural analysis. This is performed in order to find theweakest area of the pump, i.e. the location with highestmaximum principal stress. The pump will be modeledas having a rigid fixation to the ground and loaded withan operating pressure as well as the mass of the motor.

When this pump is tested, the test is performed using apressure of 24 [bar], which adds a safety factor to theoperating pressure. As can be seen on the CAD modelillustrated in Fig. 1, there are openings, where the motor,inlet pipe and outlet pipe should be mounted. Applyingonly the pressure to the inside of the pump housingis not sufficient for representing the load case, sincethe pressure also acts on the openings, while the pumpis in operation. Therefore, the pressure is balanced byapplying compensating forces on each opening. Thesebalancing forces are found by multiplying the operatingpressure by the opening area.

To best capture the complex stress state, quadraticelements are used. At first, a coarse mesh is used inorder to get an overview of the stress state, and findthe most critical areas, whereafter subsequent modelrefinement can be done. The results from the presentedmodel are shown in Fig. 2

Fig. 2 Illustration of the stress state in the pump. It can beobserved, that by far the most critical point is the volutetongue. The stresses have been normalized with the maximumvalue.

Having identified the most critical point to be in the edgeshown in Fig. 2, which is termed the volute tongue,mesh refinement can be done. As only the studentversion of ANSYS was available with a limit of 256,000nodes, a submodel of the tongue is performed. By doingthis, the rest of the pump can be meshed with a relativelyfine global mesh, while still capturing the high stressfrom the stress concentration in the volute tongue usinga very fine mesh. The results are shown in Fig. 3.

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Fig. 3 The stresses in the submodel. The values have beennormalized with respect to the maximum stress from Fig. 2.

Due to confidentiality agreements, the results cannot bestated, however, it is recognized from this submodel thatthe stress is many times the material strength, which isa problem, that has to be treated in the redesign.

Another important factor for the failure of the pump isthe eigenfrequencies of the pump. The motor mountedto the pump housing runs at a constant frequencyof either 50 or 60 [Hz] depending on the motor.The pressure in the pump is created by a rotatingimpeller inside the pump housing. The impeller ofthis pump has six blades, and the impeller increasesthe operating frequency due to the blade’s trailingedge passing the volute tongue. This is termed theblade-passing frequency. The blade passing frequencytherefore becomes 300 or 360 [Hz] depending on themotor.

The eigenfrequency analysis is also performed usingANSYS Workbench. Besides the mass of the motor,the mass of the impeller and the mass of the waterinside the pump are also included in the analysis. Themass of the water is added by increasing the density ofthe material, and thereby assuming that it acts evenlythroughout the pump. The mass of the motor is takenas the motor with the highest mass in the Grundfoscatalogue. The analysis is conducted using the samemesh as in the static structural analysis. The resultingfirst three eigenfrequencies are shown in Tab. I.

Mode 1 2 3Frequency [Hz] 106.16 480 623.44

Type Bending x-axis Bending z-axis Torsional

Tab. I The three first eigenfrequencies of the pump withdescription of the eigenmode. These three eigenfrequenciesare the most critical as they are closest to the operatingfrequency range. For coordinate system, see Fig. 2.

Some inaccuracy in the model is expected due to thefact that the piping adds additional stiffness to thesystem, added mass from the surrounding water, i.e. ifthe pump is submerged in water, and also if the analysisis conducted with a smaller motor, which results in asmaller mass, is not taken into account.

2. Material ModelingThe anisotropy the material exhibits is a consequenceof the presence of the fibers in the matrix. Thefiber dispersion is characterized by the particles beingaimlessly dispersed due to the injection molding. Onthe other hand, the orientation tends to follow a patternwhich is heavily influenced by the flow conditionsand the geometry of the part. Thus, a model, whichcan predict the behavior of the anisotropic material, isstudied.

2.1 Fiber Orientation PredictionIn the continuum, fiber orientation is described by aprobability distribution function ψ(p), where p is thedirection vector of the fiber. For a planar flow state, pis described in terms of the orientation angle φ

p1 = cosφp2 = sinφ

(1)

As a consequence of having a concentrated suspension(40% of fiber volume fraction), the interaction betweenfibers must be accounted for with the Folgar-Tuckermodel [1], where the rotational speed φ̇ is calculated.This is used in the continuity equation (2) to calculatethe rate of change of the distribution function througha control volume, as:

∂ψ

∂t= − ∂

∂φ(ψφ̇) (2)

The probability function can be presented with anorientation tensor aij , defined as the average of alldirections of p and weighted by ψ [1].

aij =

∫pipjψ(p)dp (3)

Thus, (2) can be expressed in terms of aij , and therate of change of aij can be obtained. Therefore, thecontinuity equation is to be solved numerically for the

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injection process. The software used for this purposeis Moldex3D, a finite element program in which theinjection is simulated, and from where aij is calculatedelement-wise at each time step.

