Opinion for the court filed by Circuit Judge LINN. Opinion concurring in part and dissenting in part filed by Circuit Judge NEWMAN.
*1056LINN, Circuit Judge.Robert H. Peterson Co. (“Peterson”) appeals from the final judgment of the United States District Court for the Northern District of Texas, which concluded that: (1) Peterson infringed certain claims of U.S. Patent No. 5,988,159 (“the '159 patent”), owned by Golden Blount, Inc. (“Blount”); (2) Peterson’s infringement was willful; (3) prosecution history estop-pel did not preclude the application of the doctrine of equivalents; (4) the '159 patent was not invalid; and (5) the case was exceptional, warranting the award of attorneys’ fees. Golden Blount, Inc. v. Robert H. Peterson Co., No. 3-01-CV-0127-R (Aug. 9, 2002) (“Final Judgment”); Golden Blount, Inc. v. Robert H. Peterson Co., No. 3-01-CV-0127-R (Aug. 9, 2002) (“Findings of Fact & Conclusions of Law”). The district court also determined that Blount was entitled to lost profit damages. Findings of Fact & Conclusions of Law at 8, para. 9. Because the district court did not provide findings of fact to support a conclusion of infringement, as required by Rule 52(a) of the Federal Rules of Civil Procedure, the judgment is vacated-in-part and remanded. However, because Peterson has not shown invalidity by clear and convincing evidence, we affirm that portion of the judgment. Finally, we find no evidence of Peterson raising the issue of inequitable conduct before the district court, and therefore conclude that that argument has been waived.
BACKGROUND
The patent-in-suit relates to fireplace burners and associated equipment. In particular, the '159 patent discloses a gas-fired, artificial logs and coals-burner assembly arranged to “enhanc[e] the natural burn in cooperation of the fireplace draft as well as the aesthetic beauty of the imitation burning logs, coals, and embers.” '159 patent, Abstract. The assembly is described as follows:
The present burner, assembly is the combination of an inexpensive primary gas logs burner in gas flow communication with a secondary coals and embers-burner tube positioned forward and below the primary burner which operates to enhance the natural draft of the fireplace to improve efficiency of burn and aesthetic appeal of the gas-fired artificial logs, coals and embers-burner assembly.
Id. at col. 3, ll. 54-60.
Figure 2, below, illustrates a secondary burner apparatus 100 in combination with a primary burner tube 14. A connector 102 attaches the primary burner tube 14 to the secondary burner tube 104, creating an enclosed fluid path for gas. A valve 106 is interposed in this fluid path and enables a user to adjust the amount of gas entering the secondary burner. Id. at col. 5, ll. 26-40. Secondary burner tube 104 includes a plurality of apertures, or gas discharge ports, along its length. The apertures can be evenly spaced or clustered, and permit gas to be discharged in a direction away from the opening of the fireplace. Directing the gas discharge away from the opening enhances the aesthetic beauty of the fire and improves safety. Id. at col. 5, ll. 45-63.
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Independent claims 1 and 17 are at issue in this suit, as well as dependent claims 2, 5, 7-9, 11-13, and 15-16. Independent claim 1 is representative and recites:
1. A gas-fired artificial logs and coals-burner assembly for fireplace comprising:
an elongated primary burner tube including a plurality of gas discharge ports;
a secondary coals burner elongated tube positioned forwardly of the primary burner tube;
a support means for holding the elongated primary burner in a raised level relative to the forwardly position secondary coals burner elongated tube; the secondary coals burner elongated tube including a plurality of gas discharge ports;
the elongated primary burner tube and the secondary coals burner elongated tube communicating through tubular connection means wherein the gas flow to the secondary elongated coals burner tube is fed through the primary burner tube and the tubular connection means;
a valve for adjusting gas flow to the secondary coals burner elongated tube positioned in the tubular gas connection means; and
the primary burner tube being in communication with a gas source with a gas flow control means therein for controlling gas flow into said primary burner tube.
The '159 patent issued on November 23, 1999. On December 10, 1999, Blount sent a letter to Peterson, informing Peterson of the issuance of the '159 patent and stating that Blount believed Peterson to be “marketing a device that is substantially similar to the burner assembly” claimed in the patent. The letter further stated that Blount would “take whatever steps are reasonable and necessary to prevent infringement of the patent.” Peterson ac*1058knowledged receipt of the letter on December 30, 1999. On May 3, 2000, Blount again wrote Peterson, stating:
We have inspected your EMB Series Ember Flame Booster and find it to be clearly within the scope of at least some of the claims of the subject patent. Our client views any infringement of its patent with great concern and will take necessary steps to stop any such infringement.
