Nonlinear Iteration na Time Integration¶

Uchanganuzi wa FrontISTR huendeleza suluhisho kwa kuchanganya nonlinear iterations na time integration katika kila substep ndani ya increment control ya kila step. Ukurasa huu unashughulikia miundo ya iteration iliyo kati ya step control ya nje na linear solver ya ndani: Newton-Raphson iteration, contact iteration, time integration ya dynamic analysis, na mfumo wa time advancement pamoja na nonlinear iteration katika transient heat conduction.

Muhtasari wa kipengele¶

Nonlinear solution na time integration ndani ya step zina muundo wa nested loops nyingi kutoka nje kwenda ndani. Kila layer ina jukumu lifuatalo.

Ni layers zipi zinaonekana hutegemea analysis type.

Kwa jinsi ya kuchagua contact type na algorithm zenyewe, tazama Contact na embedding. Ukurasa huu unaeleza jinsi algorithm iliyochaguliwa inavyoundwa kama iteration loops.

Jinsi ya kuchagua solution method¶

Muundo wa iteration huamuliwa kiotomatiki kutoka uchaguzi wa analysis type (Aina za uchanganuzi), uwepo wa nonlinearity (geometric nonlinearity, material nonlinearity, contact), na contact algorithm (SLAGRANGE au ALAGRANGE). Mtumiaji hurekebisha moja kwa moja upper limits za iteration count na convergence criteria.

Mwongozo wa marekebisho ni kama ifuatavyo.

• Kwa uchanganuzi wa kawaida, upper limit ya iteration count kwa kawaida inafanya kazi vizuri kwa default. Ikiwa tatizo lenye convergence ngumu linaishia “limefikia maximum iteration bila convergence”, kwanza tumia cutback pamoja (Udhibiti wa step); ongeza upper limit tu ikiwa hilo bado haliboreshei hali.
• Rekebisha convergence criteria kwa kusawazisha uthabiti wa uchanganuzi na muda wa hesabu. Kadiri criteria zinavyokuwa kali, equilibrium accuracy huongezeka lakini iteration count pia huongezeka. Kwa contact analysis au uchanganuzi wenye Lagrange multipliers, kutumia displacement-increment criterion au Lagrange criterion pamoja na residual criterion kunaweza kuboresha uthabiti.
• Kwa time-integration parameters za dynamic analysis (coefficients za Newmark-β), anza na mchanganyiko wa kawaida wa thamani na uzirekebishe ikiwa unataka kubadilisha numerical damping characteristics za vibration response.
• Kwa transient heat conduction, chagua Crank-Nicolson kwa matatizo yenye mabadiliko laini ya temperature; chagua Backward Euler unapothamini uthabiti kwa mabadiliko makali au uchanganuzi wa muda mrefu.

Newton-Raphson iteration katika static analysis¶

Katika nonlinear static analysis, equilibrium equations zenye geometric nonlinearity au material nonlinearity hutatuliwa kwa Newton-Raphson method kwa kurudia. Kila iteration huendelea kwa hatua zifuatazo.

1. Tathmini residual vector \(\boldsymbol{R}\) kwa suluhisho la sasa.
2. Unganisha tangent stiffness matrix \(\boldsymbol{K}_T\).
3. Tatua linear system \(\boldsymbol{K}_T \Delta \boldsymbol{u} = -\boldsymbol{R}\) ili kupata displacement correction \(\Delta \boldsymbol{u}\).
4. Sasisha suluhisho kwa \(\boldsymbol{u} \leftarrow \boldsymbol{u} + \Delta \boldsymbol{u}\).
5. Linganisha na convergence criterion; ikiwa imeconverge, maliza iteration. Ikiwa haijaconverge, rudi hatua ya 1.

Utatuzi wa linear system (hatua ya 3) unafanywa na linear solver. Kwa uchaguzi wa solver na preconditioner, tazama Solver na preconditioning.

Iteration control hufanywa kwa step-control parameters za !STEP. Thamani kuu za udhibiti ni zifuatazo.

Maximum Newton iteration count inayotumiwa katika maamuzi ya automatic increment/cutback kwenye Udhibiti wa step inamaanisha idadi ya Newton iterations zilizoelezwa hapa. Automatic increment huongeza time increment baada ya substep iliyohitaji iterations chache na huipunguza baada ya substep iliyohitaji iterations nyingi.