2.2 MicromechanicsTaking offset in the material properties correspondingto all the fibers aligned, the resultant properties of anyfiber orientation and fiber probability distribution can becalculated. In this project, the procedure explained in [2]is followed. The principle is to average the propertiesof the unidirectional material along all the possibleorientations weighted by the probability of orientation.The averaged stiffness tensor of the material is definedas

〈Cijkl〉 =∫ ∫

ψ(p)xipxjqxkrxlsCpqrsdp (4)

where xij is the coordinate transformation tensor ψ(p)is the orientation function, 〈−〉 means orientationaveraged magnitude and Cpqrs is the stiffness tensor.

As a result of the procedure followed to characterize thefiber orientation and the stiffness, the resultant materialwill be an orthotropic material for each point withvarying properties along the continuum. This methodis used for every 3D stiffness calculation.

2.3 Laminate ApproximationAlthough the previous method represents a good defi-nition of the material properties, it is a costly method,which requires a flow analysis and a micromechanicalpostprocessing for each change of the design. Anyiterative design process demands a flexible model onthe initial stages which allows a quicker development.

After studying the patterns of the components manufac-tured through injection molding, it turns out that thereis a common distribution of fiber orientation though thethickness. Three major regions can be clearly separated:two regions close to the walls of flow-oriented fibers anda core in between with a more random orientation, asseen in Fig. 4.

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 1 2 3 4 5 6 7 8

Ori

enta

tio

n t

enso

r co

mp

on

ents

Thickness (mm)

Orientation distribution

X-Direction

Y-Direction

Z-Direction

Fig. 4 Fiber orientation distribution of a component simu-lated in Moldex3D.

Being aware of this, it was decided to approximate theinjected components to a three-layer laminate. The corehas the properties of completely random distributionof fibers in the plane perpendicular to the thicknessdirection. The face sheets are symmetric and have theproperties of flow-oriented fibers. Knowing the materialproperties and the average direction of the flow in aregion, the only variable left is the relation betweenthe thickness of the face sheets and the core. For thiscalculation, a series of plates with different thicknessesare injected in Moldex3D and the variation of theorientation probability is extracted. Then, the platelaminate stiffness matrix is calculated through the Firstorder Shear Deformation Theory (FSDT). The laminatestiffness matrix of the equivalent three-layered platesis defined as a function of the thickness of the facesheets. Through the least-squares method, this thicknessis calculated by the optimization problem in equation (5)for each total thickness.

Minimize: f(h) =∑8i=1

∑8j=1

(Cmoldij − Ceq

ij (h))2

Subject to: −h ≤ 0h ≤ t

(5)Where Cmold

ij is the plate stiffness matrix of the dataobtained from Moldex3D, Ceq

ij is the plate stiffnessmatrix of the equivalent laminate, h is the thickness ofthe face sheets and t is the total thickness of the plate.

Finally, the laminate approximation is validated inANSYS by comparing the results with the ones with3D fiber orientation data imported from Moldex3D.The setups used are cantilever beam under tension andthree-point bending. The results compared are maximumdisplacement and maximum principal stress in staticstructural analysis and three lower eigenfrequencies inan eigenfrequency analysis. The results are compared interms of relative error in Tab. II and Tab. III.

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Plates Max. disp. 1st mode 2nd mode 3rd mode2x80x200 9% 3% 2% 3%4x80x200 1% 0% 4% 1%6x120x240 1% 1% 4% 1%7x160x350 3% 0% 3% 1%8x160x320 2% 1% 3% 1%

Tab. II Relatives errors of the considered variables betweenthe plates for the cantilever beam model. Max. disp. meansmaximum displacement.