Peterson responded that it disagreed with Blount’s assessment that the Peterson devices were substantially similar to the claimed invention and requested that Blount explain, in detail, the basis upon which Blount believed Peterson was infringing. Blount never answered, and on January 18, 2001, Blount filed this suit against Peterson, alleging patent infringement.
A bench trial was held, beginning on July 29, 2002. The district court issued its Final Judgment against Peterson on August 9, 2002, along with supporting Findings of Fact and Conclusions of Law. The district court concluded, among other things, that: (1) Peterson literally infringed each of the asserted claims, Findings of Fact & Conclusions of Law at 3, para. 16; id. at 7, para. 7; (2) if Peterson did not literally infringe, Peterson contributorily infringed or induced infringement, id. at 3-4, paras. 17-18; id. at 7, para. 3; (3) in the alternative, prosecution history estop-pel did not apply, and Peterson infringed under the doctrine of equivalents, id. at 4, para. 19; id. at 7, para. 5; (4) Peterson’s infringement was willful, id. at 5, para. 26; id. at 8, para. 10; (5) the claims of the patent are “valid,” id. at 7, para. 7; (6) Blount had established the Panduit factors and was- entitled to lost profit damages, id. at 8, para. 9; and (7) the case is exceptional under 35 U.S.C. § 285, supporting an award of attorney fees, id. at 8, para. 11. Peterson filed a timely appeal, asserting that each of the district court’s conclusions was incorrect.
This court has jurisdiction over the appeal under 28 U.S.C. § 1295(a)(1).
ANALYSIS
A. Standard of Review
On appeal from a bench trial, this court reviews the district court’s conclusions of law de novo and findings of fact for clear error. Brown & Williamson Tobacco Corp. v. Philip Morris Inc., 229 F.3d 1120, 1123 (Fed.Cir.2000). Claim construction is a question of law, reviewed de novo. Cybor Corp. v. FAS Techs., Inc., 138 F.3d 1448, 1454 (Fed.Cir.1998) (en banc). A determinátion of infringement, whether literal or under the doctrine of equivalents, is a question of fact and is reviewed under the clearly erroneous standard. Biovail Corp. Int’l v. Andrx Pharms., Inc., 239 F.3d 1297, 1300 (Fed.Cir.2001). Invalidity based on obviousness is a question of law, reviewed de novo, based on underlying factual findings, reviewed for clear error. Graham v. John Deere Co., 383 U.S. 1, 17, 86 S.Ct. 684, 15 L.Ed.2d 545 (1966). To establish invalidity, the supporting facts must be shown by clear and convincing evidence. WMS Gaming, Inc. v. Int’l Game Tech., 184 F.3d 1339, 1355 (Fed.Cir.1999).
B. Infringement
Peterson argues first that the district court erred in finding literal infringement of each of the asserted claims because the district court’s claim constructions are erroneous and the conclusion of literal infringement is not supported by any evidence. Peterson also argues that the district court’s findings of contributory infringement and induced infringement are erroneous because the district court *1059applied the incorrect standards and because there is no supporting evidence for the findings. Peterson further argues that the district court’s conclusion, in the alternative, of infringement under the doctrine of equivalents is flawed because the district court’s application reads out a claim limitation. Peterson ' also argues that prosecution history estoppel precludes a finding of infringement under the doctrine of equivalents. Blount, on the other hand, argues that the district court’s claim constructions were proper, that the district court applied the correct standards, that the district court’s infringement findings were supported by substantial evidence, and that prosecution history estoppel does not apply.
A determination of infringement requires a two-step analysis. The court must determine (1) “the scope and meaning of the patent claims asserted,” and (2) how “the properly construed claims ... compare[] to the allegedly infringing device.” Cybor Corp., 138 F.3d at 1454. Turning first to claim construction, there are two claim terms in dispute on appeal— “raised level” and “away from the fireplace opening.”
Claim 1 recites that a support means holds “the elongated primary burner in a raised level relative to the forwardly position secondary coals burner elongated tube.” The district court concluded that “raised level” meant that the primary burner was at a raised level with respect to the secondary burner, for example with respect to the centerline. Findings of Fact & Conclusions of Law at 6, para.-2. Peterson argues that the so-called “center-line” test is incorrect and that the raised level should be measured with respect to the level of the gas ports. Blount argues that the centerline test is correct, or alternatively, that any test that uses a plausible reference point, such as the top of the tube, can be used.
