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Thermal dissipation force modeling with preliminary results ... - ZARM

ARTICLE IN PRESS4 B. Rievers et al. / Acta Astronautica ( ) –Fig. 2. Disturbance acceleration **with** growing number of antenna nodes for analytical testcase 1a.sources is not an exact representation of the real emissionpattern (dominated by the radiation fins). Looking at theseissues one can formulate three criteria for an improved assessmentof the given task.(1) The resulting graph shows that a better representationof the real geometry of the modeled bodies deliversmore accurate **results**. Therefore, an exact computation ofthe thermal recoil acceleration for the Pioneer spacecraft demandsthe **modeling** of the shape of each component in avery high detail. Furthermore, interaction between the differentgeometries such as shadowing or multiple reflectionshave to be considered.(2) The assumptions made for the power distributionsof the radiation sources in the analytical test case are toosimplistic. The actual radiation pattern will differ fromthe isotropic case due to the geometry and different temperatureson the RTG surfaces. For a computation **with**higher accuracy one has to acquire the surface temperaturedistribution of the craft based on heat sources, completegeometrical shape, environmental conditions and materialproperties.(3) The **modeling** of the heat sources in the analytical testcase is only valid (**with** the simplifications mentioned above)for the start of the mission. The main power source are thePt 238 fuel rods which decay exponentially **with** a half-lifeof approx. 24 years. Furthermore, not all power generated inthe RTGs is emitted over the radiation fins but a fraction isused in the electrical compartment and then dissipated vialouver system and shunt radiator. For a precise computationof the thermal recoil at different mission times all theseissues have to be taken into account especially if one thinksof the constancy of the anomaly which seems to contradicta thermal influence (that intuitively should express itself asan exponential decay in the residuals).In order to meet the criteria formulated above new **modeling**and analysis techniques have to be developed. At **ZARM**an approach is taken that uses finite element analysis toacquire high precision surface temperature maps based onaccurate geometry models, material data and heat sourcemodels that include the dynamic behaviour of the radioactivefuel. The **results** are the exported and processed **with**numerical algorithms that use raytracing to compute the resultingsurface **force**s. These algorithms include diffuse, specularand multiple reflections and will be explained in moredetail in the following.4. Simulation and **modeling**In order to improve the **modeling** accuracy and the levelof geometrical detail that can be processed an innovativecomputing method based on FE method has been developed.The **modeling** in FE is a time consuming task. The geometryof the spacecraft for which the thermal perturbations have tobe computed has to be modeled **with** hexaedral elements inthe detail needed. The determination of necessary and **with**respect to thermal effects unnecessary geometrical featuresneeds strong experience in thermal design. The challengeis the **modeling** of different geometrical structures **with** thegiven FE brick shape. It is easy to see that brick elementscannot be used for round shapes or cut-out **with**out furtherprocessing. The necessary **modeling** step is the so-called premeshingwhere all volumes in the model are treated suchthat each single volume fulfils the requirements for mappedhexaedral meshing. Depending on the implemented detailthis can result in extensive cutting and gluing operationsthroughout the model because each new node/element constrainthas to be continued through adjacent volumes as well.After the premeshing step the mesh can be generated in thedetail needed. In general a smaller element size will lead tohigher accuracies but also the computation time rises exponentiallybecause radiation exchange is computed for eachmodel surface. With the meshing the material parametersare also assigned to the model volumes. On the mesh nodesand elements constraints such as heat generation, heat sinksor radiosity can be applied. After this the steady state solutioncan be acquired **with** an FE solver (e.g. ANSYS). ThePlease cite this article as: B. Rievers, et al., **Thermal** **dissipation** **force** **modeling** **with** **preliminary** **results** for Pioneer 10/11, ActaAstronautica (2009), doi:10.1016/j.actaastro.2009.06.009

ARTICLE IN PRESSB. Rievers et al. / Acta Astronautica ( ) – 5Fig. 3. Allocation of solid angle elements and **force**.premeshing, the meshing and the acquirement of the steadystate temperatures is shown for a test case model of the PioneerRTGs in the next section.After the surface temperatures have been computed the**results** are read into the **force** computation algorithm andthe total recoil **force** generated by thermal **dissipation** can becomputed. The total **force** is composed of three major parts:• Computation of **force** due to emission F emis .• Computation of losses due to absorption F abs .• Computation of gains due to reflection F ref .The resulting **force** can be computed by the sum of recoil**force** **with**out losses and the contributions of absorption andreflectionF tot = F emis − F abs + F ref . (16)4.1. Emission partThe **force** contribution of the emission can be computedfor each surface element **with** the equations introduced inthe previous section. Obeying Lamberts law the resultingrecoil **force** is normal to the surface of the emitting element.With the known element node positions the surface normalvectors e n (i) can be computed. Thus the total **force** resultingfrom thermal emission can be summed up over all surfaceelements in the model asF emis = ∑ i4.2. Absorption partF emis (i) =−e n (i) 2 ε A (i)A(i)σT(i) 4. (17)3 cAn exact and detailed spacecraft model also includesoverlapping geometries and surfaces that are shielded byother surfaces. Thus fractions of the radiation emitted by asurface element can be absorbed by other surface elementsand vice-versa. Therefore, for each model surface the possibleradiation exchange partner surfaces have to be determined.In the **force** algorithm this is realised by means ofraytracing, sorting procedures and angular criteria.For this a hemispheric pattern of outgoing ray-vectors isinitialised at each model surface. The ray pattern is modeledby dividing the hemisphere above the radiating surfaceinto the so-called solid angle elements dΩ **with**dΩ = sin β dβ d. (18)The vector from the radiating element centre coordinate e c (i)to the centre of a solid angle element e c,dΩ (, β) is the rayvector R(i, , β) **with**R(i, , β) = e c,dΩ(,β) − e c (i). (19)Fig. 3(left) shows the resulting allocation of solid angle elementsS ,β in a tesseral division over the hemispherical surface.To speed up the computation process, angular criteriabased on the surface normal vectors and the ray vectorsare checked first to reduce the number of surfaces and raysthat actually have to be raytraced. E.g. all surfaces which aresituated behind the element which is currently consideredactive cannot receive any radiation because of the hemisphericradiation pattern. All surfaces that pass these angular“visibility” criteria have to be processed **with** raytracing.Within this computation all surfaces in the model are consideredonce as the active element.Starting **with** the first active element all other elements inthe model that are “visible” to the active element are sortedby distance from the active element. This measure is a preparationfor the **modeling** of shadowing. Starting **with** the firstray in the pattern it is checked whether the ray intersectsthe element which is nearest to the sending element or not.If a hit is detected the ray is shut down and the next ray isinitialised. If the ray does not intersect the element the nextelement in the sorted order will be checked. Thus rays cannothit elements which are behind other surface elements.The intersection of the rays and the surface elements ischecked using barycentric coordinates [1]. First the intersectionpoint of the ray and the receiving element plane iscomputed solving the following equation:N 1 + r(N 2,1 ) + s(N 3,1 ) = e c (i) + tR(i, , β), (20)where N 1 , N 2 , N 3 are node coordinates of the receiving element.The solution to this system for r, s and t gives theintersection point P **with**P = e c (i) + tR(i, , β). (21)The receiving element surface is now divided into two triangleswhere the triangle nodes are considered as vertexnodes for the use of barycentric coordinates [1]. The nodePlease cite this article as: B. Rievers, et al., **Thermal** **dissipation** **force** **modeling** **with** **preliminary** **results** for Pioneer 10/11, ActaAstronautica (2009), doi:10.1016/j.actaastro.2009.06.009

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