Plates Stress Max. disp. 1st mode 2nd mode 3rd mode2x80x200 5% 6% 0% 2% 1%4x80x200 2% 0% 3% 2% 2%6x120x240 0% 1% 2% 0% 2%7x160x350 0% 1% 2% 1% 3%8x160x320 1% 2% 1% 3% 2%

Tab. III Relative errors of the considered variables betweenthe plates for the three-point bending model. Max. disp. meansmaximum displacement.

2.4 Anisotropic Failure CriteriaThe orientation of the fibers and the stiffness of theanisotropic material have now been characterized in ev-ery finite element. It is therefore desired to characterizethe strength in order to evaluate failure of the pump.To do this, the longitudinal and transverse strength ofthe unidirectional DFRC must be determined. Followingthe approach of [3], the longitudinal strength σcL canbe determined as

σcL = Vfσf

(1− Lc

2L

)+ σ′m (1− Vf ) , for Lc ≤ L

(6)

σcL = Vfσf

(1

2Lc

)+ σ′m (1− Vf ) , for Lc > L (7)

Here, σf is the fiber strength, σ′m is the matrix stressat fiber failure, Vf is the fiber volume fraction, L is thelength and Lc is the critical length, which is found as:

Lc = σfd

2τ(8)

where d is the fiber diameter, and τ is the interfacialshear strength. By assuming a strong interfacial bondbetween fiber and matrix, the transverse strength σcT isequal to the strength of the matrix, σm i.e.

σcT = σm (9)

In the case of isotropy of the matrix and stronginterfacial bonds, the shear strength can be determinedby the von Mises criterion

τ =σm√3

(10)

A popular and effective choice of failure criterion forCFRC materials is the Tsai-Wu failure criterion [4]. Thiscriterion can also be formulated for DFRC by includingthe fiber orientation tensor. The general Tsai-Wu for a3D stress state is formulated in contracted notation as:

Fiσi + Fijσiσj = 1 (11)

Indicating that failure occurs if the equation is equal tounity. The stresses σi and σj are the applied stressesin contracted notation. The Tsai-Wu strength tensors Fiand Fij are given by the strength of the material. Byconsidering a composite with fibers perfectly alignedalong one axis, the material will behave transverselyisotropic. The definition of the strength tensors of thetransversely isotropic DFRC, Fi = {0} and Fij willhave the following nonzero entries

F11 =1

σ2cL

, F12 = F21 = −1

2σ2cL

F22 =1

σcT 2

, F23 = F32 = −1

2

(2

σ2cT

−1

σ2cL

)

F33 =1

σcT 2

, F44 =

(4

σ2cT

−1

σ2cL

)(12)

F55 = F66 =1

τ2,

To take the fiber orientation into account, the fiberorientation tensor must be included. The procedure isto average the Tsai-Wu strength tensor, by the use ofthe fiber orientation tensor [1]. In expanded notation,this results in

〈F 〉ijkl = B1aijkl +B2(aijδkl + aklδij)

+B3(aikδjl + ailδjk + ajlδik + ajkδil) (13)

+B4(δijδkl) +B5(δikδjl + δilδjk)

Here, aijkl is the fourth order orientation tensor, δij isthe Kronecker delta and B1,...,5 are constants directlyrelated to the tensor being averaged. After calculationof 〈F 〉ijkl, the strength tensor can be contracted andis then ready for use in (11). However, to make (11)scalable, it must be linearized, since it is nonlinearlydependent on the stresses in its traditional form. Thisis done by introducing the variable γ and by rewriting(11) into a polynomial expression as follows:

γ2Fijσiσj + γFiσi − 1 = 0 (14)

From which γ can be solved. The failure index FI isthen defined, using γ as:

FI =1

γ< 1 (15)

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3. Structural OptimizationIn the following, the structural optimization methodsused, and the results obtained, will be described.

3.1 Topology OptimizationA topology optimization is performed on the pump,having applied the new material. As the pump willbe manufactured using injection molding, which isexcellent for manufacturing highly complex shapes, itis of much interest to find the optimum placement ofmaterial, no matter how complex it is. In order toperform the topology optimization, the CAD modelof the pump housing is enclosed by material, as thetopology optimization method is not able to placematerial, but to find the optimum distribution in anexisting domain. This is illustrated in Fig. 5.

Fig. 5 Rear side of the pump housing wrapped in material.The spheres illustrated by A and B are the masses of theimpeller and motor respectively.