The plain language Of the claim is relatively straightforward, and the district court correctly gave the claim term its ordinary and customary meaning. The only dispute comes from the reference point used to determine if the primary burner tube is at a raised level. Typically, when measuring whether something is higher than, or at a raised level with respect to, another object, the tops of the two items at issue are compared. There is nothing to indicate that persons skilled in the art would attribute any other or different meaning. Thus, the ordinary meaning of “raised level” in claim 1 refers to the top of the primary burner tube being at a raised level with respect to the top of the secondary burner tube. The written description and prosecution history provide no limitation or illumination on this issue. The written description merely indicates that the purpose of the positioning “enhances the natural draft of the fireplace to improve efficiency of burn and aesthetic appeal.” '159 patent, col. 3, ll. 54-60. This purpose, however, does not indicate which portion of the burner tubes should be utilized as a reference. We thus construe the limitation “at a raised level” in claim 1 to refer to the tops of the two burner tubes. ■ Claim 17 recites that the secondary burner tube is positioned below the primary burner tube. Although the district court did not separately construe the term “below,” both parties argued that the vertical limitations in claims 1 and 17 should be construed similarly. We agree, and conclude that the tops of the tubes should be used to determine whether the secondary burner tube is positioned below the primary burner tube as recited in claim 17.
Claim 17 further recites that the gas ports are directed “away from the *1060fireplace opening.” The district court construed this term to mean “any direction that does not include a horizontal component pointed toward the vertical plane of the fireplace opening.” Findings of Fact & Conclusions of Law at 7, para. 2. Peterson argues that, regardless of the ordinary meaning of the claim language, the specification requires the gas ports to be directed towards the primary burner. Blount, on the other hand, argues, that Peterson’s contention is wrong and is based solely on a single embodiment. The district court’s construction reflects the ordinary meaning of the claim language. Nothing in the specification suggests that the applicant disavowed or otherwise disclaimed any scope of coverage. In particular, the specification states that “[i]n the secondary burner tube 104, the gas is discharged in a direction away from the opening of the fireplace, or in another aspect, is directed somewhat toward or directly toward the primary burner 14.” '159 patent, col. 5, ll. 58-62. That language supports Blount’s argument that the reference in the written description to the direction of the gas ports towards the primary burner relates only to an embodiment. Peterson points to no portions of the specification that limit the ordinary meaning of the claim language. Thus, we agree with the district court that “away from the fireplace opening” means “any direction that does not include a horizontal component pointed toward the vertical plane of the fireplace opening.”
With respect to the second step of the infringement analysis, the district court did not provide any findings of fact or analysis to support its conclusion. Rather, the district court simply stated conclusions without any apparent bases. In the district court’s Findings of Fact and Conclusions of Law, its entire infringement analysis for literal infringement, contributory infringement, induced infringement, and infringement under the doctrine of equivalents is presented in six short and conclusory paragraphs. For example, with respect to literal infringement, the only discussion in the entire district court opinion is as follows: “Applying the claim construction referred to in the Conclusions of Law, this Court finds there is [literal infringement of the asserted claims].” Findings of Fact & Conclusions of Law at 3, para. 16. There is nothing to explain how the limitations of the claims, as construed, compare to the allegedly infringing device. See, e.g., Cybor Corp., 138 F.3d at 1454 (explaining that “the properly construed claims ... [are] compared to the allegedly infringing device”). The paragraphs relating to contributory infringement, induced infringement, and infringement under the doctrine of equivalents are equally conclu-sory and devoid of any analysis. In the absence of any findings, this court cannot determine whether the trial court had any evidence to support its conclusions, nor is this court able to determine whether the district court applied appropriate legal standards.
After a bench trial, a trial court must put forth the findings of fact relied upon to justify its actions. Fed.R.Civ.P. 52(a); Atl. Thermoplastics Co. v. Faytex Corp., 5 F.3d 1477, 1478-79 (Fed.Cir.1993). While Rule 52(a) “does not require elaborate, detailed findings on every factual issue raised,” the district court opinion “must include as many of the subsidiary facts as are necessary to disclose ... the steps by which the trial court determined factual issues and reached its ultimate conclusions.” Atl. Thermoplastics, 5 F.3d at 1479.
When the trial court provides only con-clusory findings, illuminated by no subsidiary findings or reasoning on all the relevant facts, as was the case here, there' is not that “detail and exactness” *1061on the material issues of fact necessary for an understanding by an appellate court of the factual basis for the trial court’s findings and conclusions, and for a rational determination of whether the findings of the trial court are clearly erroneous.