Contact iteration katika static analysis¶

Katika contact analysis, contact state hubadilika wakati wa iteration kutokana na contact, separation, na sliding za contact surfaces; kwa hiyo Newton iteration huunganishwa na update ya contact state. Muundo hutegemea contact solution algorithm.

Kwa algorithm zote mbili, contact-state search loop huwekwa nje kabisa. Katika kila mzunguko wa loop hii, baada ya inner Newton iteration kuconverge, contact state (contact/separation) hutathminiwa upya; ikiwa hali imebadilika, muundo wa stiffness matrix husasishwa na inner iteration huendeshwa tena. Loop huisha wakati contact state haijabadilika kutoka mzunguko uliopita na contact convergence criteria (contact-force na Lagrange-multiplier criteria) zimetimizwa. Upper limit ya idadi ya mizunguko hubainishwa na MAXCONTITER ya !STEP.

Kwa SLAGRANGE contact, contact degrees of freedom huingizwa katika linear system kwa standard Lagrange multiplier method, hivyo kupata muundo wa layers mbili ambapo Newton iteration iko moja kwa moja ndani ya contact-state search loop. Hakuna augmented Lagrange iteration.

Kwa ALAGRANGE contact, augmented Lagrange iteration huongezwa ndani ya contact-state search loop, na Newton iteration hufanywa ndani yake, hivyo kupata muundo wa layers tatu. Katika kila augmented Lagrange iteration, contact constraint hutathminiwa upya kwa mchanganyiko wa penalty term na Lagrange multiplier, na constraint accuracy huboreshwa hatua kwa hatua. Upper limit ya augmented Lagrange iteration count hubainishwa na AUGITER ya !CONTACT_ALGO.

Thamani za udhibiti wa contact iteration ni kama ifuatavyo.

Contact iteration count inayorejelewa katika automatic-increment decision ya Udhibiti wa step inamaanisha idadi ya iterations zinazoelezwa hapa, ikijumuisha contact-state updates. Uchanganuzi ambao contact na separation hutokea mara kwa mara huwa na contact iteration count kubwa, na hii huathiri pia automatic increment/cutback decision. Kwa contact types, pair definitions, na algorithm selection, tazama Contact na embedding. Kwa jinsi ya kutatua linear system yenye contact degrees of freedom, tazama Solver na preconditioning.

Time integration katika dynamic analysis (implicit method)¶

Implicit method ya dynamic analysis hutumia Newmark-β method kuhusianisha displacement, velocity, na acceleration za wakati unaofuata katika kila time step na kuzichanganya na equation of motion ili kupata hali ya wakati \(t + \Delta t\). Stability restriction kwa time step \(\Delta t\) ni nyepesi, hivyo structural response inayotawaliwa na low-frequency components inaweza kutatuliwa kwa ufanisi.

Time-integration parameters ni hizi mbili.

Mchanganyiko \(\beta = 1/4\) na \(\gamma = 1/2\) huitwa average acceleration method na ni chaguo la kawaida lenye unconditional stability bila numerical damping. Ili kuingiza numerical damping, chagua \(\gamma\) kubwa kuliko \(1/2\). Kwa maelezo ya kihisabati ya mchanganyiko wa parameters, stability, na error characteristics, tazama theory manual.

Ikiwa nonlinearity (geometric nonlinearity, material nonlinearity, au contact) ipo, Newton iteration hufanywa ndani ya kila time step; linear system hutatuliwa ndani ya iteration na hali husasishwa. Convergence criteria hutumia CONVERG na thamani nyingine zinazoshirikiwa na static analysis. Katika dynamic analysis yenye contact, muundo uleule wa iteration kama Contact iteration katika static analysis huingizwa katika kila time step.

Time step ya dynamic analysis kimsingi hudhibitiwa na time conditions za !STEP. Kwa fixed increments, !STEP hubainisha DTIME kama time step na ETIME kama step time span. n_step na t_delta za !DYNAMIC huchukuliwa kama defaults ikiwa !STEP imeachwa na kwa backward compatibility. Kwa nonlinear implicit method, INC_TYPE=AUTO ya !STEP huwezesha automatic increments/cutback, na time step huongezwa au kupunguzwa kulingana na hali ya Newton na contact iterations. Iwe fixed au automatic increments zinatumika, weka time step kwa kuzingatia convergence na required accuracy.