As the topology optimization is conducted using AN-SYS Workbench some limitations are present. Thetopology module in ANSYS Workbench only hasthe Solid Isotropic Material with Penalization (SIMP)method available, which is not applicable to anisotropicmaterials. It is therefore desired to perform the topol-ogy optimization as an approximation to the optimalgeometry, and then post-process the results to achieve afeasible design.

The topology optimization is done in FEM form. TheSIMP scheme used, presented in [5][6], is a sim-ple density-based topology optimization method. Themethod entails solving the 0-1 problem of topologyoptimization by introducing a continuous density vari-able ρ(x), which simplifies the problem by allowing

intermediate values between 0 and 1 as:

Ee(ρe) = ρeE∗e (16)

Here, Ee is the stiffness of a finite element and E∗eis the stiffness of the material. The continuous densityvariable, ρe, is here defined for each element. As onlythe values 0 and 1 are permissible, the intermediatevalues are penalized by inclusion of a penalty factorp, making 0 and 1 the most efficient values for thealgorithm to choose.

Ee = ρpeE∗e (17)

The value of p is usually set to 3 [7].

Constraints are then formulated with respect to stressand eigenfrequency. The stress constraint is formulatedusing the method proposed by [7]. This is done inorder to avoid the singular optima problem occurringwhen using stress constraints, as elements that have zerodensity may still have finite stress values. The solutionproposed is to relax the problem by introducing a newvariable q, which replaces p in the stress constraintexpression, where q < p. By applying this, the stressconstraint is formulated in finite element form as

σVM,e = ρqeσY,e (18)

The stress σY,e is the yield stress of the material in anelement and σVM,e is the von Mises stress computed inthe center of the element.

In this formulation, the problem becomes a very largescale problem due to the stresses being formulated aslocal constraints, creating a constraint for each elementin the model. In order to make the algorithm moreefficient, the constraints are therefore aggregated intoa global constraint. This is done by e.g. the P-normmethod

gVM =

(ne∑e=1

(σe,VM )P

)1/P

(19)

Here, P is a penalization parameter. The larger thevalue of P , the more accurate the constraint willbe. However, large values of P are computationallyinefficient. Therefore, the idea of adaptive constraintscaling outlined in [8] where a scaling parameter c ismultiplied by the global constraint. The stress constraintcan thereby be formulated as:

max(σVM ) = cgVM

ci = αimax(σVM )i−1

gi−1VM

+ (1− αi)ci−1 (20)

α = [0, 1]

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Here, i is the current iteration number and α is adamping constant, which value is chosen in order toprevent c from oscillating.

The optimization problem can then be formulated. Interms of minimizing volume subject to stress constraintsand eigenfrequency constraints, the formulation is

minimizeρ(x)

V =1

V ∗

ne∑e=1

veρe

subject tocigVM (x)

σY− 1 ≤ 0

ωd1 ≤ ωi ≤ ωd2Ee = ρpeE

∗e

0 < ρmin ≤ ρe ≤ 1

(21)

Here, V is the volume, V ∗ is the reference volume, ve isthe volume of an element, ωi is an eigenfrequency, ωd1and ωd2 are limits on the eigenfrequency. A lower boundon the density variable ρmin is included, which ensuresthat the stiffness matrix does not become singular.

ResultsMany optimization setups are investigated to deter-mine, which will provide the best results. The initialapproach is to run volume minimization subject tostress constraints only, then subject to eigenfrequencyconstraints only and then make an interpretation of theresultant geometry. The results of the stress constrainedtopology optimization are shown in Fig. 6. The next

Fig. 6 Results from the stress constrained topology optimiza-tion.

optimization run, which was done with eigenfrequencyconstraints, was unable to converge. Therefore, a multi-objective formulation is used instead, where the volumeis minimized and the third eigenfrequency is maximized.The objective is subject to a stress constraint and twoeigenfrequency constraints of limiting the frequencies

to be at least 30% below the operating frequency. Theresults are shown in Fig. 7.

Fig. 7 Results from the multi-objective optimization.

These results provide a starting point for further design.

3.2 Thickness OptimizationThe resultant geometries from the topology optimizationanalyses provide an idea on how the mass shouldbe distributed in the structure to achieve the requiredgoals. A postprocessing of the results is necessary toobtain a geometry subject to manufacturing constraintsand suitable for further analyses. One of the mainmanufacturing constraints when designing specimens,which are to be injection molded, is avoiding changesin thickness through the same surface. This is toavoid shrinkage and warpage. Therefore, the resultinginterpreted geometries should have constant surfacethickness in all the reinforcements.