Id. (quoting EEOC v. United Va. Bank/Seaboard Nat’l, 555 F.2d 403, 406 (4th Cir.1977)). In such situations, remand is necessary and proper. Id.
Because the district court’s sparse opinion provides this court with only bald conclusions for review, we conclude that the district court’s judgment as to literal infringement, contributory infringement, induced infringement, and infringement under the doctrine of equivalents is insufficient under Rule 52(a). We thus vacate those portions of the district court’s opinion and remand those issues to the district court for specific factual findings. Further, because we have vacated the district court’s judgment with respect to all aspects of infringement, we also vacate and remand the district court’s judgment as to willfulness, the exceptional nature of the case, and damages.
On remand, the district court “shall find the facts specially and state separately its conclusions of law thereon.” Fed.R.Civ.P. 52(a). Further, on remand, if the district court finds no direct infringement by Peterson, but concludes that the '159 patent is infringed, either literally or under the doctrine of equivalents, by a customer of Peterson or other party using Peterson components, the district court must then consider the claims of contributory and induced infringement to find Peterson in violation of the patent. See Carborundum Co. v. Molten Metal Equip. Innovations, Inc., 72 F.3d 872, 876 n. 4 (Fed.Cir.1995) (“Absent direct infringement of the claims of a patent, there can be neither contributory infringement or inducement of infringement.”). Contributory infringement liability arises when one “sells within the United States ... a component of a patented machine ... knowing the same to be especially made or especially adapted for use in an infringement of such patent, and not a staple article or commodity of commerce suitable for substantial noninfringing use.” 35 U.S.C. § 271(c) (2000). Thus, Blount must show that Peterson “knew that the combination for which its components were especially made was both patented and infringing.” Preemption Devices, Inc. v. Minn. Mining & Mfg., Co., 803 F.2d 1170, 1174 (Fed.Cir.1986). Further, Blount must show that Peterson’s components have no substantial noninfringing uses. Alloc, Inc. v. ITC, 342 F.3d 1361, 1374 (Fed.Cir.2003). In order to find Peterson liable for inducing infringement under 35 U.S.C. § 271(b), Blount must show that Peterson took actions that actually induced infringement. Met-Coil Sys. Corp. v. Korners Unlimited, Inc., 803 F.2d 684, 687 (Fed.Cir.1986) (“There can be no inducement of infringement without direct infringement by some party.”). Further, Blount must show that Peterson knew or should have known that such actions would induce direct infringement. Micro Chem. Inc. v. Great Plains Chem. Co., 194 F.3d 1250 (Fed.Cir.1999).
C. Invalidity
Peterson also argues that the district court erred in not concluding that the '159 patent would have been obvious in light of prior art related to an earlier assembly of Peterson, known as the F3 device, and drawings by a Mr. Jankowski. Before this court, Peterson offers merely the bare assertion that the patent claims would have been obvious. We therefore have no reason to overturn the district court’s conclusion that the '159 patent is not invalid, *1062Findings of Fact & Conclusions of Law at 7, para. 7, based on its findings that “[t]he prior art relied on by [Peterson] does- not show the same concepts” as the '159 patent claims, id. at 2, para. 7, and that the drawings failed to meet the “clear and convincing evidence” standard, id. at 2, para. 9. Therefore, we affirm the portion of the district court’s opinion holding the '159 patent not invalid.
D. Inequitable Conduct
Peterson finally argues that the district court erred in not holding the '159 patent unenforceable for inequitable conduct. Blount counters that Peterson has waived this argument, because Peterson did not raise it in its district court pleadings. Because there is no evidence that Peterson raised the issue of inequitable conduct before the district, court, and because Peterson has not suggested, nor do we find, any special circumstances in this case to militate against a finding of waiver, Peterson cannot now be heard to make the argument for the first time on appeal. See Sage Prods., Inc. v. Devon Indus., Inc., 126 F.3d 1420, 1426 (Fed.Cir.1997) (stating that, barring a few exceptions, the failure to raise an argument at the trial level constitutes a waiver of that argument on appeal).
CONCLUSION
For the foregoing reasons, the court: (1) vacates the district court’s judgment with respect to literal infringement, contributory infringement, induced infringement, and infringement under the doctrine of equivalents; (2) vacates the district court’s judgment with respect to willfulness, the exceptional nature of the case, and all damages awarded; (3) affirms the district court’s judgment finding the '159 patent not invalid; and (4) concludes that Peterson has waived its inequitable conduct argument.
AFFIRMED-IN-PART, VACATED-IN-PART, AND REMANDED.
COSTS
No costs.