Time integration katika dynamic analysis (explicit method)¶

Explicit method ya dynamic analysis hutumia central difference method kuhesabu hali ya wakati unaofuata moja kwa moja kutoka displacement, velocity, na acceleration za wakati uliotangulia pekee. Kwa kuwa simultaneous equations hazitatuliwi, computation cost kwa step ni ndogo. Hata ikiwa nonlinearity ipo, kila time step ni single-stage update bila Newton iteration.

Time step ina upper limit inayowekwa na stability condition inayotegemea minimum natural period ya mfumo (CFL condition). Time step inayozidi limit hii husababisha numerical solution kudiverge, kwa hiyo haiwezekani kutumia time step kubwa kama katika implicit method. Explicit method ni faida kwa matukio kama impact, wave propagation, na high-speed contact, ambayo kiasili yanahitaji time steps ndogo.

Ikiwa contact ipo, contact constraints hushughulikiwa kwa Forward Increment Lagrange method. Contact forces hutathminiwa kwa namna inayolingana na single-stage update ya explicit method.

Solution method ya frequency response analysis¶

Frequency response analysis hupata moja kwa moja periodic steady-state response katika frequency domain kwa modal superposition method. Hutatathmini response huku excitation frequency ikibadilishwa, bila kufuatilia time history kwa iterations. Nonlinear iterations wala time integration hazifanywi.

Kabla ya frequency response analysis, modal analysis lazima ifanywe kwa mfumo huohuo na idadi inayohitajika ya natural modes itolewe. Linear models pekee ndizo zinazotumika; uchanganuzi hauwezi kufanywa ikiwa geometric nonlinearity au material nonlinearity imewezeshwa. Kwa nafasi ya analysis type hii, tazama Aina za uchanganuzi.

Time advancement na nonlinear iteration katika transient heat conduction¶

Transient heat conduction ina time loop yake inayokamilika ndani ya !HEAT. Time step hudhibitiwa kwa njia tofauti na !STEP na !AUTOINC_PARAM za structural analysis, na material properties zikitegemea temperature, nonlinear iteration hufanywa ndani ya kila time step.

Time-integration scheme huchaguliwa kwa parameter \(\beta\) ya !HEAT.

Time step hudhibitiwa adaptively kwa kuchanganya initial time step, minimum time step, na maximum temperature change kwa step. Ikiwa temperature change inazidi DELTMX baada ya hesabu katika time step, time step hupunguzwa na hesabu hurudiwa. Uchanganuzi hukatishwa ikiwa time step inashuka chini ya minimum time step DTMIN.

Material properties zikitegemea temperature, nonlinear iteration hufanywa ndani ya kila time step. Iteration hudhibitiwa kama ifuatavyo.

Steady heat conduction haitumii time stepping; nonlinear iteration pekee hufanywa ikiwa material properties zinategemea temperature. Kwa kuwa time control ya heat conduction analysis ni huru kutoka !STEP ya structural analysis, angalia mipangilio ya !HEAT unapobadilisha time step.

Vipengee vinavyohusiana¶

• Aina za uchanganuzi — Nafasi ya kila analysis type.
• Contact na embedding — Uchaguzi wa contact types, pair definitions, na solution algorithms.
• Solver na preconditioning — Utatuzi wa linear simultaneous equations zinazoitwa ndani ya Newton iteration.
• Udhibiti wa step — Step/substep control ya nje na automatic increments/cutback.
• Mbinu ya Newton-Raphson (nadharia) — Uundaji wa iterative solution method.
• Convergence criteria (nadharia) — Mathematical definitions za convergence indicators.
• Mbinu za dynamic analysis (nadharia) — Uundaji wa Newmark-β method na central difference method.
• Transient heat conduction analysis (nadharia) — Uundaji wa time integration ya heat conduction.
• Contact analysis (nadharia) — Uundaji wa Lagrange multiplier method.
• Keyword reference: !STEP, !CONTACT_ALGO, !DYNAMIC, !HEAT.