This second step of the optimization is supposed togive a better representation of a real specimen. Forthis purpose, a FEM model is done using the moduleANSYS Composites Prep/Post (ACP) for defining theorthotropic properties and fiber orientation in eachregion. The input of this module is a geometrycomposed of surface bodies. The fiber orientation isobtained from a qualitative analysis of the flow. Then,a solid model is created, as the one shown in Fig. 8, byextruding the input surfaces.

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Fig. 8 Example of a geometry generated in ACP.

After creating the full model, the thickness of eachsurface is defined as a parameter in ANSYS Workbenchin order to perform different response studies and useoptimization algorithms in the design process.

The module ACP is intended to be used for laminatedcomposites. Thereby, it makes it very complicated torepresent complex 3D geometries. The critical regionin terms of stress is the tongue. Therefore, this modelwill only be used to optimize the part in terms ofeigenfrequencies.

4. Detailed DesignIn this section, the detailed redesign stage, which in-volves design for strength and design for eigenfrequen-cies alongside with their corresponding results, will bepresented.

4.1 Design for StrengthThe detailed design will have as main objective todownsize the stress at the volute tongue, which isapproximately seven times the tensile strength of thenew material, until a value lower than this one, whichmight make the design feasible for structural purposes.This redesign stage will comprise two phases; thefirst one consists of an optimization of the existingvolute tongue geometry, and if the target stress isstill not reached, a second phase involving majorgeometrical changes following the engineering intuitionwill be done. The first optimization is carried outchanging the basic geometry of the tongue, whichare the three roundings that conform this part, seeFig.9. Different bounds are fixed for each roundingand a Surface Response Optimization (using a Multi-Objective Genetic Algorithm) is executed with the aimof minimizing stress until an optimum geometry isobtained. The values for the optimized tongue are: 4.8[mm] (side rounding) and 0.2 [mm] (for both upper and

lower roundings). The value of the stress in the tongueis lowered approximately a 50 % of the initial value.

Fig. 9 Roundings of the volute tongue chosen for theoptimization.

This reduction of stress is not enough to comply withthe requirements of the redesign, which suggests thatthe volute tongue should be treated alongside with thewhole pump in the design. Subsequently, an intuitiveredesign will be accomplished to solve the issue. Beforemodifying the pump housing any further, a closer lookat the deformed state of the pump is taken to identifythe cause of the stress accumulation in the tongue. Thisfurther investigation on the deformation made clear thatthe walls of the pump expand outwards, see Fig. 10.As a result of this phenomenon, the tongue, which actslike a joint between both walls, carries all the stressescaused by this deformation. Thereby the stress in thetongue is not only dependent on the geometry of thispart, but of the entire pump.

Fig. 10 Deformed shape of the pump (up), undeformed shapeof the pump (down).

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This observation clearly indicates that a reinforcementalong the shoulder of the pump (which undergoes thehighest deformation, see Fig. 10) has to be implementedfor preventing expansion of the pump, see Fig. 11. Aniterative process, adding ribs and changing the shape andconfiguration of the reinforcements, takes place until afeasible model that can withstand the load case statedis reached.

Fig. 11 Pump housing reinforced.

4.2 Design for EigenfrequenciesAfter creating a geometry from the results of the topol-ogy optimization, a thickness optimization is run in or-der to fulfill the modal requirements. This optimizationfailed, which shows that the initial interpreted geometryis not feasible, the second and the third eigenfrequenciesare not separated enough to maintain both on the safeside, the second below the blade-passing frequency andthe third one above. Consequently, it is decided to startan iterative design process in which other results fromthe topology optimization and engineering intuition areused to perform the necessary changes in the geometry.

Two new variables are defined to track the designprocess:

• Objective: The difference between the second andthird eigenfrequencies measured as a percentage ofthe required difference.

• Improvement: The change of the objective, definedabove, between following subiterations.

An iteration is a step where the resultant design issatisfactory, and a subiteration is a step when the outputis unsatisfactory. The result history of the design processis shown in Fig. 12. The design was guided by the studyof the mode shapes, trying to stiffen the third modewhile leaving the second mode unaltered.

-10%

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 1 2 3 4 5 6 7 8

Req

uir

ed s

epar

atio

n (

%)

Design step

Design history

Objective Improvement

Fig. 12 History of the variables used to track the designprocess.

It can be seen that the initial changes improved thedesign rapidly but only caused minor changes duringthe last steps. When, after several steps, the designresult remains the same, the iterative design process isconsidered finished. Then, the resultant model is subjectto a thickness optimization to separate the second andthe third eigenfrequencies as much as possible. The bestcandidate found is:

1st mode 2nd mode 3rd modeFrequency [Hz] 188.4 209.4 413.6

Tab. IV Results from the thickness optimization.

4.3 Final DesignAfter optimization has been done for the stress andeigenfrequency separately, the results from these arecombined to a final design. As a result of thecombination of the optimized results, the final geometryis modeled, such that it should fulfill the requirementsthat both designs demand. The new design is illustratedin Fig. 13.

Fig. 13 Model of the final design.

However, after analyzing the injected pump, it is noticed

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that both requirements cannot be achieved under thesame model. Thus, it is chosen to comply with thestrength, since the reduction obtained, a reduction ofa factor of 6.6, is substantially better than the resultsobtained for the eigenfrequencies. The stress has beenevaluated in the Tsai-Wu failure criterion established.The failure in the tongue is illustrated in Fig. 14.

Fig. 14 The failure index in the tongue.

It can be noted, that the stress in the tongue still fails.However, by reducing the safety factor, i.e. such thatthe operating pressure is used, or by investigating, if theorientation of the fibers can be improved in the tongue,this problem could be resolved.

The eigenfrequencies are shown in Tab V.

2nd mode 3rd mode 4th modeFrequency [Hz] 174 300 423

Tab. V Eigenfrequencies of interest of the final design.

Small changes to the geometry to push the 3rdeigenfrequency off the operating frequency is, asobserved, required. Additionally, harmonic analysiscould help establish, how close the eigenfrequency canbe allowed to be to the operating frequency values.

5. ConclusionThis project concerns changing the material of a pumphousing from metal to composite in order to eliminatethe problem of corrosion. Material modeling of the PPS-GF40 material was done. Here, the stiffness and strengthof the anisotropic material were characterized by theprediction of the fiber orientation state after injectionmolding simulations of the pump housing.

Topology optimization was performed on the design,using the new material, however, the topology optimiza-tion method was inadequate, as it was based on isotropicmaterials. Postinterpretation of the generated design wastherefore done to comply with the requirements.

From the interpretation of the topology optimizationgeometry, further design and parametric optimizationfor the stress state in the pump housing’s critical point,the tongue, and for the pumps eigenfrequencies weredone. The findings were put together in a final designproposal. In this design, the stress was reduced by afactor of 6.6 in the tongue. The final proposed design didnot comply for eigenfrequencies, however, as one modewas critical when compared to the operating frequency.It was found, that a compromise had to be made betweento move this eigenfrequency away from the operatingfrequency, as this would increase the stress.

AcknowledgementThe authors of this work gratefully acknowledgeGrundfos for sponsoring the 7th MechMan symposium.

References[1] S. G. Advani and C. L. Tucker, “The use of

tensors to describe and predict fiber orientation inshort fiber composites,” Journal of RheologyVolume 31, Issue 8, 1987.

[2] J. Schjødt-Thomsen and R. Pyrz, ContinuumMicromechanical Modelling of Composites andCellular Materials, Part 1: Elasticity. AalborgUniversitetsforlag, 2003.

[3] F. V. Hattum and C. Bernardo, “A model topredict the strength of short fiber composites,”Polymer Composites Volume 20, Issue 4, 1999.

[4] S. W. Tsai and E. M. Wu, “A general theory ofstrength for anisotropic materials,” CompositeMaterials Volume 5, 1971.

[5] M. P. Bendsøe, “Optimal shape design as amaterial distribution problem,” StructualOptimization Volume 1, vol. 1, pp. 193–202, 1989.

[6] M. Bendsøe and O. Sigmund, TopologyOptimization - Theory, Methods and Applications.No. ISBN: 3-540-42992-1, Springer, 2003.

[7] M. Bruggi, “On an alternative approach to stressconstraints relaxation in topology optimization,”Struc Multidisc Optim Volume 36, 2008.

[8] C. Le, J. Norato, T. Bruns, C. Ha, andD. Tortorelli, “Stress-based topology optimizationfor continua,” Struc Multidisc Optim Volume 41,2010.

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